# Category Archives: Probability

## The Monty Hall Paradox And Borges’ GARDEN OF FORKING PATHS

Nota Bene: this is still very much a work in progress. I have not yet achieved that mental state at which I can indulge, at least for a while, in the delusion that I have achieved the maximum point of crystalline clarity.’ I am not responsible for any brain damage anyone reading this stuff may incur. My ignorance of the topic is vast, and this is an exercise in writing to learn.

Here is what I will be trying to do: 1) I will be trying to show that probability should be explicated in terms of a person’s body of knowledge (the person’s ignorance will be incredibly important, but this is to be articulated in terms of the person’s knowledge). To do this, I will be using the concept of possible worlds that are epistemically accessible relative to a knower. This means that at the core probability is epistemic in nature. Eventually, I will be trying to use this perspectival character to illuminate the red apple/yellow apple example. (At the moment, this makes more sense to me than to you, dear reader.) I will use what I will call the ‘Monty Hall Shell Game’ to show that the fact that objects have histories has to be included in any adequate explication of probabability.

Throughout the challenge will be to avoid uselessly going in circles by explicating probability in terms of other probabilistic concepts, such as ‘randomness’ and ‘likelihood’. My reader will judge how successful I am.

**********

Some random scattered notes.

February 10, 2020: The Monty Hall Shell Game leads us very directly to the problem of the reference class.

The ‘Switched-Choice shell’ is very much ‘Table #6’ in the Pho restaurant. And fits in very nicely with the knowledge/ignorance stuff. Ignorance as the hinge.

January 09, 2020: I am now suffering under the delusion that ‘I’ve got this’. Doubtlessly this delusion will go away once I take my meds. The only problem with taking my meds, however, is that the nice purple rhinocerous grazing peacefully at my feet while I blog (he’s my blogging rhino) also goes away (he doesn’t like my meds). I like my blogging rhino. I don’t want him to go away. So I won’t take my meds. So here it is — one more try, this time with feeling.

I start by stating my ignorance. One of the things my vast ocean of ignorance encompasses is quantum mechanics. All that I know of quantum mechanics — precious little — comes from various popularizations, including Feynman’s lectures on physics. But I don’t think I risk too much by saying that quantum physics relies heavily on probabilities less than 1 and greater than 0. The electron circling around this nucleus has a certain probability of jumping one nano-second from now to a higher or lower energy level. We don’t know that the electron will shift orbits one nano-second from now; if we did, the probability of its doing so would be 1, not less than 1, and its shift would be certain. Nor do we know that it won’t shift orbits one nano-second from now; if we did, the probability of its doing so would be 0, not some number greater than 0, and its not shifting would be certain. Our citing probabilities greater than 0 and less than 1 reflect our uncertainty as to what the electron is about to do, that is to say, our not knowing, our (relative) ignorance. [Ignorance is a relative not-knowing.]

In the larger-than quantum realm, As what’s-his-name shows here, All we can do — at least at present — is state a probability. We are ignorant of any “hidden variables” that might give us a deterministic explanation of why the electron moved to a different energy level. We may be ignorant of them because they don’t exist (if I am not mistaken this is or is closer to the standard view); or it may be that, while they exist, we are currently unable (and may forever be unable to) to uncover them. Whichever is the case, I definitely lack certainty regarding what the electron will do next. Lacking certainty, I am in thrall to a degree of ignorance regarding the electron’s next move. (and may never be able to

01/12/2020: Aha Erlebnis for the day: The person behind the counter who takes my order for Pho hands me a sign saying ‘6’, and tells me “You are at table #6.”

November 21, 2019: Am I getting closer to ‘crystalline clarity’ — what one achieves when the individual pieces of one’s argument are obvious or close to trivially obvious, but the sum is not.

The probability in a particular situation equals one’s chances in that situation. ‘Ones chances’ already starts to make probability perspectival. One’s chances in that situation depend upon, hinge upon what knowledge one implements. What one excludes from consideration, for example, directs one’s eyes from, the pool of possible choices. One can’t implement knowledge obviously if one doesn’t have the knowledge. So the probability in a given situation depends upon the knowledge. Knowledge is a necessary condition for the probabilities of a given situation being, say, 1 in 2 instead of 1 in 3. And it is difficult to separate out knowledge as an event in one’s brain from the implementation of knowledge — say, in how one flips the coin.

And of course ignorance.

But how can we be sure it is 1 in 2 in the normal shell game? Intuition fails us in the Monty Hall case, after all — how can we rely on intuition in this case? Spoiler: we can. But go through the Monty Hall shell game anyway — we will see that knowledge affects Smith’s chances at two points: first, Elizarraraz’ implemented knowledge changes Smith’s chances; second, Smith’s implemented knowledge changes his chances. ffffffgggggg

Another link: dissolving the reference class problem.

Here in outline form are the points I intend to make (assuming I succeed — success if not totally guaranteed at the moment).

1. Assuming a deterministic universe, ignorance is a necessary condition for randomness (i..e, probabilities less than 1 and greater than 0). Normally this is a “mix” of ignorance and knowledge. Ironically, my own ignorance comes into play in trying to argue this. There will be various lacunae in my argument which, I hope, will steadily decrease over time.
2. There are (at least) two components to randomness: the sample space and the likelihood of each element will ‘come up’ in an “experiment”. (I am using the standard, unfortunately misleading terminology one can find by googling ‘probability’. This terminology distinguishes between ‘events’ — subsets of the sample space set whose members are possible outcomes (and, I will argue, possible realizations) — and ‘experiments’ — actual outcomes/realizations appearing in time and space.)
3. In the normal shell game, Morgenstern’s increase in knowledge suffices to decrease the size of the sample space. Her ignorance increases that cardinality. The likelihood of winning the peanut changes accordingly. In the Monty Hall shell game, Elizarraraz’ knowledge also decreases the sample space.
4. The cardinality of the sample space also depends upon the background of common sense which specifies that certain things count as belonging to the set and other things do not. Sample spaces are “subjective” and highly perspectival in character.
5. Normally the actual outcome of an experiment is something that can be taken in at once. It is restricted to the present. This particular card was drawn. The flipped coin came up heads. The die came up 6. The peanut appeared when the shell was turned over. This is the realm of what could be. The shell Smith is about to turn over could be hiding a peanut, or it could be hiding nothing but empty air.
6. But I would like to expand members of a sample space to include what I will call possible realizations. This particular possible path through time, when realized, resulted in Smith’s now turning over the shell with the peanut. Before it was just one of 13 possible paths that could have threaded its way through time; now it is an actually accomplished, actually realized path. It is the path that made it into actuality among the 12 other paths that could have been.
7. The idea of calculating the probability via a tree (shown on countless Google-able sites) came to be via the Borges short story GARDEN OF THE FORKING PATHS; but also the idea of treating all the possible paths as members of a set came to me from that story.
8. I try to define likelihood as an idealization of the numbers one gets after repeating an experiment a very large number of times. (Flip a fair coin 10 million times using a particular standard method; it comes up heads 5 million and 1 times and tails 4 million and 999 times; we idealize that to ‘the coin is equally like to come up heads as tails.) The likehood depends partly upon the method one uses (using Emo’s particular technique of cheating; always switching one’s choice in the Monty Hall shell game; always taking the path on the right at the end); it also depends partly upon the cardinality of the sample space. So likelihood is partly perspectival and “subjective”, partly anchored in the objective world. The more subjective, the less the weight of the anchor. So no sharp distinction between “natural probability” and “subjective probability”.
9. If one does not “carry over” the knowledge from the past that would let one identify one shell as the ‘initially selected’ one, there is no method one can use that would raise | alternatively lower one’s chances from 50/50 to 2/3 | alternatively 1/3. The probability in this case is 50/50, so making sense of the strong intuition that switching would not make any difference. The ability to give oneself the higher chances depends upon carrying over into the present information from the past. It requires the depth of the past. 50/50 is what you get when you are stuck in the depth-less present. This point is going to require some finessing. From this perspective, the chances are 50/50. From this other perspective, the chances are 2/3 | 1/3.
10. Two pictures: First, I am staring at the two remaining shells with no way to distinguish them. I am restricted to the present and to the near future (what could be). I cannot remember which was the shell I had initially chosen — the shells are too much alike. (This is a shell game, after all.) My chances are 50/50. Second, I do remember and can identify the shell I had initially chosen. The past and its “possibilities” — its could have beens — is opened up to me. My chances are 2/3 | 1/3, depending on how I use the information.
11. Want to end with a contrast with Searle’s illustration.
12. The upshot: assuming a deterministic universe, a ratio or “mix” of ignorance/knowledge is required for the existence of randomness. Randomness is perspectival in character through and through.

[Different games/practices will have different rules/constraints which will determine what the sample space is and its cardinality. To play a probabilistic game/engage in a probabilistic practice one must have a certain amount of knowledge (‘a peanut is hidden under one of the shells’; this shell is the one initially selected by Smith); for there to be a game/practice at all there must be this item of knowledge. Without the knowledge there are only latent games — latent probabilities. And within the confines of the game/practice there must be an ignorance that poses an obstacle to one’s achieving the point of the game/practice. Rules/constraints plus knowledge/ignorance determine the sample space and what game is being played. For it to be a game/practices of chance, ignorance is required. This outline is currently a mess; I hope to clean it up later.

The picture of a set of forking paths in the forest is useful because it encapsulates Smith’s moving from past to future; preserving knowledge of which shell was initially selected is necessary in this movement. Path dependence. So not just a matter of which shells are staring one in the face at the moment.

Some examples: Smith knows that there is a peanut under one of the shells. He turns over the shells one by one until he uncovers the one hiding the peanut. The probability that he will find the peanut is 1. Even here there is ignorance.

It must make sense to say of the game/practice: ‘were the game repeatable.’

“Uncertainty” aka randomness is a “mix” of knowledge and ignorance.

Do this and you will win ~10 million times | alternately 5 million times out of 15 million times. The probability given this practices is 2/3 | alternately 1/3. The probability is relative to the practice (which could be a game). The practice is taking place within an arena (horizon, world) that defines what counts as an item that is eligible to be “in play”. This arena makes the item relevant to the practice (the shells piled up to the side don’t count in the game). One has knowledge, but one is also ignorant. Theoretically, one might be so dumb as to not be able to exploit that knowledge in the practice one is engaged in … unlikely in the case of knowing that shell #1 is empty. One knows things about the shells. Some of these things will affect one’s chances; others will not. Shell #1 is empty | alternatively hides the peanut. Of the two paths you or I (second or first person will be important) comes across at the end, one is on the left and the other is on the right. I or you may or may not be able to exploit either item of knowledge, but both will affect one’s chances if exploited. ]

What is the point of the arguments that are about to follow? These arguments are one snippet in an attempt to get clear in my mind regarding the nature of probability. (Yes, I know, this is absurdly ambitious. You may be a bit less inclined, gentle reader, to break out in raucous laughter if you keep in mind I am just trying to arrive at the point at which, in a doubtlessly delusional state, I suffer from the strong conviction I have gotten clear in my own mind regarding the nature of probability. Once achieved, this strong conviction will doubtlessly evaporate like a mirage as I increase my knowledge of the field. Or if I take my meds.)

The reason I want to get clear in my own mind about the nature of probability because I think this is necessary in order to uncover at least one relation that makes the antecedent relevant to the consequent in relevant indicative conditionals. I expect to be making changes to this post as time goes on.

[What is the conclusion I am heading towards with all the verbiage below? This: the existence of a probability greater than 0 but less than 1 has as both its necessary and sufficient condition a ratio of ignorance/knowledge within a given perspective, itself limited by a background of common sense. Probability within these two limits is perspectival down to the very root for this reason; it could not exist within the “perspective” of an infinite mind that does not suffer any ignorance at all, partly because such a Mind would not enjoy any perspectives at all. Given a deterministic universe, this is the only way there can be probabilities between 0 and 1 noninclusive. This places me in the camp of those who, like Keynes, regard probability as “subjective”, but I hope to do justice as well to the rather hard-edged intuition expressed by ‘just don’t take your subjective probabilities to [Vegas.’ ]

[Probability is not a frequency, although this is a factor; it is not a “credence”, i.e., belief, thought that is also a factor. Probability is always and everywhere epistemic in nature, playing in the arena of knowledge/ignorance and evidence.]

[In the clearest cases, the role knowledge/ignorance plays in determining such a probability is easiest to see in the case of independent events; but dependent events, as in the case of the Monty Hall puzzle, can increase/decrease the probability of a given event. ]

The Five Factors I will be considering. These are the factors — 5 of them: 1) P’s knowledge that there is a peanut hidden under one of the shells. 2) P’s ignorance (alternatively, knowledge of) of which shell hides the peanut. 3) The true description of the shell (‘one of the three’; shell #1, shell #2, shell #3; the remaining shells, not #1; ‘shell #n, under which I placed the peanut’; (Monty Hall) the shell I initially chose; (Monty Hall) the shell I did not initially choose; (Monty Hall) shells #1, #2, #3); in the Monty Hall case the (true) description behaves like the placard I’m given in the Pho restaurant that turns the table I choose into ‘table #6’; 4a) the method of selection, and 4b the knowledge that enables this method. 5) The idealized frequency, which determines equal or non-equal likelihood.

The Scene. A Shell Game Is Set Up. Let me begin by describing the scene. In an apple and cherry orchard in Iowa, a table has been set up on which a number of shell games will be played. God only knows how it got there — maybe a tornado dropped it there — but not far from the table is a giant conch shell. The sky above is clear. Unknown to and hidden from the people in and about to enter the orchard, but within view should one occupy the right vantage point, a tornado is touching down intermittently across the Missouri River, in Nebraska.

I describe the scene this way because this scene is a situation. A situation is partially defined by what is hidden from one and unknown to one, and by the information that is available to one. Situations will become important in later posts because some versions of Relevant Logic rely on them rather than on possible worlds. I describe this particular one now because I will be returning to it later.

Elizarraraz sets up a shell game: Elizarraraz (although this is not relevant to the example, the name, taken from the name of my landlord in Chicago from whom I was renting a studio for some years, is Ladino for ‘poor king’. Ladino is the Sephardic counterpart to Yiddish, and in Elizarraraz’ case the name, and his paternal ancestry, comes from Mexico. Although they were not officially allowed to, a number of conversos managed to emigrate to Latin America in order to place a more comfortable distance between themselves and the Spanish Inquisition. Just thought I would provide my made-up characters with concrete backgrounds. Don’t ask though what Elizarraraz is doing in an orchard in Iowa. But I digress) … as I was saying, Elizarraraz sets up on the table a shell game with three shells and a single peanut.

[These are the factors — 5 of them: 1) P’s knowledge that there is a peanut hidden under one of the shells. 2) P’s ignorance (alternatively, knowledge of) of which shell hides the peanut. 3) The true description of the shell (‘one of the three’; shell #1, shell #2, shell #3; the remaining shells, not #1; ‘shell #n, under which I placed the peanut’; (Monty Hall) the shell I initially chose; (Monty Hall) the shell I did not initially choose; (Monty Hall) shells #1, #2, #3); in the Monty Hall case the (true) description behaves like the placard I’m given in the Pho restaurant that turns the table I choose into ‘table #6’; 4a) the method of selection, and 4b the knowledge that enables this method. 5) The idealized frequency, which determines equal or non-equal likelihood.]

The shells are labelled in order 1, 2, and 3 [3: true description]. The labels help here because the shells are so similar in appearance that telling them apart may prove difficult even for Elizarraraz. The shells so labelled are individuated by the trajectories they undergo in four-dimensional space-time. As anyone who has read any of the popularizations of Relativity Theory knows, an entity with mass is always moving in at least the temporal dimension even if it is at rest in the remaining three spatial dimensions. [Temporal movement of the position here.] An entity with mass is always delineating a trajectory of movement. The trajectory shell #1 is undergoing is unique to that shell and lets us pinpoint that shell as distinct from every other individual in the universe; the same holds true for shells # 2 and #3.

Employing a randomizing device of some sort (say, an app on his Android designed to output 1, 2, and 3 randomly), Elizarraraz places the peanut under the shell selected by his randomizer. Naturally, he knows under which shell the peanut is hidden.

By ‘random’ I mean ‘cannot be predicted on the basis of the relevant previous happenings’, where I will leave both ‘cannot’ and ‘relevant’ as undefined primitives. An example may clarify what I am getting at. It is plausible to suppose that the next staggerings of a drunk person cannot be predicted just on the basis of his past movements (‘the relevant previous happenings’). Those staggerings count as random. Of course, I would expect an omniscient being to be able to trace a deterministic but humungously long causal chain which includes input from the motion of this atom five seconds after the big bang and the motion of this other atom from the alpha centuri star…, but this deterministic account would not be restricted to just the previous movements of the drunk. For our story, perhaps a bit tongue in cheek, we can make the output of Elizarraraz’ Android randomizing device follow the amblings of this or that drunk person, dividing the ground they are walking on into pie-like slices, so that it if the drunk goes in one direction the device outputs ‘1’, ‘2’ if they go in a different direction, and ‘3’ if they veer off into the remaining direction.

Notice that ignorance is key to the notion of randomness — the drunk’s staggerings are random because we cannot know his next motion on the basis of his previous motions.

[At some point need to discuss how I end up trying to explicate probability in terms of other probabilistic concepts. I will have to face up to this apparent circularity sometime. We have a family of probabilistic concepts; each member of that family will aid in illuminating the others.]

Smith arrives on the scene: Smith (although this is not relevant to the example, the name is English for ‘smith’ as in ‘blacksmith’. But you knew that already) arrives on the scene. Smith is about to play what I will call, for reasons that will become clear shortly, the ‘normal’ shell game. He is to select a shell and turn it over to see if it is hiding the peanut. He is to get just one chance.

Smith knows that there is one, and only one, peanut hidden underneath one of the three shells. (Elizarraraz, who is a reliable conduit of information, has told him this.) [1) P’s knowledge that there is a peanut hidden under one of the shells] Otherwise, he is unlikely to be playing the game, otherwise, though of course he might idly turn over a shell out of sheer boredom and without any expectations at all. Of course, Smith does not know under which of the three shells the peanut is hidden. It could be under shell#1 (this is a possibility), under shell#2 (another possibility), or under shell#3 (yet another possibility). [2: P’s ignorance (alternatively, knowledge of) of which shell hides the peanut]

Two possibilities — two might be’s/could be’s | alternatively might not be’s| possibly is not’s — attach themselves to each shell. Obviously each shell could be/might be hiding the peanut, and each shell could be/might be hiding nothing but air and a piece of table.

But we can go deeper than this. Let me elucidate these possibilities with reference to epistemically accessible possible worlds. A world w0 is

A world w0 is epistemically accessible from w for an agent S (in w) iff S knows nothing that would rule out the hypothesis that w0 = w. Then, p is epistemically necessary (for S) iff p is true at all possible worlds that are epistemically accessible from w (for S).

Fitelson, p. 3

The possible world w0 in which the peanut is under shell #1 is epistemically accessible for Smith, as is the world w1 in which the peanut is not under shell #1. In neither case does Smith know anything that would rule out the hypothesis that the world in question (w0 or w1) is identical with this actual world. He does not know because the shells are hiding from him their true peanut-hiding circumstances, and he does not have the means — say, a special ray gun that came in the mail with his special decoder ring — to know how things stand in this regard. Smith is ignorant in this regard.

Let me call an ‘epistemic possibility relative to P’ any state of affairs such that P knows nothing that would rule out this state of affairs being comprised by (contained in, included in, part of) the actual world. I use ‘comprised by’ to mean ‘contained in’, ‘included in’, ‘part of’. I use ‘comprise’ to take advantage of the image of encirclement: a possible world ‘encircles’ the possibilities in that world, so to speak. A state of affairs can either obtain or fail to obtain. I will leave the concept ‘state of affairs’ as an undefined primitive, except to say that, whatever else it is, it is something that either obtains or fails to obtain.

Following Roderick Chisholm, let me identify propositions with states of affairs [state this more precisely]. So every possibility for P in a given possible world is a true proposition in that world, i.e., a state of affairs in that world whose obtaining would not, given what P knows, rule out that world’s identity with the actual world. So a proposition maps to every epistemically accessible possibility in an epistemically possible world relative to P. The proposition expressed by the sentence ‘the peanut is under shell #1’ maps to that possibility; likewise the proposition expressed by ‘the peanut is not under shell #1′ maps to that possibility in another possible world.

So I am in effect explicating ‘possibility’ [the concept] in terms of Smith’s knowledge/ignorance. In the totality of what Smith knows, there is no item of knowledge that would rule out the peanut’s being under | not being under shell #1. If we take everything Smith knows: {k0, k1, k2, k3….kn}, the intersection of that set with the following two sets is the null set, {}. These two sets are: the set comprising any knowledge anyone (for example, Elizarraraz) may have that would rule out the peanut’s being under shell #1, and the set comprising any knowledge anyone may have that would rule out the peanut’s not being under shell #1. [Relate to accessible possible worlds] Most obviously, these items of knowledge would be, respectively, ‘person P knows that the shell does not hide the peanut’, and ‘P knows that shell #1 does hide the peanut’. In this way, I explicate the concept ‘possibility’ in terms of the set comprising everything Smith actually knows, and the set things of regarding the peanut vis a vis shell #1 that any entity could know. Since this explication starts off from the notion of possible (describable) worlds epistemically accessible to Smith, it is not surprising that we end up explicating the concept ‘possibility’ in terms of knowledge. More informally, in terms of Smith’s knowledge/ignorance, where the ignorance is captured by what some person could know.

Of course, this ‘could’ is a bit awkward because it is a possibility concept. We end up trying to explicate the notion of a possibility in terms of another possibility. The way around this, I think, is introduce a (possibly — ha!) fictional entity, a being that is omniscient, that is, who actually knows everything. Ditto for ‘possible world’ in terms of ‘what could be described.’

Given that Smith does not know the peanut-hiding status of shell #1, there are at least two epistemic possibilities relative to Smith regarding that shell. There cannot be fewer than two because then either Smith would have to have to know that the possible world in which the peanut is under the shell is not identical with the actual world (so eliminating the possibility that the peanut is under the shell), or he would have to know that the possible world in which the peanut is not under the shell is not identical with the actual world (so eliminating the possibility that the peanut is not under the shell). In short, he would have to know either that the peanut is under the shell or that it is not, but not (of course) both.

So there cannot be fewer than two possibilities if Smith does not know how things stand in the peanut-hiding department with regard to shell #1. There are only two such possibilities provided we take only the presence of the peanut under the shell into consideration. (Say, we are not interested in whether the peanut is still in its shell or not.) So given Smith’s ignorance, the cardinality of the set of shell-hiding possibilities of shell #1 is 2. [The corresponding proposition is a disjunction: either the peanut is under shell #1 or it is not. How things stand with regard to shell #1 hiding | not hiding the peanut.]

Suppose now that Smith, taking advantage of an opportunity while Elizarraraz is not looking, surreptitiously turns over shell #1 and discovers that the peanut is there. Acting quickly enough so as to remain undetected, he covers the peanut with shell #1 again.

So far I have been talking about what I will call ‘subjunctive possibilities. The peanut may or may not be under shell #1. Its presence under the peanut is a matter of ‘could be, may be, might be, could go either way’. This hews, I think, to our ordinary notion of a possibility as something that is not yet actual.

But that the peanut is actually under shell #1 still counts as a possibility according to the definition given above, because Smith knows nothing that rules out the identity of the corresponding possible world with the actual world. That is good, because the corresponding possible world is the actual world. Treating this actuality as a kind of possibility — a non-subjunctive possibility — accords with what is, I think, the ordinary intuition that if a state of affairs is actual, it is also possible.

[The set comprising this particular non-subjunctive possibility comprises a single member.] Given Smith’s knowledge, the cardinality of the set of shell-hiding epistemic possibilities relative to Smith of shell #1 is 1.

That the peanut is not under shell #1 does not count as a possibility according to the definition given above, because Smith does know something that rules out the identity of the corresponding possible world with the actual world: namely, that the shell is under shell #1. The corresponding epistemically possible world relative to Smith is, remember, the possible world described by the proposition ‘Everything apart from shell #1 is the same as this actual world, and the peanut is not under shell #1’. Given Smith’s knowledge, the set comprising the set of not-shell-hiding epistemic possibilities shell #1 is null. So cardinality of the set of not-shell-hiding epistemic possibilities relative to Smith of shell #1 is 0.

So given what Smith’s knowledge now is, the cardinality of the set of epistemic possibilities relative to Smith with regard to shell #1’s hiding | not hiding the peanut is either 1 or 0. Smith’s newly-acquired knowledge collapses, so to speak, the set of relevant epistemic possibilities from a minimum of 2 to either 1 or 0. We move from what I will call ‘subjunctive possibilities’ — from maybe, could be, might be, it could go either way (or any number of ways — to ‘here it is in front of our eyes in broad daylight — it was never impossible’.

Above, I had Smith possessing the knowledge that one and only one of the three shells hides a peanut. With this knowledge, Smith knows something that rules out there being two peanuts hidden there, or three…or six (supposing each shell can hide two peanuts). Without this knowledge, the number of possibilities relative to Smith would increase. There could be zero, or one, or two, or three…. peanuts. Smith’s background knowledge regarding the relative sizes of the shells and of peanuts, however, rules out the identity of the actual world with any possible world in which the shells hide one hundred thousand peanuts. That the shell hides one hundred thousand peanuts is not an epistemic possibility relative to Smith. While I am not totally sure how many possibilities this generate — is there a vague border here? — I do know that there are now more than six. And while a peanut being under one of the shells is one of the possibilities, that the peanut is under one of the shells is not. For this possibility presupposes a true, known description [true description] — the peanut placed under one of the shells by Elizarraraz. A peanut that now has a definite history, a trajectory through time that can be used to identify it. ‘A peanut’ does not specify any such thing. Remember that a possibility is relative to a proposition. The state of affairs with a selection, a specification. The possibility that (zero, one, two, … some reasonable n….) peanut is hidden.

Before Smith cheated and peeked, the total number of subjunctive possibilities that the peanut lies hidden under a given shell is 3. {shell#1p} u {shell#2p} u {shell#3p} = {shell#1p, shell#2p, shell#3p}. The cardinality of this set is 3, as is the cardinality of the set of subjunctive possibilities that under a given shell the peanut does not lie: {shell#1np, shell#2np, shell#3np}. These two sets comprise the “sample spaces”, i.e., “possibility spaces” for the odds that Smith will find | not find a peanut when he turns over any given shell.

Given that Elizarraraz used a randomizer to select which shell to place the peanut under, and given that there is just one peanut, the chances that Smith will uncover a peanut are 1/3 — the number of peanuts distributed among the shells divided by the cardinality of the corresponding possibility space. The chances that Smith will fail to uncover a peanut are 2/3 — the number of “peanut absences” divided by the cardinality of the corresponding possibility space.

In order for a probability to be assigned to the event [‘event’ used technically here; do explain], a likelihood also needs to be assigned. In the case of the shell game set up by Elizarraraz, it is equally likely that any one of the three shells hides the peanut, since Elizarraraz employed a randomizer to select under which shell to place the peanut. For now, I will rely on the intuition that employing the randomizer will result in an equal likelihood; I won’t try to put this claim on a solid foundation. So the probability that Smith will uncover the peanut (before he surreptitiously peeked) is 1 in 3. Correspondingly, the chance that he will uncover just some tabletop unburdened by the weight of a peanut is 2 in 3. I will use ‘probability’ as substitutable for ‘chance’ (at least for now), so the probability that Smith will uncover | fail to uncover the peanut is 1 in 3 and 2 in 3 respectively.

In this case, whether there actually is an equal likelihood can be tested by further iterations of the shell game. If, for any given one of the three shells the number of times — the frequency at which — the peanut turns out to be under any given shell is sufficiently close to one third of the total number of iterations of the game, and if the iterations are sufficiently large, we will say that the chances that Smith will select the shell hiding the peanut are 1 in 3. At least for now I will leave the weasel phrases ‘sufficently close’ and ‘sufficiently large’ as they are, without attempting to get more precise.

I assume that the greater the number of iterations, the closer to 1/3 will these chances, this probability, come. After ten million iterations, I do believe, (look at the computer simulations of the Monty Hall game), that number will come very, very close to an exact 1/3. the At the time of this writing, I am ignorant how the numbers will behave. Will the deviations, under or over, always be getting less and less? Or will the variation, at some point, become random, with (to borrow the idea from Taichi Ohno’s Toyota Production Process) any deviations above the Upper Control Limit or below the Lower Control Limit indicating that some outside factor has intervened?

This is all I will have to say for the moment about frequency.

If there is a glitch in the randomizer one possibility will have more weight than the the other. There will not be an equal likelihood that the peanut is under any given one of the three shells — as we would see, as the game is iterated, should the peanut end up being under shell # 1 2 out of 3 times. Frequency pretty much provides the last word on likelihood. In the case of the Monty Hall Game, what shows conclusively that the player should switch doors is the fact that the switched door hides the car 2 out of 3 times.

In this on-line tutorial on probability, probability is conceptualized as having at least two “dimensions”, so to speak: the possibility (sample) space and the likelihood. The likelihood gets represented as a larger area. Above, I have suggested a third dimension: knowledge/ignorance (as constituted by the union of two sets of “knowledges”). Now I would like to show how knowledge determines the cardinality of (the size of) the possibility space.

Now Morgenstern, who is in cahoots with Smith, enters the scene. Morgenstern (the name is German for ‘morning star’) hails from Brooklyn and was in MY COUSIN VINNIE. Initially, Morgenstern does not know that Smith had turned over shell #1 and discovered the peanut there. Smith signals to his in-cahoots-ee that shell #2 is empty.

Before Smith signaled to her that shell #2 is empty, Morgenstern was dealing with a possibility space with a cardinality of 3. As with Smith when he first played the shell game, Morgenstern deals with three subjunctive possibilities corresponding to the propositions ‘the peanut is under shell #1’; ‘the peanut is under shell #2’; and ‘the peanut is under shell #3’. For each proposition, Morgenstern knows nothing that would rule out the identity with the actual world of possible worlds in which the peanut lies under shell #1 | alternatively shell #2 | alternatively shell #3. All of those worlds are epistemically accessible relative to Morgenstern from the actual world. Because the set of these subjunctive possibilities has a cardinality of 3, the possibility space Morgenstern is dealing with, identical with this set, also has a cardinality of 3.

turned up empty. At this point, her chances are 1 in 3 of winning the peanut. (This is so both if she desires the peanut, or, though not desiring the peanut, she selects a shell anyway.) But now Smith finds a way to signal to her that shell #1 is empty.

Assuming that Morgenstern opts to use her knowledge by mentally removing shell #1 from the pool of possible choices comprising the could be’s and may be’s, and arbitrarily/randomly selecting a shell from the remainder, her chances of winning the peanut have increased from 1 in 3 to 1 in 2.

Of course, Morgenstern might opt not to use her knowledge. She might be guilt-stricken, for example, by her having received a signal from Smith, for example, and employ a randomizer. This would make her chances 1 in 3. Or she might choose to utilize her knowledge in a non-standard way and purposely lose by selecting either shell #2 or shell #3, making her chances 0. But given a desire to win, or at least not to lose, and barring any unusual events (say, the tornado crossing over from Nebraska and preventing Morgenstern from actually selecting a shell), her chances of winning the peanut are now 1 in 2.

Clearly, then, Morgenstern’s chances and Smith’s chances are determined by their knowledge and by their ignorance — assuming of course any number of things, such as leaving the Nebraskan tornado out of the picture, as well as any guilt on Smith’s and Morgenstern’s part, their desire or lack thereof to actually use their knowledge, and God knows what else. The situation I am interested in abstracts from all of this, leaving just Smith’s and Morgenstern’s knowledge and their ignorance. ffffggg

[Reference class. Knowledge determines the sample space]

fffggdfdfdfdfd

The proposition expressed by the sentence ‘the peanut is under shell #1’ maps to that possibility; likewise the proposition expressed by ‘the peanut is not under shell #1′ maps to that possibility in another possible world.

likelihood and frequency ffddssdasdsdsdsdssdsd

These numbers of course presuppose a number of factors:

Smith may be ignorant of an indefinite number of states of affairs vis a vis the shells laid out on the table. If there are grains of sand strewn on the table, for example, he may be ignorant of, say, the number of grains of sand shell #1 is hiding. 3 grains is one possibility as ‘possibility’ is articulated above; 1 is another possibility; 5 is yet another, and so on. I think it would be difficult, to say the least, to specify here the cardinality of the set of epistemic possibilities relative to Smith. And I haven’t mentioned other epistemic possibilities relative to Smith, such as the epistemic possibility that shell #3 is hiding a ruby.

But in the case of the shell game set up by Elizarraraz, the possibilities of interest to Smith and Elizarraraz and to any other player in the game are sharply demarcated. There is just one object of interest hidden by one of the three shells, the peanut. For each of the three shells, there are just two possibilities, two states of affairs that may or may not obtain and regarding which Smith knows nothing that would rule out the identity of the possible worlds comprising those states of affairs with the actual world: the peanut is under the shell, or the peanut is not under the shell. Now a state of affairs bears a one to one relation (possibly, identity) with a proposition. So relative to the proposition expressed by the English sentence ‘The peanut is under shell #1, or the peanut is not under shell #1’, there are two possibilities, two states of affairs related in the appropriate way to Smith’s knowledge and the holes therein to make them count as epistemic possibilities relative to Smith. The cardinality of the set of possibilities corresponding to this proposition is 2.

Stated another way, before Smith turns over shell #1, things could go either way; and the cardinality of the set of ‘ways things could go in the peanut getting hidden (not hidden) by shell #1 department’ is 2.

Doubtlessly going from an indefinite cardinality to the specific cardinality named by ‘2’ has something to do with human practices generally and with games specifically. Leave that thought to the side for the moment — I will be getting back to it.

Before Smith turns over the peanut, there are two epistemically possible (relative to Smith, which henceforth will be understood) states of affairs — two possibilities. Smith does not know anything that would exclude, rule out the state of affairs described by ‘shell #1 is the one of the three that hides the peanut’ from inclusion in the actual world, and he does not know anything that would exclude, rule out the state of affairs ‘shell #1, hiding nothing but empty air plus a brief expanse of table top, is not the one of the three shells hiding the peanut.’ These two epistemic possibilities will be in play until Smith turns over shell #1 and finds out which one actually obtains. Things could go either way. The cardinality of the set of ‘ways things could go’ here is 2. The set of possibilities described by ‘shell #1 was hiding (not hiding) the peanut is 2. There are two possibilities with regard to how things might go in the shell #1 hiding the peanut department. The possibilities are open.

After Smith turns over the shell, that the shell was hiding the peanut is an epistemic possibility for Smith because Smith still knows nothing that would rule out this state of affairs being comprised by the actual world. This is good, because this state of affairs is comprised by the actual world. Speaking a bit neologistically, this state of affairs is an “actual possibility”. There is just one way the things actually are in the shell #1’s hiding the peanut department. Here, the set of ‘ways things actually are in the shell’s #1 hiding the peanut department’ is 1. So an “actual possibility” has a cardinality of 1. The set of possibilities described by ‘the peanut was under shell #1 has a cardinality of 1. With the knowledge Smith gained by turning over the peanut, the number of possibilities has decreased from 2 to 1. We have gone from a state of open possibilities (things could go either way) to a state of one closed possibility (the actual state of affairs). This is a contracted, closed possibility. Things can no longer go either way.

That the shell is not hiding the peanut is no longer a possibility, because Smith now knows something — namely, that the peanut is under shell #1 — that would rule out this state of affairs being comprised by the actual world. Given this, it is an epistemic impossibility for Smith — the shell could not have been not hiding the peanut during the period of time it was hiding the peanut. So after Smith exposed the peanut, the set of possibilities described by ‘the peanut was not under shell #1’ is the null set. Its cardinality is 0. This is a contracted, closed impossibility. Things can no longer go either way.

Set of possibilities relative to a proposition.

Regar uing how things stand with respect to the peanut and shell #1, both possibilities cannot obtain at the same time. Either the shell is hiding the peanut, or it is not. One state of affairs has to fail to obtain. I will distinguish between I shall call ‘contracted possibilities’, which have a cardinality that is either 1 or 0, and ‘uncontracted possibilities’, which have a cardinality greater than one. Let me explain.

Let me take uncontracted possibilities first. Before Smith turns over shell #1, there are two epistemic possibilities relative to Smith. Either the shell is hiding the peanut, or it is not. Articulated this way, shell #1’s hiding the peanut | alternatively not hiding the peanut are, before Smith turned over that shell, both epistemic possibilities relative to Smith. After Smith turns over the shell, that the shell is hiding the peanut is an epistemic possibility for Smith because Smith knows nothing that would rule out this state of affairs being comprised by the actual world. That the shell is not hiding the peanut is no longer a possibility, because Smith now knows something — namely, that the peanut is under shell #1 — that would rule out this state of affairs being comprised by the actual world. And, of course, given Smith’s knowledge that there is one and only one peanut lurking somewhere in that group of three shells, that shell #2 or shell #3 is hiding the peanut are also no longer epistemic possibilities for Smith.

Notice that this account presupposes that the epistemically accessible world is also (classical) logically accessible, and physical-lawfully accessible from the actual world. Classical logic’s law of the excluded middle has to obtain for these possibilities, and objects of a certain size, such as peanuts, cannot suddenly find themselves the next moment on the nose of the Mona Lisa. I assume that physically-lawful possible worlds are a subset of classically-logical possible worlds, and epistemically possible worlds are a subset of physically-lawful possible worlds.

From Smith’s perspective, the moment before he turned over shell #1, his exposure of what was underneath could have have revealed the peanut, or it could have revealed nothing at all except some slight expanse of tabletop. It could have gone either way. Both states of affairs: ‘peanut underneath’ or ‘nothing underneath’ were posited only; Smith had no information or knowledge yet which one actually obtained. One of them must obtain (law-of-the-excluded-middle necessity), so one of Smith’s positings will hit, will get fulfilled by, an actual state of affairs when he turns over the peanut; and as it turned out this was the peanut’s lying underneath shell #1. This was a factual state of affairs that became exposed when Smith turned over the peanut. But the other state of affairs will remain a mere ghostly positing, the content of an unfulfilled representation. For a while, it was a merely possible state of affairs — and, to add insult to injury, it is now, at least at the moment, an impossible state of affairs. (Add all temporal qualifications. Also we are speaking of the larger-than-quantum realm.)

actual or possible, with regard to which Smith (more generally, a knower P) knows nothing that would rule out the hypot Phesis that the world which contains this state of affairs is identical with the actual world. Defined this way, the state of affairs in which the peanut is not under shell #1 is a possibility — here a merely possible possibility — and the state of affairs in which the peanut is under shell #1 is also a possibility — here an actualized possibility. I define things this way in order to agree with the intuition that if a state of affairs is actual, then it is clearly possible. Henceforth, unless I explicitly say ‘actualized possibility’, I will mean by ‘possibility’ what I am a bit awkwardly calling a ‘merely possible possibility’.

The set of things that Smith does not know (the set of his ignorances, so to speak), is the set of things that could turn out either way (part of a description of a world that is identical with the actual world or not identical with it, which it is is hidden). I will call these things ‘possibilities’. A possibility is the ‘it’ in the sentence ‘it could turn out either way’. Conceived this way, a possibility is explicated in terms of the ignorance of a particular knower, a subject S. A possibility is a could-be/might-be (alternatively, a possibly-is|possibly-is-not/might-be|might-not-be) or a particular not-knowing, a particular ignorance.

Smith could turn over a shell. Having himself placed the peanut under one of the shells (say, shell #2), Elizarraraz of course knows — is in possession of the information that — which shell he placed the peanut under. For Elizarraz, this state of affairs comes already unmasked. And For Elizarraraz, nothing rules out the hypothesis that the possible world in which the peanut is under shell #2 is identical with the actual world. When a particular possibility becomes or is already exposed, known, and becomes/already is known, there is, of course nothing in the union of the set comprising this possibility plus the set comprising all the other states of affairs known to Smith plus the set comprising what can be gained by the handwave ‘and everything else that is the case’ to rule out the hypothesis that this possible world is identical with the actual world. This is good, because this world is identical with the actual world. So the relation of epistemic accessibility is reflexive, given that the actual world is epistemically available for P from itself. Later, I think I can show this is the foundation of the implication that has been so troubling to me, If p Then p. However that may be, I will call ‘an actuality’ any state of affairs in the actual world, known by Smith or not, and a ‘certain actuality’ a possibility that has become or already is ‘exposed’, or ‘known.’

If we are talking about possible worlds, we are talking about possibilities. To talk about epistemically possible worlds is to talk about a particular kind of possibility — one that will, I think, make sense of the notion of probability. Smith’s not knowing under which shell the peanut lies generates six could be’s/might be’s — the peanut could be/might be under shell #1….shell #1 could be/might be empty. Each of these is a possibility. The absence of the information available to Smith creates these six what I shall call ‘subjunctive’ possibilities. The availability to Elizarraraz of the information that the peanut is under shell #2 generates what I will call a ‘non-subjunctive’ possibility. A non-subjunctive possibility is an exposed (to P) actuality, an actuality the information regarding which is available to or “exposed” to P.

[This will make probability an epistemic concept — one to be analyzed in terms of knowledge]

[The context determines the meaning of ‘is under the shell’]

The state of affairs described by a subjunctive possibility always carries a probability of less than 1. What the probability is of, say, the peanut beiung under shell #1

That a state of affairs is actual means that is possible. That there are three shells on the table is an actuality means that it is a realized possibility. andLet me say, then, that ‘a possibility’ expressed by a proposition p (e.g., the peanut is under shell #1 | alternatively | the peanut is not under shell #1) consists in a person P’s lacking information that would decide (‘rule out the hypothesis’) whether the possible world in which p is true is identical with the actual world. A possibility, then, is relative to a knower, a person P to whom information is available, not hidden. If Elizarraraz knows, for example, that the peanut is under shell #2 (he was the one who placed the peanut there, after all), then the peanut’s being under not being under shell #2 is not a possibility. And its being under shell #2 is not a possibility, but a certainty. I think I might reveal a wondrously intuition for the blindingly obvious if I said that there is a pool of three candidates for the shell that is hiding the peanut. (Henceforth ‘candidate shells’.) Only these three shells are candidates; the shell that is lying on the beach, say, in Santa Cruz, California at the exact GPS location xxxxxx is not a candidate. Clearly, then, whatever background of commonsense or practices that excludes the Ventura shell is a necessary condition for delimiting the pool of candidate shells. Only these three shells could be/might be hiding the peanut; it would be nonsensical to ask of any other shell ‘could this one be hiding the peanut?’

Let’s take a look at this ‘could be/might be.’ Smith’s not knowing, his ignorance, is another necessary condition for it’s being the case that the peanut could be under any one of these shells. For if he somehow came to know (say, he bribed Elizarraraz to tell him) that the peanut was under, say, shell#1, it would no longer be the case (from Smith’s perspective) that the peanut could be, is possibly, under shells #2 and #3.

Nor, I daresay, would it any longer be the case, from Smith’s perspective, that the peanut could be, is possibly, under shell#1. For it is now a certainty, a given, that the peanut is under that shell. Smith has moved from ‘the peanut could be/may be/might be under shell #1′ to ‘the peanut is under shell#1.’ Certainty is the antithesis of could be/may be/might be.

And of course, for Elizarraraz, who knows under which shell he placed the peanut (and continues to know, assuming some out-of-the-ordinary event has not occurred such as a sudden onset of amnesia), it was never the case that shell#1 could be hiding the peanut. Nor was it ever the case that, for him, the peanut could be under the other shells. He was never in a state in which it would make sense for him to say to himself (or to anyone who, he knows, also knows the location of the peanut) ‘the peanut could be under this shell, or that one (or perhaps this other one)’ The ‘or’ with the split it implies is needed for the ‘could be’ to make sense.

Ignorance, not-knowing, is a necessary condition for it’s being the case that, from a perspective, the peanut could be under this shell or that one. But knowledge that there is a peanut under one of the shells is also a necessary condition. [Explain the various cases. Knowledge that there is not a peanut under one of the shells. Knowing that there could be a peanut under one of the shells, based on some frequency of peanuts happening to be under shells. But ‘a peanut could be under one of the shells’ is not the same as ‘the peanut could be under this shell or under that shell or under that other shell.’ Anyway, what we will arrive at is a mix of knowing and not knowing — a middle state between complete certainty and total ignorance — is a necessary condition for probabilities less than 1 and greater than 0.]

This ‘could be/may be/ might be (henceforth just ‘could be’) is clearly perspectival in character and relative to a person, depending as it does on what each person knows/doesn’t know. From Smith’s perspective, there are three ‘could be’s’. From Elizarraraz’ perspective, there are zero. Likewise, from the viewpoint of an entity that knew everything that is knowable, there would also be zero ‘could be’s’ in this situation.

[I am avoiding the temptation to say that these ‘could be’s are ‘mind dependent’, or to call them ‘possibilities.’ I have the definite feeling that doing either creates the potential for some serious confusion.]

I think it would be uncontroversial to say that, assuming there is an equal likelihood that any one of the three shells is the one hiding the peanut, Smith’s chances of winning the peanut (henceforth ‘Smith’s chances’) in this situation are 1 in 3.

Let me attempt to clarify the notion ‘equal likelihood’ as follows. Were Smith to play the normal shell game [some number close to ten million and a multiple of 3] times under the same conditions (each time Elizarraraz employing a randomizer to decide which shell to place the peanut under), the number of times shell #1 was hiding the peanut will be very, very close to exactly 1/3 of that number; and ditto for shells #2 and #3. Repeat this a number of times and that ratio might from time to time actually be exactly 1/3; but the ratio is more likely (a probabilistic word again) to be just a little bit over or a little bit under. For all I know, the overages/underages might even approach 1/3 asymptotically, but here I have to confess my current ignorance (there is that concept ‘ignorance’ again). I will just say that at a certain point we say the ratio if ‘close enough for government work’, and round up (or down) and say there is an equal likelihood that the peanut is under any one of the shells. For now, of course (and perhaps forever) I will leave ‘close enough for government work’ undefined. An idealization.

[Much more to say about this. Three quail chasing one another around a bush. ]

The ‘could be’ means that Smith does not have a basis on which to act, the action here being a selection of a shell, choosing a shell. His choice of shells would be arbitrary. He could, in fact, just as well use some sort of randomizer device — say, a computer program randomly outputting 1, 2, or 3 — to make his selection.

Assuming an equal likelihood, I was saying, that the peanut is under any one of the three shells, Smith’s chances of winning the peanut are 1 in 3. These chances depend upon his ignorance of which shell hides the peanut, giving him no non-arbitrary or non-random way of selecting the peanut and no knowledge giving him a basis, no matter how treacherous (more on this later) on which to decide. Smith’s chances and what he knows/doesn’t know. As it happens, Smith does possess some knowledge vis a vis the shells and the peanut — but this knowledge is not necessary for his chances being 1 in 3. For Smith would have the same chances were he to idly turn over a shell out of sheer boredom, without knowing that one of the three hides a peanut.

And of course if Smith did know, as a result of bribing Elizarraraz, that the peanut is under shell #1 , his chances of winning the peanut are 1 if he acts on that knowledge by selecting that shell, and 0 if — perhaps because he feels guilty — he purposely selects either shell #2 or shell #3 in order to avoid an illegitimate “win”. In the first case, he uses his knowledge to intentionally — there is no way this cannot be intentional, given the knowledge — make his chances of winning the peanut 1; in the second case, he uses his knowledge to intentionally make his chances 0. Whichever his choice is, Smith would have a firm basis for making it.

Of course, if the tornado that had been, unbeknownst to the players of the shell game, touching down in Nebraska, suddenly crossed the river and touched down in the orchard just as Smith was reaching over to turn over shell #1, his chances of winning the peanut would not be 1. So we have to say ‘barring bizarre events like this’, Smith’s chances are 1. Another way to make the same point is to say that ‘restricting the scene to a situation that excludes the tornado across the river in Nebraska, Smith’s chances are 1. A situation, remember, is a part of a possible world (in this case, one possible world, namely, this actual world). A possible world includes everything; a situation does not. At some point I will probably need to explain situations in terms of something like ‘information structures’ (see Mares), but I am getting ahead of myself. fffggggg

But back to Smith who is ignorant vis a vis what is the case with the shells and the peanut. As he is supposed to do given the rules of the game, Smith turns over a shell — say, shell #1. As it turns out, shell #1 was hiding nothing except empty air (plus a certain stretch of table wood). Smith returns the shell to its previous position (carapace side up).

Now Morgenstern (German for ‘morning star) enters the scene.  (She hails from Brooklyn and she was in MY COUSIN VINNIE.) Initially, she does not know that shell #1 turned up empty. At this point, her chances are 1 in 3 of winning the peanut. (This is so both if she desires the peanut, or, though not desiring the peanut, she selects a shell anyway.) But now Smith finds a way to signal to her that shell #1 is empty.

Assuming that Morgenstern opts to use her knowledge by mentally removing shell #1 from the pool of possible choices comprising the could be’s and may be’s, and arbitrarily/randomly selecting a shell from the remainder, her chances of winning the peanut have increased from 1 in 3 to 1 in 2.

Of course, Morgenstern might opt not to use her knowledge. She might be guilt-stricken, for example, by her having received a signal from Smith, for example, and employ a randomizer. This would make her chances 1 in 3. Or she might choose to utilize her knowledge in a non-standard way and purposely lose by selecting either shell #2 or shell #3, making her chances 0. But given a desire to win, or at least not to lose, and barring any unusual events (say, the tornado crossing over from Nebraska and preventing Morgenstern from actually selecting a shell), her chances of winning the peanut are now 1 in 2.

Clearly, then, Morgenstern’s chances and Smith’s chances are determined by their knowledge and by their ignorance — assuming of course any number of things, such as leaving the Nebraskan tornado out of the picture, as well as any guilt on Smith’s and Morgenstern’s part, their desire or lack thereof to actually use their knowledge, and God knows what else. The situation I am interested in abstracts from all of this, leaving just Smith’s and Morgenstern’s knowledge and their ignorance. ffffggg

Sufficient conditions: knowledge and desire. Ignorance I think will be a necessary condition. n this case, her knowledge has increased her chances, but her relative ignorance still keeps those chances from becoming 1 (or 0). ffggggg

And, as stipulated, Smith is ignorant of which shell is hiding the peanut. As a first approximation, let me say that Smith’s ignorance is a necessary condition for his chances being 1 in 3. (Another of course would be the need for an equal likelihood of the peanut’s being under any one of the shells.) For surely if somehow he knew that the peanut is not under, say, shell #1, his chances would be 1 in 2, not 1 in 3. And if he knew that, say, shell #3 was the one hiding the peanut, then his chances would be 1 given a desire to win the peanut, and 0 given a desire to avoid winning the peanut. Smith’s ignorance/knowledge is, let us say as a first approximation, a necessary condition for his chances being such and such.

There are at least x number of conditions that must be fulfilled if this is to be so.

1. There must actually be a peanut under one of the shells. However, even though Smith knows this because Elizarraraz told him so and Elizarraraz is a reliable source in this case, this particular item of Smith’s knowledge is not necessary for his chances of uncovering the peanut to be 1 in 3. It is a factor that makes him (considerably!) more likely to play the game.

I will, however, venture the claim that this particular item of Smith’s knowledge does constitute evidence that, for any of the three shells, the peanut lies under that shell. Since Smith’s chances are not greater than 1 in 2 (I will call this the standard for up-to-par evidence) that for any given shell he selects the peanut will be under it, I will call this sub-par evidence. Even though it is sub-par, however and probably in most situations will count as unacceptably weak, it is evidence nonetheless.

2. Smith must be in a certain state of ignorance, which I will try to specify this way. Normally, he will be ignorant of which shell hides the peanut. If he does know — say, he has x-ray vision, or is gifted with psychic intuition, or he has bribed Elizarraraz to tell him — then he must refrain from utilizing that knowledge, say, by employing a randomizer to determine which shell he selects, thereby rendering inoperative his knowledge of which shell is hiding the peanut and preventing his knowledge from guiding his action.
3. The reference class must be clearly delineated. Only these three shells are in play — not these plus shells that Elizarraraz has swept away to the side of the table, nor these plus a certain shell on the beach in Ventura, California. In the case of a game, the reference class is circumscribed by common sense. In the case of less artificially circumscribed situations — see Graham Priest on the Sherlock Holmes story THE RED-HEADED LEAGUE — the boundaries of the reference class are less obvious.
4. Each shell must be equally likely to be hiding the peanut. What ‘equally likely’ means is to be determined by frequency. An idealized frequency. Repeat the shell game ten million times using the same procedures (Elizarraraz uses a randomizer to place the shell, and so on) and under the relevantly same conditions, and the number of times shell #1, say, turns out to have been hiding the shell will be very close to 1 in 3; likewise for shell #2 and #3. Possibly the ratio will be approaching a precise 1 in 3 as a limit, as I believe Reichenbach thinks. Maybe it will sometimes get closer — maybe even hitting the exact ratio from time to time –, sometimes get further away in a random manner. Of these things I am ignorant at the time of this writing. Whatever is the case, at some point we say ‘this frequency is close enough for government work’, and round up (or down) to 1 in 3 and pretend we have a precise figure — just as we may take a tiny dot and pretend it is a Euclidean point with no size. In both cases, we are performing an idealization, an act of the imagination. There is an artificiality here — just as there is an artificiality in the reference class getting restricted to just these three shells in the middle of the table.

[Utilized knowledge is a sufficient condition for Morgenstern’s having a 1 in 2 chance. But it is not a necessary condition for Smith’s having a 1 in 3 chance. Without knowing that a peanut is underneath one of the shells, Smith might idly turn one over — and this would be enough for him to have a 1 in 3 chance. There is no opportunity here for Smith’s knowledge to affect his chances. Of course his chances are zero if he does not play the game.]

[Smith’s ignorance is a necessary condition for his having a 1 in 3 chance. Ignorance of something — either of the peanut condition of each of the shells, or of what shell the randomizer will tell him to select. Ignorance of which shell he is about to select. Come to think of it, though, I think it is the absence of utilized knowledge that is key, even in the case of the randomizer. God might tell him what number the randomizer is about to produce, but Smith is not able to ensure the number names the shell hiding the peanut. ]

[ As long as That is to say, the probability there is a peanut underneath that particular shell is 1/3, and the probability that there is not is 2/3. [I THINK this move from ‘chances’ to ‘probability’ is unproblematic, but the alert reader may want to pay careful attention to this move because I will be relying on it quite heavily, at least in terms of rhetoric.] Smith’s chances are determined, first, by the number of shells at play on the table; second, by the number of shells within that set of whose peanut status (hiding or not hiding a peanut) he is ignorant; third, by the fact (just stipulated here for the moment, of course) that the peanut is equally likely to be lurking under any one of the shells.

Smith knows that a peanut is lurking underneath one of these shells on the table that are in play. Smith’s knowledge that there is a peanut came to him through a reliable channel of information. The reliability is the marked case, so to speak; we always expect chaos, noise, and unreliability to be the default. The reliability cannot be completely taken for granted; the channel of information has to be checked periodically if it is to retain our confidence. Has Elizarraraz suffered a personality change and becomes less reliable? Is the doorbell’s ringing still a reliable signal that someone or something is depressing the button outside?

But what is taken for granted, what is automatically “assumed” — maybe ‘lived’ or ‘breathed’ are more apt words — is that these three shells set up by Elizarraraz on the table are the ones in play. It is not lurking under one of the shells that, say, are strewn on the ground, rejected by Elizarraraz for one reason or another; nor is it lurking under one of the shells that Elizarraraz has swept to the side on the tabletop, excluding them, perhaps arbitrarily, from the three he has grouped together on the table. Or rather, if one of those does happen to hide a peanut, Smith’s uncovering it is not going to win him the game. It is the three shells on the table that are in play, not four (the three plus one on the ground, or one in the pile swept to the side), not 5 (those four plus a shell on the coast of California near Ventura). This is so completely, utterly taken for granted that never in his life will Smith ever check to make sure it is still true or bother to affirm it, no more than the fish needs to check that the water in which it is swimming is still there. This “understanding”, which I will call ‘the background’, is already part of the scene the moment Smith walks into it. I discuss the background in just a little bit more detail in the appendix.

The background delimits the reference class or sample space (I will use the two terms interchangeably), i.e., those things the count of which serves as the denominator in the division determining, on the assumption of equal likelihood, the probability of an event. Coin landing heads or coin landing tails. The seven days starting from Monday last week (three of which were wet and four of which were warm). The number of people walking around in London whose coats display a shiny cuff on the right and a worn area on the left elbow on the day a certain Mr. Jabez Wilson consults with Sherlock Holmes regarding a mysterious Red-Headed League (see Graham Priest, LOGIC A Very Short Introduction, Chapter 11).

Smith’s knowledge that the peanut is under one of the shells puts, along with the background of understanding (which background includes understanding what shell games are and that the point is to try to win them) the three shells on the table and only those three shells into play. Without this knowledge, the three shells would not be in play. They would be just three shells lying on the table. They would not form the reference class of shells that could be/might be hiding a peanut.

For if Smith does not know that one of these shells hides a peanut, doesn’t even know there is a game to be played here (say, he just fell out of the milk truck yesterday), and is just idly overturning things, bottle caps, shells, whatever, what the reference class is becomes a bit ambiguous. The ‘case of the missing reference class’ makes its baleful appearance again. Is the reference class all the over-turnable things on the table (shells, bottle caps, large leaves, say?) What if he has some sort of weird tendency to overturn things, without realizing the point of a shell game? In that case, even if for the moment there are just three over-turnable things on the table (the three shells), the cardinality of the reference class could change at any moment — say, Elizarraraz opens of bottle of beer and puts the cap convex-side-up on the table, or a leaf happens to blow onto the table? Shouldn’t the reference class then be ‘over-turnable things that could end up on the table during such and such a timespan?

In the special case of a game with definite rules as interpreted against the background of common-sense understanding, or in a reference class already specified (Graham Priest’s week), there is no problem in determining what the reference class is. Once determined, its cardinality is stable to one degree or another (just three shells on the table; just seven days in the week.) But in less structured situations there is very definitely a problem: there does not seem to be any clear, non-arbitrary way to determine what the reference class is and therefore its cardinality. In the case of Sherlock Holmes and THE MYSTERY OF THE RED-HEADED LEAGUE, should we take the reference class to be the people wearing coats in London? Persons of the male persuasion wearing coats in London? In the United Kingdom as a whole? Should we also include France? There is no clear answer to these questions. Were Smith ignorant of the fact that a peanut is hiding under one of the shells on the table, there would likewise be no clear, non-arbitrary way to determine the reference class.

Once Smith comes to know that there is a peanut under one of these shells Elizarraraz has grouped together on the table as pieces in a game that Smith could choose to enter, he is drawing a conceptual boundary around those shells. Inside that boundary is the reference class comprising the pieces that are in play in the game. Outside that boundary is everything that is not this reference class — everything in the world that is not a piece at play in this game. The rules of the game presuppose the ability to draw this conceptual boundary, and therefore also presuppose the knowledge plus the background understanding that makes that knowledge possible.

Pull this one string — knowledge that a peanut lies hidden underneath one of the shells, and a bunch of other strings get pulled out as well. A peanut lies hidden underneath one of the shells, and its being there is part of a game the point of which is to try to uncover the peanut in a situation in which one does not know under which shell the peanut lies. To possess these “strings” of knowledge is to draw a conceptual boundary around the three shells and just these three shells. Now is the time to start discussing ignorance. Knowledge, made possible by the background of understanding, determines the reference class. Ignorance brings could be/might be — in other words, probabilities between 0 and 1 — into the world. ffffffggggghhhhhhiiiiijjjjjjjjkkkkkkklllllll

knowledge that a peanut is lurking under one of the shells on the table (he just doesn’t know which one) helps determine his chances by helping to determine the reference class, i.e., what can count as ‘being in play’ in the shell game. This knowledge is the tip of an iceberg comprising a completely taken for granted background of ‘understanding’. Smith knows that the peanut is lurking under one of these shells, the shells set up on the table by Elizarraraz. It is not lurking under one of the shells that, say, are strewn on the ground, rejected by Elizarraraz for one reason or another. Or rather, if one of those does happen to hide a peanut, Smith’s uncovering it is not going to win him the game. One can go on and on describing this background; I will stop here. But I do want to point out before I go on that this is a shared knowledge, a shared understanding, and a shared background. It is not limited to Smith. This knowledge and the background against which it occurs delimit the reference class and enable us, in the case of this particular game, to say that this reference class has a definite cardinality. It is the three shells on the table that are in play, not four (the three plus one on the ground), not 5 (those four plus a shell on the coast of California near Ventura). and the renders him more willing to play the shell game. Assuming that any given one of the shells is equally likely to be chosen by Smith, these (the knowledge, understanding, and background) are what determine (along of course with whatever determines the equality of likelihood) that Smith’s chances of winning the peanut are 1 in 3. ffffggggghhhhiiiiiiii

As he is supposed to do given the rules of the game, Smith turns over a shell — say, shell #1. As it turns out, was hiding nothing except empty air (plus a certain stretch of table wood). Smith returns the shell to its previous position (carapace side up).

Ignorance is necessary for could be/might be. Here is why. ffffgggggghhhhhiiiii

Although the rules of the game do not allow this, no divine censor will prohibit Smith from thinking: “If I turn over any one of the remaining shells, my chances of revealing the peanut would be 1 in 2.” And I think I think most people would find Smith’s evaluation of what his chances would be to be uncontroversial. One would be highly likely to have a strong intuition that this is so. Warning: a challenge to this intuition (which I will be making partly as a matter of due diligence, partly as an excuse to discuss the Monty Hall paradox) will be coming up shortly.

Now Morgenstern (German for ‘morning star) enters the scene.  (She hails from Brooklyn and she was in MY COUSIN VINNIE.) Initially, she does not know that shell #1 turned up empty. She does know, however, that one of the shells hides the peanut, because a reliable source of information, Elizarraraz, told her. At this point, the chances of her winning the peanut are 1 in 3.

Unbeknownst to Elizarraraz, however, Smith and Morgenstern are colluding to increase their chances of winning the peanut. (They have agreed beforehand to split the peanut if one of them wins. This is a truly sought-after peanut.) Smith has found a way to signal to her that shell #1 is in fact empty. Given this additional knowledge, Morgenstern’s chances of winning the peanut are now 1 in 2. ‘Given this knowledge.’ I should say, rather, that given this knowledge and the desire, the ability, and the choice to implement that knowledge, Morgenstern’s chances are 1 in 2. [I emphasize the ‘implement’ part to steer away from the idea that what determines probability is something locked up inside the skull and not an ‘extra-mental’ phenomenon. Be sure to bring this out at some point. The bias of the steering wheel in favor of this direction has made me uncomfortable]

For this additional knowledge to increase her chances (by reducing the size of the sample space — more on that later), she has to implement that knowledge. Suppose, for example, that she is so guilt-stricken by her cheating that she is unable to turn to her advantage the knowledge that shell #1 is not hiding the peanut. She uses a randomizer instead to decide which of the three shells to select. In that case, her chances are still 1 in 3.

But given the proper set of circumstances (she has a desire to win the peanut; she is not paralyzed by guilt; some other strange psychological complex causes her to have the randomizer make the choice for her), the additional knowledge will be implemented, and the implementation will result in an increase in her chances to 1 in 2. You might see the implementation in her behavior — for example, the way her eyes circle around just shells #2 and #3, avoiding shell #1.

Were Elizarraraz to decide to cheat and employ his knowledge of under which shell the peanut lurks to win the peanut, his chances of winning the peanut would be, of course, 1. For he knows which shell to directly reach for. There is no possibility [excluding possibilities at the margin — tornado, mini-stroke, and so on. — A situation is an idealization.] that he may be in error as to which shell is hiding the peanut. His action would be guided by his having certainty, that is to say, a credence of 1, which again, is a probability of 1. The terms used to describe probability have knowledge built into them.

The final “use case” I want to consider is the situation in which a person has no knowledge with which to guide them. Jones (Welsh for something) arrives on the scene, but she does not know that there is a peanut under one of the shells. She figures something is under one of the shells — something is up, given that Elizarraraz, Smith, and Morgenstern are all standing around the table, after all. But she does not know whether one of the shells is hiding a peanut, a ruby, a walnut, or none of the above — they are all hiding just empty air. fffffgggggghhhhhhhhhhh

Information/absence of information determines the sample space, along with, obviously, what the information is about: Smith (although this is not relevant to the example, the name is English for ‘smith’ as in ‘blacksmith’. But you knew that already) enters the scene. He knows that there is a peanut hidden underneath one of the three shells. (Elizarraraz, who is a reliable conduit of information, has told him this.) Smith is about to play what I will call, for reasons that are about to become clear, the ‘normal’ shell game. He is to select a shell and turn it over to see if it is hiding the peanut. He is to get just one chance. I think it would be uncontroversial to say that Smith’s chances of winning the peanut are 1 in 3. That is to say, the probability there is a peanut underneath that particular shell is 1/3, and the probability that there is not is 2/3. [I THINK this move from ‘chances’ to ‘probability’ is unproblematic, but the alert reader may want to pay careful attention to this move because I will be relying on it quite heavily, at least in terms of rhetoric.]

This number is the result of two factors: first, the sample space, and second, the likelihood that any of the members of that sample space will become an actual, and not just a possible outcome (Smith selects the shell that is hiding | alternatively not hiding the peanut). In this particular case (the normal shell game), the sample space is the set of possible outcomes of turning over any of the shells in play on the table. Shell #1 hides the peanut, or shell #2 hides the peanut, or shell#3 hides the peanut. So the sample space Ω is :

{ shell#1p, shell#2p, shell#3p }

or, to show explicitly that if, say, shell #1 happens to be hiding the peanut, the remaining shells are perforce empty:

{ { shell#1p, shell#2p, shell#3p }, { shell#1p, shell#2p, shell#3p }, { shell#1p, shell#2p, shell#3p } }

where the superscript ‘p’ means the shell is hiding the peanut and the superscript ‘p‘ means the shell is not hiding the peanut.

Let me get some terminology out of the way. I will be relying on the standard google-able terminology of probability theory. The sample space is the set of possible outcomes of an “experiment”. An “event” is a subset of this superset, i.e. the sample space. For example, the subset ‘shell #1 hides the peanut and shells #2 and #3 do not’ is an event. The term is a bit unfortunate, because ‘event’ usually connotes — at least to my ears — a concrete happening occurring (or having occurred or occurring in the future) in space and time. Here, however, an ‘event’ is an abstraction — a subset, and not a concrete happening occurring in space and time. But whatever. An “experiment” is by contrast a concrete action, such as turning over shell #1 and discovering it to be hiding the peanut | alternatively hiding just empty air.

There are at least two factors that determine the “size” or cardinality of a sample space — the number of members it has. These factors are 1) what I will call ‘the shared background of common sense’ and 2) a person’s individual ignorance/knowledge. Both of these factors are, in one sense or another, “subjective” and perspectival.

1) Shared background of common sense: A sample space is, I have said, a set of possible outcomes of a given activity. But what determines what is eligible to count as a ‘possible outcome’? The answer to this question will help determine the “size” or cardinality of a sample space. There are a number of factors that contribute to answering this question.

This shared background of common-sense comprises social practices, rules, deeply-ingrained dispositions to count certain things as relevant and other things as not relevant, the stability and predictability of physical objects (at least on the post-quantum level). I address these sub-factors in an order suggested to me by Wittgenstein’s ON CERTAINTY, going from the most vulnerable to change to the least vulnerable, from the least deeply taken for granted to the most deeply.

1a) Rules of the game: Taken-for-granted rules govern practices in general and games in particular and help to define these practices and games. Drawing a standard recognized card from a pack of cards normally counts as an eligible outcome; drawing a scrap of paper that may strayed into the pack normally does not. The sample space for the cards has a cardinality of 52, not 52 plus the one scrap of paper. Flipping a coin has two possible outcomes, heads or tails. The coin’s landing on its edge is not a possible outcome, at least not if the normal rules that apply to the practice of flipping a coin are in force. It might be a possible outcome in a different game. In the normal practice of flipping a coin, the sample space is the set with two members: coin lands heads or it lands tails. In a non-normal practice, the sample space might have three members: The same holds mutatis mutandis for throwing a die. In the normal practice, the sample space comprises six members. But should anyone be skilled enough to make the die land on one of its edges invent a new, non-normal practice, the sample space would comprise 12 members. Winning the peanut is the point of the normal shell game set up by Elizarraraz, not uncovering a particular grain of sand — no matter how exquisite that grain is. So if none of the three shells covers the peanut, but one covers a grain of sand, the cardinality of the sample space will be 0, not 3.

Extrapolation of the rules to form a different game: It may make sense to talk about the cardinality of a sample space of a game that is merely possible, and not actual. Suppose, for example, that no game — call it the non-Monty-Hall shell game — currently exists with the following rules. The three shells, one of which is hiding the peanut, are placed on the table as before. But now the player has two chances, not just one, to try to win the peanut. As before, the player’s selection(s) are made without any action taking place that depends upon knowledge of the peanut | peanut-less state of the shells. Even in the absence of an actual game like this, one can, clearly, see that after the first selection the sample space would now be restricted to two members. Merely possible rules suffice to determine a sample space and its cardinality.

1b) What counts as an eligible item in play is determined by the background of common sense: What items that are ‘in play’ in a practice are also taken for granted. The shells Elizarraraz has placed on the table are the ones that are in play in the normal shell game he is setting up. The shells that are in a pile a few feet away from him are not in play. Nor is this or that shell on the beach 1,500 miles away to the east, or 2,000 miles to the west.

1c) As part of the background of common sense, the stability of the physical world plays a role: Even more deeply taken-for-granted is the stability of the physical world — at least on the post-quantum level. We assume for example — except for a few Twilight Zone moments — that the peanut stays under whichever shell it is under and is not going to behave like the electron which (according to my undergraduate chemistry TA), for all one knows, might be on the nose of the Mona Lisa. Additional to the three shells in the sample space that might be hiding the peanut, there is not also a fourth shell, sometimes one underneath the Mona Lisa or stuck on her nose that might be hiding the peanut, sometimes one on the floor of the Farnese Palace underneath the Carracci ceiling.

1d) Cases in which ‘What size is the sample space?’ does not have a clear answer: For the moment, dear reader. let me refer you to Graham Priest’s treatment of Sherlock Holmes’ induction that Jabez Wilson is likely to do a great deal of writing given the smooth patch on Wilson’s coat at the elbow. (LOGIC A Very Short Introduction, Graham Priest, Oxford, Oxford University Press, 2000, pp. 78-85).

All of these examples concern what is relevant to a particular problem, practice, or game. Naturally, this should raise some concern in the non-comatose reader that I may be going in a circle. For what I eventually hope to gain at the end of these ruminations is an account of at least one relation that makes p relevant to q in a relevant IF p THEN q conditional. For the moment, I will beg for mercy by pointing out that there circles and there are circles; some circles are narrower than others; some wider circles give one a more expansive view of the territory and constitute the scenic route.

2) A person’s ignorance/knowledge:

Let’s return now to Smith, who is about to turn over the shell he has selected. He is ignorant of which of the three shells is hiding the peanut, but he knows that the peanut is lurking behind one. Let’s stipulate that each shell is equally likely to be hiding the peanut. What ‘equally likely’ means I will be turning to shortly; for now, let’s just take it as a primitive. In this situation, Smith’s chances of winning the peanut are clearly 1 in 3; his chances of turning up just empty air are 2 in 3.

Gun on the shelf that will fire later in the story [1]: Obviously, Smith does not know if the shell he is about to turn over hides the peanut or not. Whatever method or non-method he uses to select the shell (he likes the slightly tawnier sand-color of this one; he rolls a 3-sided die; he just picks one), he does not choose based on any knowledge, or on any evidence of what might be | might not be lurking underneath the shell.

Let’s be Smith for a moment — he is, after all, Any Person/Every Person.

Might and could be defined by ignorance.

Smith knows that a peanut lies hidden underneath one of the three shells. (And of course it follows from his knowing that the peanut is under one of the shells that it is under one of the shells.) But Smith does not know under which That shell could be, might be shell #1, or it could be /might be shell #2, or it could be/might be shell #3. Were Smith come to know that the peanut is under, say, shell #3, it would no longer be the case that the peanut could be/might be under shell #1, and it would no longer be the case that the peanut could be/might be under shell #2. These would no longer be possibilities, that is to say, possible outcomes, could be’s or might be’s. [Currently unredeemed intuition] A possibility’ requires a combination of knowledge and ignorance. Remove the ignorance, and the possibility no longer exists. And since, in this particular case, it is Smith’s ignorance that obtains, the possibilities are such, are possibilities only from Smith’s point of view. ffffffffffffffffffffffff

Smith turns over the shell — say, shell #1. It was hiding nothing except empty air (plus a certain stretch of table wood). Smith returns the shell to its previous position (carapace side up).

Now were Smith allowed to repeat the game, but this time with the two remaining shells, #2 and #3 — the two shells the contents of which he is still ignorant — his chances of winning the peanut would surely be 1 in 2. That this is so should be clear intuitively. Of course, the Monty Hall shell game which I will be discussing shortly, tends to generate the exact same false (fasle yes — but with certain caveats) intuition. Can we rely on our intuition in this case? Spoiler: yes. But I will get to that after the long, boring disquisition on the Monty Hall shell game that will ensue shortly.

Now Morgenstern (German for ‘morning star) enters the scene.  (She hails from Brooklyn and she was in MY COUSIN VINNIE.) She does not know that shell #1 turned up empty. She does know, however, that one of the shells hides the peanut, because a reliable source of information, Elizarraraz, told her. The peanut is still under one of the remaining shells. Using a randomizing device, Elizarraraz has shell #2 selected for him. He points to that shell #2 and asks both Smith and Morgenstern what are the chances the peanut is under that shell. For Smith, surely, the answer is 1 in 2. For Morgenstern, the answer has to be 1 in 3. For Elizarraraz, who knows where he put the peanut, the answer has to be either 0 or 1. Were Elizarraraz to point to shell #1, the answer for both him and Smith would have to be 0. What the probabilities are differs from the perspectives of each of the three because the sample space differs for each given what each knows.

[Likelihood: Naturally, both the ‘let’s say’ and the ‘equally likely’ cry out for some finessing, given that the whole point of this exercise is to state what probability/randomness is. Explaining randomness, a probabilitistic concept, in terms of other probabilistic concepts (‘likelihood’) does seem a bit unpromisingly circular. But before I turn my attention to this threatening circularity, I want to focus for a moment on the concept of a sample space. ]

Elizarraraz is not ignorant of which shell the peanut lies. He knows that it is under shell #3, since he was the one who placed it there. From his point of view, it is not the case that the peanut could be/might be under shell #1, and it is not the case that it could be/might be under shell #2. From his point of view, it is certain that the peanut is under shell #3.

Future outcomes. Elizarraraz is ignorant, however, of which shell Smith is about to select. That shell could be, might be shell #1, or it could be /might be shell #2, or it could be/might be shell #3. Were Elizarraraz none of these could be’s/might be’s would be the case. Were Elizarraraz to know that at some future time tsubn that Smith will choose, say, shell #1, then it would be certain that at tsubn Smith will choose that shell. Knowing that p implies the truth of p. But of course it could be true that that Smith will select shell #1 at tsubn and Elizarraraz not know that. In that case, Smith’s selecting that shell remains a possibility from Elizarraraz’ point of view. But what if Elizarraraz does not know because it is not certain — it is not a 100% probability — that Smith will select shell #1 at time tsubn? Suppose it makes sense to say that even from the “view from nowhere,” that of an an entity that knows every true proposition, this entity does not know that Smith will select shell #1 because there is no fact of the matter — there is only a certain probability. Time tsubn comes along, either Smith selects shell #1 or he does not — he selects another shell or no shell at all. Smith just selects the shell, say, even though no previous cause establishes a 100% probability. The event just pops up. nd then Elizarraraz cannot know that Smith will select shell #1 at that time, and Smith

Suppose no peanut was lurking under that shell — say, shell #1. Smith now knows that there was no peanut under shell #1. In at least some sense of the term ‘certain’, he is now certain that shell #1 was not the one hiding the peanut. He has the information that shell #1 was not hiding the peanut. But he knows that (is certain that, has the information that) there is a peanut lurking under one or the other of the remaining shells, #2 and #3. I have, and I think most people will have, the strong intuition that the probability the peanut is under shell # 2 (alternatively shell #3) is 1/2. For the original sample space of 3 has been reduced to 2, and each outcome is, we say, equally likely.

At this point, Smith confronts two possibilities. A possibility is a possible outcome. Possibility #1: the peanut lurks under shell #2 and shell #3 is empty. Possibility #2: the peanut lurks under shell #3 and shell #2 is empty. To talk about ‘a possibility’ here is to say the following: because Smith knows there is a peanut under one of the shells (he just doesn’t know which one), there is a peanut under one of the shells. For if one knows that p, then p is a true proposition (or, better, a state of affairs that obtains [I follow Chisholm in identifying propositions with a proper subset of states of affairs]. From Smith’s point of view, the peanut could be under shell #2 or shell #3; that is to say, he doesn’t know which one. So, at least in cases like this one, [yes, I know, this needs to be more sharply defined] ‘a possibility’ requires a combination of knowledge and ignorance. Remove the ignorance, and the possibility no longer exists.

From Smith’s point of view, it is no longer the case that the peanut could be under shell #1. Its being under shell #1 is no longer a possibility for Smith. And the probability that it is under shell #1 is now 0. Were Elizarraraz to turn over the shell that does hide the peanut (say, shell #3) (and were Smith to see the peanut that had been hiding there, and were nothing at fault in Smith’s visual apparatus), it would no longer be the case that, from Smith’s point of view, the peanut could be under shell #3. It is under shell #3. Its being under shell #3 is no longer a mere possibility, but a certainty. Again, remove the ignorance, and the possibility no longer exists. From Smith’s point of view, the probability that the peanut is under shell #3 is now 1.

When Smith turned over shell #1 and discovered it to be empty, he decreased the size of the sample space from three possibilities (the peanut is under shell #1 and shells #2 and #3 are empty; the peanut is under shell # 2 and shells #1 and #3 are empty; the peanut is under shell #3 and shells #1 and #2 are empty) to just two (the peanut is under shell #2 and shell #3 is empty; the peanut is under shell #3 and shell #2 is empty). A sample space is a set of possibilities; the cardinality or “size” of the space is the number of possibilities it has as members. The metaphor of ‘a space’ is apropos here because a given space, a room, for example, can contain items, just as set “contains” its members. If a sample space contains n possibilities and each possibility is equality likely, then the probability of each event (subset of the sample space) must be expressible as a ratio with n as the denominator. If the size of the sample space is six, for example, the probability of each event must be expressible as 1/6, 2/6, 3/6, 4/6, and 5/6.

When the possibilities involve physical entities, such as a number of shells one of which hides a peanut, it is easy to think of the size of the sample space as equal to the number of those entities. Later, however, I intend to show that the sample space can include possible as well as actual entities.

Now Morgenstern arrives on the scene.

A likelihood is assigned to the sample space: A likelihood is a number that can find | alternatively fail to find an anchoring in the real word — and this to varying degrees. One can anchor a likelihood by repeating an experiment a very large number of times. If one flips a coin ten million times, for example, and the coin comes up heads five million times and tails five million times, one may perhaps be allowed to say that the coin’s turning up heads and its turning up tails are equally likely events. If one flips the coin twenty million times and the coin lands heads 10,000,001 times and lands tails 9,999,999 times, one may perhaps be allowed to say ‘this is close enough for government work — I will say the two events (landing heads and landing tails) are equally likely’. I regard as highly credible the idea that, no matter how often the coin if flipped, the numbers will rarely be completely even. At the time of this writing I have no idea — I lack the mathematical depth — whether the numbers will converge on some 50/50 limit which we could then use to assign a likelihood to the sample space in an objective manner; or whether the numbers will vary randomly, with upper control and lower control limits ala Taiichi Ohno (in which case I will be stuck in a circle trying to define ‘probability’ in terms of concepts that are themselves probabilistic), or whether the numbers themselves just vary randomly (oh my Gosh, there is that concept ‘randomness’ again) without being subject to a statistical analysis like this. I propose that the way to get out of this circle is to treat ‘equally likely’ as a concept that has vague boundaries but is nonetheless not empty.

But maybe we are not entitled to be confident about this intuition. The Monty Hall paradox shows rather clearly that our intuition in these matters cannot always be accepted at face value. Let me briefly describe the Monty Hall paradox.

The name of the paradox comes from a television game show hosted by a certain Monty Hall. The show employed doors hiding cars and goats, but I prefer to stick with shells hiding either a peanut or empty air. The game proceeds as it does with the non-Monty-Hall shell game, but with this difference. After Smith has selected a shell, he does not turn it over to see if it hides the peanut. Instead, Elizarraraz turns over one of the peanuts. The peanut he turns over has to meet two criteria: first, it cannot be hiding a peanut; and second, it cannot be the shell (initially) selected by Smith. Elizarraraz then gives Smith the choice of either sticking with his initial selection, or switching to the remaining shell (that has not yet been turned over).

One can be forgiven for having the strong intuition that neither strategy has any advantage over the other. As one pictures the two remaining shells with the mind’s eye, may seem completely obvious that Smith’s chances of winning the peanut are 50/50 if he sticks with his initial selection, and 50/50 if he switches. The sample space, after all, would seem to comprise just two possibilities, just as does the sample space of the non-Monty Hall game. Possibility #1: the one shell either hides the peanut, in which case the other shell hides just empty air; or (possibility #2) the former shell hides empty air, and the latter shell hides the peanut. This is what could turn up, what could be very shortly in the near future.

But, as it will turn out, this is not the sample space of the Monty Hall shell game. And Smith’s chances of winning the peanut are not 50/50 regardless of his strategy, but 1 in 3 if he opts to stick with his initial selection, and 2 in 3 if he opts to switch. As if that were not (at least initially) counter-intuitive enough, it remains true that Smith’s chances of winning the peanut are 50/50 if he chooses by flipping a coin which of the remaining two shells to select; and his chances of choosing his initial selection |alternatively| choosing the shell that was not his initial selection are also 50/50. How can all of these propositions be true at the same time? How can the ‘2 in 3′ be true at the same time the ’50/50’ is true? And what can we learn about the nature of probability from the co-truth of these propositions?

Taking my cue, first from Judea Pearl, then from Luis Jorge Borges, I will prove the ‘1 in 3’ vs. ‘2 in 3’ probabilities for sticking with the initial choice vs switching. Then, after proving the 50/50 cases, I will show how these are compatible with the 1 in 3 and the 2 in 3.

Computer simulations of Monty-Hall-type games (for example, the one available online here or here) show definitively that Smith’s chances of winning the peanut are 1 in 3 if he sticks with his initial choice and 2 in 3 if he switches. One of the simulations I linked to repeats the game ten million times. Few, I think, would dispute that these simulations show that the chances are 1 in 3 | 2 in 3. But they won’t suffice to give one any intuitive sense why those are the chances. No Aha Erlebnis will be coming from just observing the simulations.

A table listing all of the possibilities, all the possible cases, goes some way, I think, towards giving one this intuitive sense. As shown in the table below (a modification of the table presented by Judea Pearl in his BOOK OF WHY (BOOK OF WHY, p. 191), which in turn is taken from Marilyn vos Savant’s column from the 90’s), there are nine distinct possibilities, nine possible cases. Each of the nine cases is equally likely. One can then start to see why the computer simulations would give Smith a 1/3 chance of selecting the shell with the peanut if he sticks with his initial choice, and a 2/3 chance if he chooses the remaining shell.

The table, however, is not perfect as a device for generating the desired Aha Erlebnis giving one to see that Smith’s chances are only 1 in 3 if he sticks with his initial choice. One may want to see rows 1, 4, and 7 in the table as each comprising two possibilities, not one, rendering problematic the math that gives us the 1/3 and 2/3 probabilities. One would be wrong, of course; nonetheless, it remains true that the table is burdened as an Aha-Erlebnis-generating tool by this complication. Also, the table does not show why the 50/50 chances (initially and perhaps even non-initially) seem so powerfully intuitive.

Listing out all the possibilities in the form of a tree, gives us a picture, another way of showing the 1/3 and 2/3 probabilities without the burden of this complication. We can picture repeated plays of the Monty Hall shell game as a trunk branching off into a number of branches. Doing so will nail down the 1/3 and 2/3 probabilities quite conclusively, though perhaps without generating an Aha Erlebnis, a concrete intuition.

Picturing the game this way will also provide at least a start at an explanation why the conclusion that the chances are not 50/50 seems so paradoxical. The idea of treating the game this way came to me in a flash of insight after reading Jorge Luis Borges’ short story THE GARDEN OF FORKING PATHS. (“You are so smart!” at work, though sometimes I suspect they mean this in a ‘you have a wonderfully intuitive sense for the blindingly obvious’ way), but, of course, essentially the same idea has occurred to other people, as one can see here and at numerous other places on the internet. I would like to think, however, that I have my own twist on the idea. Anyway, onto the chart shown below and an explanation of what it shows.

The Monty Hall Shell Game Considered As Conceptual Sleight Of Hand: In the chart shown below, Elizarraraz (employing a randomizing device) chooses which shell to place the peanut under (tanned orange). In order to make the chart readable, I show just Elizarraraz’ choice of shell #1. The possible choices that ensue from the “space” that would open up if Elizarraraz placed the peanut under this shell are, I claim, canonical. That is to say, they comprise a piece (shell #1) of the larger picture that enable one to draw conclusions about the larger picture (all three shells).

A moment later, Smith comes into the scene and, employing a randomizing device, makes his initial selection of a shell (pink). Elizarraraz then turns over one of the shells, employing, not a randomizer, but his knowledge of which shell Smith has selected and which shells are empty (baby-aspirin orange). Those shells Elizarraraz cannot turn over are crossed out by red lines.

Finally, using a randomizer, Smith decides either to switch shells or stick to his initial choice. The decision to switch is shown (for reasons that will become clear when I get to the ‘forking paths’ metaphor) by the bolded arrow. The winning shell (Smith gets the peanut) is shown by the darker viridian or “sea-glass” green color of the oval symbol picturing the shell. The losing shell is shown by the lighter viridian green, which looks like a light blue.

[Each oval represents a possible outcome (for example, Smith initially selects shell #1). Until we get to the culminating possibilities (represented by the green ovals), each possible outcome opens up (and sometimes closes down) what I will call a ‘possibility trail’, i.e., a “trail” in which one possible outcome follows another. Smith’s initial choice of shell #1, for example, opens up a path in which Elizarraraz turns over shell #2, which in turn forks into two paths, one leading to Smith’s winning the peanut and the other leading to his losing the game; and opens up another path in which Elizarraraz turns over shell #3, which path in turn forks into…; and results in a dead end, in which Elizarraraz is constrained by the rules of the game from turning over shell #1. ]

[Each fork opens up what I shall call a “cone” of possibility paths. Elizarraraz placing the peanut under one of the shells opens up three such cones, not labelled here. Smith’s choosing a shell opens up three cones, which I label A, B, and C. The paths in cone A culminate in four different possible outcomes; the paths in cone B and cone C each culminate in two possible outcomes. ]

[Cones A, B, and C match with rows 1, 2, and 3 respectively in the table shown previously. Each cone/row constitutes a wider sample space whose “places” or “slots” are themselves narrower “sample spaces” whose “places” are still narrower samples spaces defined by the forks and, ultimately, by the possible ending outcomes. These narrower sample spaces would (note the subjunctive mood) succeed one another in time; one such sample space, one set of possibilities would open up for example were Smith to initially select shell #1. There are two final sample spaces in cone A. These sample spaces begin, respectively, at Elizarraraz’ possibly turning over shell #2, or his possibly turning over shell #3, and include their ending “leaf” possibilities: shells #1 or #3; or shells #1 or #2 respectively. Both of these final sample spaces are included as places in the sample space comprising cone A. The sample space that is cone A is defined by the fork that gets generated by Smith’s possibly making the initial selection of shell #1. Cone A in turn, along with cones B and C, are included in the sample space that is generated by Elizarraraz’ possibly placing the peanut under shell 1.]

If Elizarraraz has placed the peanut under shell #1, then of course Smith has only a 1 in three chance of winning if he sticks by his initial choice. For in this case he will win the peanut only if that initial choice was shell #1. But the chances shell #1 was his initial selection are just 1 in 3. So his chances of winning by sticking with his initial choice are also just 1 in 3. It follows that his chances should he switch will be 2 in 3. If this conclusion is not already already intuitive to you, gentle reader, I think it will become more intuitive once I start laying out the forest of forking paths picture.

Suppose that Smith, compulsive gambler that he is, plays the Monty Hall Shell Game ten million times. At the end of each game, he is presented with just two shells. One was initially selected by him, the other not. Now suppose that the shell that was initially chosen is marked as such; ditto the shell that was not initially chosen. If Smith sticks to a strategy of of chosen the shell he did not initially select, he will win 2/3 of the time and lose 1/3 of the time. Conversely, if he sticks to a strategy of sticking to his initial choice, he will lose 2/3 of the time and win 1/3 of the time.

Now suppose the markings ‘initial choice’ and ‘not initial choice’ are removed from the shells — and, because the shells looks so similar, Smith cannot remember which one he had initially selected. No labels ‘shell #1’, ‘shell #2’, ‘shell #3’ have been applied to help guide him. Smith has to flip a coin to decide on which shell to select. I think it is clear from the chart that Smith will win the peanut 1/2 the time by flipping a coin. This 50/50 probability is, I think, what makes the Monty Hall Shell Game so drastically counter-intuitive. One looks at the two shells, each of which could be hiding the peanut, and (correctly) sees a 50/50 chance should they flip a coin.

But notice that in the game, Smith is not asked to flip a coin to decide between the two remaining shells. Instead, he is asked either to stick with his initial choice or to switch. That is the Monty Hall Shell Game, which presents Smith with a 2/3 (alternatively, 1/3) chance of winning. He is not asked to flip a coin to decide between the two remaining shells. That is a different game altogether, one that results in a 50/50 chance of winning. Let me call this other shell game the ‘Monty Hall With-A-Final-Coin-Toss-Added-In-At-The-End-For-Good-Measure Shell Game.’

We base the figure 2/3 | alternatively 1/3 on what WOULD happen were the Monty Hall Shell Game played 10 million times, adopting one or the other of the two available strategies. This provides confirmation. But it does more than that, because it provides a way to define randomness that does not rely on the concept of ‘equal likelihood’ or some other ‘probability function’. It gives us a way to define it in a non-circular fashion. So: ignorance/knowledge in the context of what WOULD happen plus idealization.

If Smith is to be able to play the Monty Hall Shell Game, he needs to know which of the two shells remaining in the penultimate step was his initial selection and which shell was not — the actually or possibly switched-to shell. Smith needs to have this information in order to play the game. The rules require keeping track of what happened in the past — there has to be a trail, a path, so to speak, leading from the past to the present. If Smith loses this trail — say, all shells have the tendency to look alike to him, and no one — Elizarraraz or anyone else — bothers to inform him which is which — then Smith has no available evidence to base his choice on except for flipping a coin. The ‘Monty Hall With-A-Final-Coin-Toss-Added-In-At-The-End-For-Good-Measure Shell Game’ is the only one he can play. Not exactly the same as the original game described above, the Non-Monty-Hall shell game, but now has the same 50/50 chance of winning the peanut.

Information has to leak, so to speak, from the past to the present and be available to Smith in the present. It has to exist, has to be available, and has to be picked up and used by Smith. This means that there is a dependency between events (‘event’ here used synonymously with “experiment”) that happened in the past (which shell Elizarraraz turned over) and the probability of possible events (‘possible event’ here is used synonymously with the standard probability term ‘event’) in the present. The 1/3 alternatively 2/3 probabilities inherent Smith’s sticking with his initial choice alternatively switching shells in the final step of the Monty Hall Shell Game depend upon Elizarraraz’ having turned over a shell in the past. This is completely unlike the standard coin flipping scenario, in which any later coin flipping event is independent of any earlier one because no earlier event affects the probability of any later event.

Even if everyone playing either game has gotten completely confused by the similarity of the shells, the information is, I will assume, still present. It is just much less available — much harder for anyone to pick up — much harder to the point of practical impossibility. And even if the information regarding which shell Elizarraraz turned over is still present in Smith’s mind, Smith is not likely to be playing the game armed with the chart below in his mind. It is a rare person who would be able to do so. Thus Smith is likely to think of the two remaining shells as a situation calling for a coin flip yielding a 50/50 chance. [Point of these paragraphs: one more case of the perspectival character of probability. The probability depends upon the information present — or at least available.]

Dependency trails. The present — what could be now — vs. what could have been, which includes all these possible dependency trails. The sample space as including the trails. “Room” made by the constant shell-game like shifting. What could have been as in a way “present” now ala Borges. Evidence. Turning over the shell changes the probability just as it does in the original shell game. Perspectival character — what the probability is depends upon whether one takes the present ‘could be’ perspective or the past ‘could have been’ perspective. Paradoxical because one tends to take the could be perspective.

Which is which will differ frequently as Smith makes his ten million plays of the game. In the case in which Elizarraraz has placed the peanut under shell #1, the initial choice will sometimes be shell #1 and the switched-to shells either shell #2 or shell #3; sometimes the initial choice will be shell #2 and the switched-to shell will be shell #1; sometimes the initial choice will be shell #3 and, again, the switched-to choice will be shell #1. If we imagine labels getting applied each time to the initial-choice shell and the switched-to shell, those labels will be constantly moving back and forth between the three shells. They will be “orthogonal” to the labels ‘shell #1’, ‘shell #2’, and ‘shell #3’, should those labels also be applied to the shells.

So which game is being played — and what the rules are for each — matters for what the probabilities are. ffff

[Since in both these games the designations ‘shell #1’, ‘shell #2’, ‘shell #3’ drop out of the picture, one may get the feeling that these are similar to the shell game as traditionally played, in which a slick operator switches the peanut between hard-to-distinguish shells by slight of hand. But here, of course, one is not trying to force their eyeballs on three actual shells in an attempt to keep from getting fooled within a single playing of the game. Shell stays the same; peanut surreptitiously moves. Instead, one is dealing with labels which stay the same even as the shells they apply to change with each new playing of the game. [How come 2/3 probability when only 2 shells remaining?]]

Under one description for the shells, the chances of winning the peanut are 50/50. Under another description (shell not initially chosen; shell initially chosen), the chances are, respectively, 2 in 3 and 1 in 3. But these are (at least at any given time) the same shells. What accounts for the difference? The difference, I think, lies in the history of how the shells got there. And in explaining this, Borges short story THE GARDEN OF FORKING PATHS will prove useful.

Enough of the shell games. Let me now apply a completely different picture, one inspired by Borge’s short story THE GARDEN OF FORKING PATHS. This picture will be of a forest containing within it a multitude of forking paths. It will, I propose, make it easier to articulate certain aspects of the paradox I am trying to make sense of.

The chart above was originally drawn as a graphic tree depicting the Monty Hall Shell Game. But now lets draw it as depicting thirteen forking paths in a forest. Smith will be walking the paths fifteen million times (he is an indefatigable hiker).

Here the sample space comprises paths cut into the forest. Just as Smith’s overturning one of the shells in the Non-Monty-Hall Shell game reduced the sample space from 3 to 2 (should the shell prove empty), the ten paths that lead to dead-ends (the clearings marked with a red X) reduce the sample space from 18 paths to just 8. Information in the shell game corresponds to dead ends in the forking paths. Certainty one will not go any further in the forest case. In the shell game the shells still in play are met by Smith’s ignorance; here where the remaining forks lead to is what meets Smith’s ignorance.

These paths are in a parallel universe which mirrors our universe, in which Smith is playing the Monty Hall Shell Game. The ovals in the chart above, which used to represent choices (Smith’s or Elizarraraz’), now represent clearings in the forest. The arrows, which used to represent ‘go on to the next step’ now represent paths leading from one clearing to the other. Which clearing Smith ends up in, and which path he takes, is determined by the choices he and Elizarraraz take in the shell game in our universe. So the forking paths picture will be a bit science-fiction-y; nonetheless, my hope is that it will result in a gain in intuitive clarity (certain points will be easier to make) which will make up for its contrived character. Think of it as like the Mercator projection which serves as the standard in maps of the world. In this projection, certain features are captured at the expense of distortions in the areas of the land and water masses mapped.

Each oval represents a clearing in the forest. Each arrow represents a path leading from one clearing to the next. There are three different starting clearings which map to Elizarraraz hiding the peanut under shell #1 alternatively shell #2 alternatively shell #3; above, only the clearing corresponding to his hiding the peanut under shell #1 is shown, since I take this to be canonical. Three paths fork of, or, more precisely, trident off from the starting clearing. If Smith takes the path to the left, These of course map onto Smith’s initially selecting one of the three shells. If Smith takes the path on the left, hink of the arrows in the chart above as depicting Let me first describe the forking-path interpretation in just enough detail to let me state the two points I want to make. Then I will lay out the interpretation in more adequate detail. We will be having Smith walk the paths…maybe ten million times would be cruel and unusual punishment, but enough times that a frequency becomes apparent. The paths end in a forest clearing which contains something stupendous which I will leave to the reader’s imagination. Maybe it is a glorious vision of a topless Channing Tatum clearing brush. Maybe it is seeing Edward in full shining resplendent crystalline display. Maybe it is seeing a gorgeously feral Jacob — another graceful son of Pan! Or maybe it is just an extra-special peanut that outshines any other peanut. Whatever.

When Smith, walking down the path for the x number of times, comes to the final fork in the path, he can do one of two things. First, he can select the path by flipping a coin. Or, second, he can adhere to a right-hand/left-hand strategy: always choose the path on the right (alternatively the left).

I think it is plan from the graph that if he chooses by flipping a coin, he will arrive at the clearing with the special prize (a view of Channing Tatum, or the extra-special peanut) one half the time. If he adheres to the right-hand/left-hand strategy, he will arrive at the clearing with the special prize two thirds of the time if he always takes the path on the right, or just one-third of the time if he always takes the path on the left. Always taking the path on the right corresponds, in the Monty Hall Shell Game, to Smith’s switching, and always taking the path on the left corresponds to his sticking to his initial choice.

The different strategies lead to different probabilities. In a short while, I will relate these differing probabilities to those of the Non-Monty-Hall Shell Game played by Smith and Morgenstern. I intend to show that just as knowledge (or lack of knowledge) accounts for the difference in probabilities in the Smith and Morgenstern case, the related concept of evidence (or lack thereof) accounts for the difference in probabilities in the forking path case (and in the Monty Hall Shell Game).

But given the difference in the probabilities established by the different strategies, one can explain why the Monty Hall Shell Game seems so paradoxical to about everyone, at least at first. For when one imaginatively confronts the choice faced by Smith (stick to the initial choice of shells or switch), one surreptitiously thinks of the choice in terms of a ‘let’s flip a coin’ scenario. This scenario is, after all, easy to picture imaginatively. The alternative is to have the choice guided by something like the graph above. This graph is, naturally, not at all easy to picture.

Let me now turn to a fuller explanation of the above chart, interpreted either as a tree (the Monty Hall Shell Game) or as a set of forking paths.

I think I have fulfilled my promise to use the forking paths picture to nail down even more firmly the 1/3/2/3 stick with the initial choice/switch probabilities. Now let me show how this picture helps explain why this result seems, at least initially, so counter-intuitive.

Now after Smith has traveled down one or another of the paths in one or another of the three possibility cones, he is presented with two shells (in cone C, for example, either shell #1 or shell #3). The peanut could be under either of those shells. At the time of this writing (September 8, 2019 — I note the date because particular pieces of my autobiography have in the past turned out, somewhat surprisingly, to be philosophically fruitful), it seems absolutely clear to me from looking at the chart that Smith’s chances of winning the peanut are 50/50. Later I may try to nail this intuition down more firmly by coding my own simulation of the Monty Hall shell game.

But note that what I am ascribing a 50/50 chance to is the peanut’s being under (for example) shell #1 or shell #3. I am not ascribing a 50/50 chance to the peanut’s being under the Smith’s initial choice of shells or his switched choice. The descriptions ‘initial choice shell’ or ‘switched choice shell’ have no meaning in this narrow sample space delimited by what could be, i.e., by the present and the potentialities of the (presumably) near future.

To get these descriptions, we have to go deeper than what could be and move into what could have been. We have to move into the past. Smith could have chosen shell #2, but he has chosen shell #3, which in turn made shell #2 the only possible choice of shells for Elizarraraz to turn over, which in turn left Smith with a final choice of shells #1 and #3. Were Smith to go back in time multiple times to his initial choice of shells but with his randomizer determining different choices — or, less science-fictionally, were he to repeat the Monty Hall shell game a large enough number of times, he would end up winning the peanut 1/3 of the time by sticking to his initial choice, and 2/3 of the time by switching.

The probabilities are determined by the sample space. When the descriptions ‘initial selection shell’ and ‘switched choice shell’ make sense, the sample space embraces three possibilities, the three possibility cones, one of which culminates in Smith’s winning the peanut should he stick to his initial choice, and two of which culminate in his winning the peanut should he switch choices. That’s the sample space that counts when those descriptions are meaningful. When those descriptions don’t make sense because we are restricted to what could be, that is, to the present because the sample space is restricted to the present, to what is facing Smith now, and to a narrow snippet of the near future, the sample space comprises only two possibilities: the peanut is under this shell or under that other one.

Were Smith told, when confronted with the two shells, to choose one of two strategies: switch or stick with the initial choice, neither strategy would make any sense at all unless he had access to enough of the past to let him identify which shell was his initial choice; or unless someone who was keeping track told him. And even then his adopting one strategy or the other would be incompletely rational unless he had plotted out all the cones with the possible paths that could have been, including both the paths that led to the present situation and the paths that ended up as dead ends. He would be better off not worrying about which shell was his initial choice and just flipping a coin.

What the sample space is, and therefore what the probabilities are, depends upon which game is being played — flip a coin, or stick-with-the-initial-choice-or-switch. Different sample space, different game; different game, different sample space. Although Pearl’s point in the following may be a bit different from what I have just described, his actual words still fit with my point. (Maybe there is another Borges story about something similar.) Pearl notes:

The key element in resolving this paradox is that we need to take into account not only the data … but also … the rules of the game. They tell us something about the data that could have been but has not been observed.

BOOK OF WHY, p. 192

When confronted with just the two remaining shells in the present, it is easy to forget that these are two different games.

Thinking about the the different cones containing different possible paths requires a certain amount of time, patience, and wetware power and bandwidth. Considering the possibilities when confronted (perceptually or imaginatively) with just two shells requires much less time, patience, and wetware power and bandwidth. This fact, plus the fact that it is perhaps not so obvious when staring at the shells that the descriptions ‘initial choice’ and ‘switching choice’ cannot be applied to the shells if one’s time horizon (and the resulting sample space) are too narrow are, I submit, at least one reason the actual probabilities of the Monty Hall shell game seem at first so drastically counter-intuitive.

As Pearl notes, there are probably 10,000 different reasons, one for each reader, why the actual probabilities of Monty Hall game seems so counter-intuitive. To return for a moment back to cars, goats, and doors:

Even today, many people seeing the puzzle for the first time find the result hard to believe. Why? What intuitive nerve is jangled? There are probably 10,000 different reasons, one for each reader, but I think the most compelling argument is this: vos Savant’s solution seems to force us to believe in mental telepathy. If I should switch no matter what door I originally chose, then it means that the producers somehow read my mind. How else could they position the car so that it is more likely to be behind the door I did not choose?

BOOK OF WHY, pp. 191-192.

The specter of mental telepathy is doubtlessly one reason the result seems so counter-intuitive; one’s tendency, resulting from the limitations on human mental power, to be perceptually/imaginatively restricted to what could be as opposed to what could have been is another. I won’t try to judge here whether one is more compelling than the other, especially since I have not yet wrapped my head around Pearl’s account of causality.

Now back (finally!) to the point of bringing up the Monty Hall puzzle in the first place. Regarding the non-Monty-Hall shell game, I asked what makes us so sure the probability is now 1/2 that the peanut is under one of the remaining shells after Smith has turned over one of the shells which turned out to be empty. Why should we trust our intuition in this case, when our intuition regarding the Monty-Hall case were initially so far off? Well, let’s provide a table of the possibilities.

Shell #1Shell #2Shell #3Shell Uncovered by SmithFormer Possibility Converted to Actuality
peanut empty empty 1 yes
empty peanut empty 1 no
empty empty peanut 1 no
peanut empty empty 2 no
empty peanut empty 2 yes
empty empty peanut 2 no
peanut empty empty 3 no
empty peanut empty 3 no
empty empty peanut 3 yes

There are two independent events a work here: Elizarraraz randomly placing the peanut under one of the three shells, and Smith’s randomly turning over one of the shells. Neither event affects the probability of the other. If we then eliminate the rows in which Smith happened to turn over the shell containing the peanut (as marked by ‘yes’ in the column ‘Possibility (that the shell hides the peanut) turned into actuality (yes, the shell did hide the peanut), we get 6 rows. Each of the three pairs of rows describes a probability: if Smith finds that shell #1 was hiding nothing except empty air, then row 2 (the peanut is under shell #2) and row 3 (the peanut is under shell #3) describe the situation. Since both rows describe equally likely possibilities, the chances are 50/50 that shell #2 hides the peanut, and the chances are 50/50 that shell #3 hides the peanut.

Our initial intuition is therefore vindicated. Smith’s turning over one shell and finding it empty changes the probability the peanut is lurking in any one of the remaining shells from 1 in 3 to 1 in 2. (It sure is nice to have a wonderfully intuitive sense for the obvious.) The probabilities changed because the sample space changed, just as changing the Monty-Hall game from ‘switch or stick with the initial choice’ to ‘flip a coin’ changed the probability of winning the peanut from 2/3 (if Smith switches) to 50/50. The probabilities in the Monty Hall case changed because the sample space relevant to the game Smith was playing changed. Having the ability to describe one of the remaining shells as ‘the initial choice’ expanded the sample space needed to support this description from two possibilities regarding each shell’s hiding or not hiding a peanut to three possibility cones each containing one or more possible paths to the current situation.

Now Morgenstern (German for ‘morning star) enters the scene, after Smith has put back the shell he turned over.  (Say, this is shell #1) She does not know that shell #1 turned up empty. The peanut is still under one of the remaining shells. Elizarraraz points to shell #2 and asks both Smith and Morgenstern what are the chances the peanut is under that shell. For Smith, surely, the answer is 1 in 2. For Morgenstern, the answer has to be 1 in 3. For Elizarraraz, who knows where he put the peanut, the answer has to be either 0 or 1. Were Elizarraraz to point to shell #1, the answer for both him and Smith would have to be 0. What the probabilities are differs from the perspectives of each of the three because the sample space differs for each given what each knows.

From Elizarraraz’s perspective, there is no hiddenness, no ignorance given how things stand with regard to the peanut under shell situation, because his knowledge is complete regarding that situation. Obtaining within that perspective is certainty: either a probability of 1 or of 0. I will go out on a limb and say that within that perspective there is no sample space at all.

Uncertainty, a probability greater than 0 but less than 1, can exist only given a particular ratio of local ignorance and local knowledge. If one’s local knowledge of the peanut under shell affair is 0 (one does not even know if there is a peanut under one of the shells) and even Elizarraraz has forgotten if he has placed a peanut under one of them or not, one can appeal to a (possibly hypothetical) infinite (or at least extremely large) Mind that does know, in which case the probability is either 0 or 1. Or one can appeal to a brute, currently unknown fact of the matter, in which case, again, the probability that the peanut is under any given shell is either 0 or 1.

But if there is to be a probability greater than 0 or less than 1 within anyone’s perspective — including the Infinite (surely impossible for that one) or at least Extremely Large Mind’s — there has to be some ignorance, some hiddenness as well as some knowledge. For an omniscient God, everything has either a probability of 1 or 0. Ignorance/knowledge is a necessary condition for such probability in between 0 and 1.

It is also a sufficient condition for there being, within a particular perspective, for there being such a probability. All that Morgenstern needs to know is that there is a peanut under one of the shells, and all she needs to be ignorant of is which one, for there to be, within her perspective, of a probability of 1 in 3 that the peanut is under this shell, or that one, or the one remaining one. The probability is 1 in 3 within this perspective because Morgenstern’s ignorance/knowledge determines the sample space.

Knowledge/ignorance suffices for the existence of a probability between 0 and 1. But other factors help determine what exactly that probability is. In the non-Marty-Hall shell game, we need only to take into account the increase in Smith’s knowledge in determining the size of the sample space when he turns over one of the shells and discovers it to be empty. The probability the peanut is under one of the shells increases from 1 in 3 to 1 in 2 because the two events — the placement of the peanut under one of the shells and Smith’s turning over one of the shells — are both random and independent.

But in the Marty Hall shell game, Elizarraraz’s turning over one of the shells doubles the probability that switching will win the prize from 1 in 3 to 2 in 3. It therefore constitutes evidence that the peanut is likely to be under the shell that wasn’t Smith’s initial choice, whether Smith is in a position to utilize this evidence for not. Since, prior to the final step in the Monty-Hall shell game, the only difference between it and the non-Marty-Hall shell game is that in the former Elizarraraz’s turning over one of the shells is, because of his knowledge, not random and is independent of neither his placement of the peanut under one of the shells nor of Smith’s initial selection of one of those shells, it follows that this lack of independence is another factor in addition to Smith’s knowledge/ignorance helping to determine the specific probability of Smith’s finding a peanut if he switches (sticks with the initial choice). By itself, all his knowledge/ignorance does by itself is guarantee a probability of at least 1 in 2 should he switch (stick with the original choice) ; given the additional factor of a lack of independence in the event of choosing which shell to turn over, that probability increases to 2 in 3 (decreases to 1 in 3) should he switch (stick with his initial choice).

At the time of this writing, however, I am unable to say anything more succinct and more sophisticated regarding why this should be so other than ‘look at the chart shown above; given the all the ovals crossed out because Elizarraraz’s choice of shells to turn over was neither random nor independent of the other events, this is how all the possibilities panned out — all three of the possibility cones, and all of the possible trails within those cones. Stay tuned.

[Present circumstances. A sample space is a set of possible outcomes of a given activity governed by a set of definite rules, or at least limited by certain definite conditions. These rules or conditions determine what is eligible to count as a possible outcome. Flipping a coin has two possible outcomes, heads or tails. The coin’s landing on its edge is not a possible outcome, at least not if the normal rules that apply to the practice of flipping a coin are in force. It might be a possible outcome in a different game. In the normal practice of flipping a coin, the sample space is the set with two members: coin lands heads or it lands tails. In a non-normal practice, the sample space might have three members: The same holds mutatis mutandis for throwing a die. In the normal practice, the sample space comprises six members. But should anyone be skilled enough to make the die land on one of its edges invent a new, non-normal practice, the sample space would comprise 12 members. Drawing a standard recognized card from a pack of cards normally counts as an eligible outcome; drawing a stray scrap of paper normally does not. Uncovering the peanut in the normal shell game counts as a possible outcome; uncovering a grain of sand does not — not even if this were a truly extraordinary grain of sand. ]

Different games (say, not realizing the point of the normal shell game) different sample space because what is eligible to count as a possible outcome differs. Assume for example — except for a few Twilight Zone moments — that the peanut stays under whichever shell it is under and is not going to behave like the electron which, for all one knows, might be on the nose of the Mona Lisa. ffff

[At least for now, I will leave the concept ‘randomness’ as an unanalyzed primitive, explicated, not by a real, concrete example, but by a (vaguely described) ideal one. A fair 6-sided die would be suitably random if, after a very large number of throws, the average ratio of the times each number came up, divided by 6, remained sufficiently close to 1/6. And yes, I will leave ‘sufficiently’ undefined. ]

[Information/absence of information determines the sample space, along with, obviously, what the information is about. The probability function is derived from an idealization of a large number of experiments. An experiment occurs when an outcome in time and space is obtained.]

Some of the things I was looking at while writing this drivel…er…I mean, this Classic of Western Civilization.

2) A standard tutorial on probability

3) Judea Perl

[Present circumstances. A sample space is a set of possible outcomes of a given activity governed by a set of definite rules, or at least limited by certain definite conditions. These rules or conditions determine what is eligible to count as a possible outcome. Flipping a coin has two possible outcomes, heads or tails. The coin’s landing on its edge is not a possible outcome, at least not if the normal rules that apply to the practice of flipping a coin are in force. It might be a possible outcome in a different game. In the normal practice of flipping a coin, the sample space is the set with two members: coin lands heads or it lands tails. In a non-normal practice, the sample space might have three members: The same holds mutatis mutandis for throwing a die. In the normal practice, the sample space comprises six members. But should anyone be skilled enough to make the die land on one of its edges invent a new, non-normal practice, the sample space would comprise 12 members. Drawing a standard recognized card from a pack of cards normally counts as an eligible outcome; drawing a stray scrap of paper normally does not. Uncovering the peanut in the normal shell game counts as a possible outcome; uncovering a grain of sand does not — not even if this were a truly extraordinary grain of sand. ]

Today’s homage to Plato’s SYMPOSIUM is Channing Tatum. Again. Who would want anything more?

## Apple Math, Comprising Some Basic (Doubtlessly Ninth-Grade Level) Probability Theory

Nota Bene:  This little bit of math is the keystone in my attempt here (still in draft status)  to provide a sharp, clear articulation of the concept of relevance as that concept pertains to Relevant Logic.  Here I invited members of the online Physics Forum to point out any mistakes in the math should I have made any.  Since no one there pointed out any such mistakes, I will assume that the math is correct.  Naturally, should it turn out that I did make mistakes in the math, I will be royally pissed.  🙂

This post belongs to the ‘I invite anyone and everyone to tear this to pieces, should they uncover any missteps’ category.

The subject here isn’t roses (this is an obscure allusion to a movie I saw in my childhood), but wormy and non-wormy red and yellow apples.

In discussing the subject of apples, I will be using the following terms: ‘set’ (which I will leave as an undefined primitive); ‘sample space’ (which term is I think self-explanatory); ‘event’ (which I will be using in an extremely narrow and a bit counter-intuitive technical sense, following the standard nomenclature of probability theory); ‘experiment’ (ditto); ‘state of affairs’ (which I will be leaving as a primitive); and ‘proposition’ (which I will define in terms of states of affairs).

Wormy Red Apple Image courtesy of foodclipart.com

First Situation:  All Of The Red Apples Are Wormy; Only Some Of The Yellow Apples Are:  Let’s start with the following situation (henceforth ‘situation 1’):  There is an orchard in Southwest Iowa, just across the border from Nebraska. In the orchard there is a pile of apples comprising 16 apples.  Eight of the apples are red.  All of the red apples are wormy.  Eight of the apples are yellow.  Of these yellow apples, four are wormy.

Let’s suppose that the DBA in the sky has assigned an identifying number (doubtlessly using the Apple Sequence Database Object in the sky) to each apple. This lets us write the set of apples in the pile — the Sample Space Ω — as follows:

The Sample Space Ω =

Ω = { a1rw, a2rw, a3rw, a4rw, a5rw, a6rw, a7rw, a8rw, a9yw, a10yw, a11yw, a12yw, a13yw, a14yw, a15yw, a16yw }

where a1…an indicate the numbered apples, and the superscripts r, y, w, and w indicate a red apple, a yellow apple, a wormy apple, and a non-wormy apple respectively.

An ‘event’ is a (not necessarily proper) subset of this set. It represents the set of possible outcomes should one draw an apple from the pile. This particular red apple is drawn; this other particular red apple is drawn; this particular yellow apple is drawn, and so on. Contrary to the ordinary sense of ‘event’, an ‘event’ here is not something concrete, happening in space and time, but abstract — a set.

Eyes shut, someone has randomly drawn an apple from the pile. They have not yet observed its color. Why their having not yet/having observed the apple matters will become apparent later [promissory note]. Following the standard nomenclature, I will call actually drawing an apple — a concrete outcome that has come forth in space and time — an ‘experiment’.

Now I show that….

E is the event ‘a red apple gets drawn from the pile’, which =

E = { a1rw, a2rw, a3rw, a4rw, a5rw, a6rw, a7rw, a8rw }

F is the event ‘a wormy apple gets drawn from the pile’, which =

F = { a1rw, a2rw, a3rw, a4rw, a5rw, a6rw, a7rw, a8rw,a9yw, a10yw, a11yw, a12yw}

And of course the intersection of E and F, E ∩ F, the set of apples that are both red and wormy =

{ a1rw, a2rw, a3rw, a4rw, a5rw, a6rw, a7rw, a8rw}

I will be assuming that each apple in Ω has an equal probability of being drawn.

The conditional probability that the apple drawn from the pile is wormy given that it is red is 1, as you can see from the following steps:

P( F | E ) = P( E  F ) / P(E)

P( E  F ) = |E  F| / |Ω| = 8/16 = 1/2

P(E) = |E| / |Ω| = 8/16 = 1/2

So:

P( E  F ) / P(E) = 1/2 / 1/2 = 1

So:

P( F | E ) = 1

The conditional probability that an apple drawn from this pile is wormy given that it is red is 1.

Now P(F) = 12/16 = 3/4.  Since P(E) = 1/2, P(E) * P(F) = 1/2 * 3/4 = 3/8.  So in this case P(E  F) != P(E) * P(F),  since 1/2 != 3/8.  But two distinct events are independent of one another if and only if

P(E  F) = P(E) * P(F)

So in this case E and F are not independent events.   The probability that the apple is wormy given that it is red increases to 1 from the 3/4 probability given just the draw from the pile, before observing whether the apple drawn is red or yellow.  (Conversely, the probability that the apple is red given that it is wormy increases to 2/3 from 1/2 given just the draw from the pile.)

When the probability of an event is 1, that event is certain, as opposed to ‘just likely’. The concept of certainty is, of course, intimately bound up with the concept of knowledge, an entanglement I hope to examine shortly. But whatever the relation is, the event of this apple’s turning out to be red moves the event of its being wormy from a mere likelihood to a certainty. And whatever the relation of certainty to knowledge is, this certainty surely provides a foundation for knowing that this apple is wormy. In this limited situation (“situation 1”), the apple’s turning out to be red is potentially telling — namely, that it is wormy. It increases our (potential) knowledge.

When this apple drawn at time t0 (the experiment that takes place at that time) turns out to be red , the state of affairs ‘this apple is red’ obtains at t0. I will label this state of affairs ‘p’. Similarly, I will call q the state of affairs that obtains at t0 when this apple is wormy. In situation 1, the fact that the probability of F given E is 1 means there is no way that p can obtain at t0 and q fail to obtain at t0. For the moment, at least, I will refrain from unpacking ‘cannot fail to obtain’, except to link this notion to the probability of an event being 1.

I like to identify propositions with states of affairs that obtain at a particular time. So p is the proposition that the apple is red, and q is the proposition that the apple is wormy. States of affairs obtain or fail to obtain; propositions are true or false. So I am now moving from talking about states of affairs obtaining (failing to obtain) to propositions being true or false. If, gentle reader, you would rather not identify propositions with states of affairs obtaining at some time, just add whatever verbiage is necessary to identify a proposition that corresponds to the state of affairs just mentioned.

In situation 1, whenever p is true q cannot fail to be true. This means that the proposition If p Then q is true, for it satisfies the truth table in Classical Logic for If Then propositions. In situation 1, If p Then q remains true even when p is false (the apple is yellow) and q is false (the apple is not wormy); when p is false and q is true (the apple is wormy); and of course the proposition is true when p is true and q is true. The only time the proposition is false is when p is true and q is false.

What is more, in situation 1, p is relevant to q. For p maps to the event E given which the probability of F, to which q maps, [talk some more about this mapping business] increases from 3/4 to 1, i.e., from mere likelihood to certainty. p inherits this ‘increasing q to certainty’ property. That one proposition/state of affairs (that the apple is red) p increases the probability of another proposition/state of affairs (that the apple is wormy) q surely renders p relevant to q. It is a sufficient condition for p’s relevance to q. It therefore renders If p Then q true in both Relevant Logic (which demands that the antecedent be relevant to the consequent) and in Classical Logic.

I submit, then, ‘increasing the probability of q to 1’ as a candidate for the relevance-making relation that p bears to q when p is relevant to q. This relation is a candidate, that is, for those If Then propositions that can be treated in a probabilistic manner. It is not a candidate for the relevance of the antecedent to the consequent in the proposition ‘If the length of side A of this right triangle is 2 and the length of side B is 3 (neither A nor B being identical with the triangle’s hypotenuse), then 13 is the length of the hypotenuse.’ For even though the antecedent here excludes any other possibility other than the hypotenuse having a length of 13 (just as the apple’s turning out to be red excludes in situation 1 the possibility of it’s not being wormy), there is nothing in the mathematical proposition that invites treatment in terms of chance and draws.

That the probability increases to 1 renders the proposition ‘If E then F’ true — at least in this circumscribed situation (this particular pile in this particular orchard for this particular stretch of time, which stretch of time will come to an end should a non-wormy red apple happen to roll into the pile). Within this situation, the apple will always be wormy should it turn out to be red. The ‘all’ in ‘all the red apples are wormy’ guarantees the truth of the conclusion as long as this ‘all’ lasts. Taking the increase in probability combined with the guarantee (the increase is to 1) together suffice to make ‘If this apple is red, it is wormy’ a true proposition in relevant logic, since the conclusion meets the truth-table standard of classical logic and meets the additional condition demanded by relevant logic, namely, that the antecedent be relevant to the conclusion. F will never fail to be true should E turn out to be true, a state of affairs that is a sufficient condition for the proposition ‘If E then F’ to be true.

I submit, then, that at least in those states of affairs that allow for a probabilistic treatment, the relevance of p to q consists in p’s increasing the probability of q to 1. [tie p and q to E and F.] Naturally, not all p’s and q’s will allow for a probabilistic treatment. Mathematical propositions don’t allow for such a treatment, for example. We should perhaps not assume that what makes p relevant to q is the same in all cases of IF THEN propositions is just one type of relation. But at least in the case of those propositions that do allow for a probabilistic treatment, we can see that increasing the probability of q to 1 given p is a strong candidate for the relevance-making relation, given that this increase suffices to render p relevant to q.

At least in those cases that do admit of a probabilistic treatment, increasing the probability of q to 1 is also a necessary condition for p’s being relevant to q.

Second Situation:  All Of The Red Apples Are Wormy, As Are All Of The Yellow Apples

When all the apples are wormy, the color, either red or yellow, of the apple becomes independent of its worminess. Thus the aforementioned sufficient condition for relevance is absent. Maybe some other relation could render p relevant to q here, but I am at a loss for what it could be. So until someone can point out such a relation, I will therefore go out on a limb and say that dependence is a necessary, as well as a sufficient, condition for the relevance of p to q in cases similar to the wormy apple case. This provides support — though clearly not support achieving the level of certainty — for the original intuition. vvggggg

A paradox or at least weirdness comes to the fore. I deal with this by examining the nature of probability. Assuming a deterministic universe (at least on the post-quantum level) probability is perspectival — on either a global or a local level. The example can seem paradoxical because one is assuming the position of someone who knows everything about the apples. A local orchard god, so to speak. But that is just one perspective. Thus the original intuition is vindicated.

If just a credence, there are no relevant IF THEN propositions from a God’s-eye’ point of view. (Actually, no perspective at all). Possible worlds (complete) vs. situations (partial).

Today’s homage to Plato’s SYMPOSIUM is this image of a young boxer appearing on the cover of a computer book.

I have to admit that this is the only computer book I have ever bought just for its cover.

How can anyone get anything done, much less study computer science and ninth-grade math, with beauty like this walking the earth?

Update 11/12/2018:  Made one revision for the sake of clarity.

## Shells And Peanuts Again (And Again…And Again…In A Never-Ending GROUNDHOG DAY)

So one more time — but this time with feeling:  following Relevant Logic, we can avoid Classical Logic’s paradoxes of Material Implication, according to which the following statements are true…

1) If Cliff lives in Houston, Texas, then the earth has just one moon

2) If Cliff lives in Orange County, California, then Paris, Texas is the capital of France

…by insisting that the antecedent p be relevant to the consequent q.  The question of course now is:  what is the relation that makes p relevant to q?  In my previous post, one can, if they are sufficiently drunk, just barely make out the answer:  ‘whatever condition c along with (in the case of subjective probability) knowledge k makes the conditional probability of q equal to 1 given p is what makes p relevant to q.   Sometimes this ‘whatever’ is identical with an INFORMATION THAT relation (p is information that q); sometimes it is not.

( When the relation is identical with the INFORMATION THAT relation, c is the channel of information that allows p to be information that q. When the relation is not identical with the INFORMATION THAT relation, c consists in background conditions, especially causal laws, which, just as in the channel-of-information case, make the conditional probability of q given p 1. My current claim is that even when the relation is not identical with an INFORMATION THAT relation, it has a structure in common with the INFORMATION THAT relation.)

What I propose to do now in the next several posts is go through the> various examples I’ve mentioned previously (shell games, children with measles, wormy red apples, the ringing of defective doorbells, and so on) and a) work out when, in the example, the IF-THEN relation is identical with an INFORMATION-THAT relation and when it is not, and b) see what strange conclusions arise from this account of the relevance-making relation.  Maybe some of these will be so awful that one would prefer Classical Logic’s paradoxes of Material Implication.

In this post I propose to work through Dretske’s famous shell game example.  In that example, one will remember, a peanut is hidden under one of four shells.  I know from whatever reliable means that there is a peanut under 1 of the shells.  This knowledge reduces the probability that (a | the ) peanut is under shell #4 from 1 in whatever billions to just 1 in 4. Maybe my waffling here between ‘a’ and ‘the’ opens up a can of worms; I am unsure. I turn over shell #1.  There is no peanut under that shell.  The conditional probability that the peanut is under any given one of the remaining shells is now 1 in 3.  I turn over shell #2.  Empty.  The conditional probability that the peanut is under any given one of the remaining shells is now 1 in 2.  I say:

If shell #3 is empty, Then the peanut is located under shell #4

And what I say is surely true!  True, true, twue!!!!!  For if shell #3 turns out to be empty, then the conditional probability that the peanut is under shell #4 is 1.  The condition c that makes this conditional probability 1 given p is the characteristic that objects have — at least those objects large enough to be immune to whatever quantum weirdness — of persisting in one place unless molested.  The electron (at least according my remembered ((and almost certainly garbled in my memory)) pronouncement of a chemistry TA I had as an undergraduate) one finds orbiting this or that particular atom could have been on the nose of the Mona Lisa before getting observed, and might be there again a moment later.  But the peanut is not going to jump around like that, leaping to shell #1 one moment while unobserved, and onto the nose of the Mona Lisa the next moment.  It is going to stay placidly and inertially where it is — under shell #4 — while one turns over shell #3 and observes it to be empty.  Given this background fact about objects the size of peanuts, shell #3’s proving to be empty rules out the possibility that the peanut is not under shell #4.

Here the relevance-making factor — what makes the IF-THEN statement I uttered true — is also that factor that would make shell #3’s turning out to be empty INFORMATION THAT the peanut is located under shell #4.

But let’s turn back the clock.  I am now back at the point at which I am turning over shell #1.  Empty.  If I now jumped the gun and said (as if this were the movie GROUNDHOG DAY ((which I have not seen, by the way)), in which one atrocious day gets repeated again and again so that…”The phrase “Groundhog Day” has entered common use as a reference to an unpleasant situation that continually repeats, or seems to.”):

If shell #3 is empty, Then the peanut is located under shell #4

what I say would surely be false. Or at least it must be false if what I said in my first paragraph is true.  For were I to turn over shell #3 and discover it to be empty, the conditional probability that the peanut is located under shell #4 would not be 1, but 1/2.  So the same IF-THEN statement would be true at one time, and false at another.  And it would be true relative to my knowledge k at one time (I know that shells #1 and #2 are empty), and false relative to my lack of that same knowledge at a different time.

Not coincidentally, the (possible) emptiness of shell #3 being information that the peanut is located under shell #4 is something that is true at some times and false at other times, and is relative to one’s knowledge (or lack thereof) in exactly the same way.  In this particular case, what makes the If p Then q statement true is identical with what makes p information that q.

Now turn back the clock yet one more time (I warned you that this is another iteration of GROUNDHOG DAY).  This time I already know from a reliable source of information, even before I have turned over any shells, that the peanut is located under shell #4.  I turn over shell’s #1 and #2 as before.  Both are empty, as before.

But now, shell #3’s proving to be empty upon turning it over would no longer be INFORMATION THAT the peanut is located under shell #4.  This is so for at least two reasons.  First, according to Information Theory, “old information” is an oxymoron.  It is not information at all.  Shell #3’s turning out to be empty is not going to tell me, inform me, show me, that the peanut is under shell #4 because I already have this information.

Second, to generate information is to effect a reduction in possibilities.  In Dretske’s example of an employee selected by a succession of coin flippings to perform an unpleasant task, the eventual selection of Herman out of 8 possible choices reduced the number of possibilities from 8 to 1.  The selection of Herman generates INFORMATION THAT Herman was selected because of this reduction in possibilities.  But in my situation, already knowing that the peanut is located under shell #4, the number of possibilities regarding where the peanut is located is already just 1.  Turning over shell #3 to prove that it is empty does not reduce the number of possibilities from 2 to 1 — that number was 1 in the first place.  So in my situation shell #3’s proving to be empty does not generate, is not information that, the peanut is located under shell #4.

That the number of possibilities is in my situation just 1, not 2 means of course that the conditional probability that the peanut is located under shell #4 is not 1/2, but 1.  This means that shell #3’s proving to be empty does not make the conditional probability that the peanut is located under shell #4 equal to 1.  For that conditional probability was already equal to 1.  We are supposing that I already know that the peanut is located under shell #4, but I would not know this if the conditional probability were not already 1.  The very strange conditions that would have to obtain to make the conditional probability say, 1 in 2 would rule out this knowledge.  The peanut would have to exist under both shell #3 and shell #4 at the same time while unobserved, then “collapse” to a single location under one of the shells upon turning over the other shell and observing its empty condition.  So to say that I already know the location of the shell is to say that the conditional probability the peanut is at that location equals 1.

Now in the first paragraph of this screed I said (maybe ‘pontificated’ is the better word):

…whatever condition c along with (in the case of subjective probability) knowledge k makes the conditional probability of q equal to 1 given p is what makes p relevant to q.

Here my knowledge k (the peanut is located under shell #4) presupposes certain conditions c (the peanut does not exist in a kind of locational smear when unobserved, only to collapse to a single location when an observation is made).  Here p (shell #3 proves to be empty) along with k and the presupposed c definitely does not make the conditional probability of q equal to 1.  This conditional probability was, given k and its presupposed c, already 1.  So in my situation, p is not relevant to q.

So were I, in my situation of already knowing that the peanut is located under shell #4, to  utter GROUNDHOG-DAY-wise:

If shell #3 is empty, Then the peanut is located under shell #4

My statement would be false for exactly the same reason that the following is false:

If Cliff lives in Houston, Texas, then the earth has just one moon

In both cases, the antecedent is irrelevant to the consequent by failing to make the conditional probability of the consequent 1, rendering the corresponding IF-THEN statement false.  The antecedent “If shell #3 is empty” is in my situation irrelevant to the consequent “the peanut is located under shell #4” in exactly the same way that “Cliff lives in Houston” is irrelevant to “the earth has just one moon.” (In exactly the same way?  Yes, at least according to the perhaps narrow definition of relevance I postulated above.  But does this narrowness weaken my claim?  Might the emptiness of shell #3 be relevant to the peanut’s being located under shell #4 in some ((perhaps)) vague way even given my knowledge k?)

To re-iterate (this is a GROUNDHOG DAY post after all), the shell statement is false in my situation for exactly the same reason that “shell #3 is empty” fails to be information that “the peanut is located under shell #4.”  In this particular case, the relevance-making condition which is lacking is identical with an INFORMATION THAT relation.

If so, however, one is faced with a consequence that may strike some as at least equally unappealing as the paradoxes of Material Implication.  (Warning:  I am about to wallow in more GROUNDHOG DAY iterations.)  For when I utter:

If shell #3 is empty, Then the peanut is located under shell #4

the statement I utter is false, but when you hear:

If shell #3 is empty, Then the peanut is located under shell #4

and your situation is such that you have seen both shells #1 and #2 are empty and you do not know that the peanut is located under shell #4, the statement you hear is true!  The same statement is both true and false at the same time, given different situations.  Put another way, what is true or false (at least for a certain class of IF-THEN statements) is not the statement, but the statement as it shows up in a particular situation.

At least in the case of subjective probability, then, truth is relative in much the same way that Galilean motion is relative.

On a purely autobiographical note, I am not sure this relativity bothers me any more than Galilean relativity (there is the possibility of an ultimate reference frame) or for that matter Einsteinian relativity (there is no ultimate reference frame which would assign a single value to the speed of a moving object) does.  The idea that a person walking inside a flying jet is moving at a speed of 1 mile per hour relative to the reference frame of the jet but at a speed of 501 miles per hour relative to the reference frame of the earth (suppose the jet’s speed is 500 miles per hour) is perfectly intuitive even though it means a contradiction is true (the person is both moving at a speed of 1 mile per hour and is not moving at a speed of 1 mile per hour).

Likewise, the contradiction of claiming that (GROUNDHOG DAY alert):

If shell #3 is empty, Then the peanut is located under shell #4

Is both true and false at the same time seems to me to be intuitive if one casts it as a matter in which a conclusion’s following (not following) from its premise hinges upon what other knowledge or evidence one has (does not have).  But I do suspect that some would prefer to this relativity of truth and the attendant tolerance of contradiction the weirdness of Classical Logic’s Material Implication which arises from treating Implication as purely truth functional.

This statement (GROUNDHOG DAY alert):

If shell #3 is empty, Then the peanut is located under shell #4

is variously true or false — even at the same time — depending upon the already-existing knowledge (or lack of it) of the person uttering or hearing the statement.  By contrast, the following statement is true regardless of what anyone knows, and true in any situation:

If the peanut is located under shell #4, Then the peanut is located under shell #4

In other words:

If p Then p

That the peanut is located under shell #4 clearly suffices to make the conditional probability that the peanut is located under shell #4 1.  So according to my account of what makes p relevant to q, p is relevant to p. p is relevant to itself.  p is in a relation to itself.  I am of course beginning to sound very weird (or maybe weirder) and very Hegelian…and I am beginning to wonder if I can get out of this weirdness by talking about 1-place relations, which are perfectly respectable mathematically.  (Not just 1-place relations!  0-place relations are also quite respectable mathematically!  What is more, Chris Date’s Relational Algebra recognizes two 0-place relations, TABLE DEE which is identical with the that weird proposition in logic TRUE, and TABLE DUM, which is identical with the equally weird proposition in logic FALSE!!!!!!!)

In this section of my post, I will decide that I am Relational-Algebra-weird by treating “If p Then p” as a 1-place INFORMATION THAT relation.  This in turn is part of my larger project to go through each example of IF-THEN statements I’ve adduced in previous posts and decide whether the relevance-making RELATION is in that particular case an INFORMATION-THAT relation or not.

Remember that to generate information is to reduce the number of possibilities to one.  When Herman is selected through 3 successive coin flips out of 8 candidates to perform the unpleasant task, the number of possibilities is reduced from 8 to 1.  The probability of Herman’s getting selected was initially 1 in 8, then became 1.  Whenever any event occurs, some states of affairs comes to obtain, some thing acquiring some property, the probability of that occurrence goes from 1 in (some usually gargantuan number) to just 1.  So any occurrence of p (Herman’s getting selected, shell #3 proving to be empty, a ruby having formed through whatever geological processes exactly one mile underneath where I happen to be sitting now typing this disreputable screed into a WordPress blog, the doorbell’s ringing) generates information.  Sometimes the occurrence of p generates information that q (that the peanut is under shell #4…that someone or something is depressing the button outside….).  But whatever else the occurrence of p generates information about, it generates at the very least the information that p.  Herman’s selection generated the information that Herman was selected, whether or not this information gets transmitted from the source situation in which the selection occurred (the room where the employees performed 3 coin flips) to the situation which is waiting for the information (the room where the boss is sitting).  When the information does get transmitted from source to receiver, the INFORMATION THAT relation is a 2-place relation comprising two situations, source and receiver.  When the information does not get transmitted, but stays where it is in the source, the INFORMATION THAT relation is a 1-place relation, comprising simply the source situation.

When the relevance-making relation that makes If p Then q true is an INFORMATION THAT relation, the occurrence (obtaining, existence) of p generates the information that q.  We have just seen that the occurrence (obtaining, existence) of p generates the information that p. So we get:

If p Then p

as a 1-place INFORMATION THAT relation.  Rather than saying, rather weirdly and rather Hegelianishly, that p is related to itself by virtue of being relevant to itself, we simply say that there exists a 1-place relation comprising the source at which the information that p was generated, and only that source.  This remains an INFORMATION THAT relation even though nothing ever tells me, informs me, shows me that, for example, a ruby exists exactly 1 mile beneath where I am now sitting, typing this disreputable screed into WordPress, or that the peanut is in fact underneath shell #4.  It is just a 1-place, not a 2-place relation, and an INFORMATION THAT relation to boot.

So in all of the following,

If a ruby exists exactly 1 mile underneath where I am now sitting, Then a ruby exists exactly 1 mile underneath where I am now sitting

If the peanut is located underneath shell #4, Then the peanut is located underneath shell #4

If Herman was selected to perform the unpleasant task, Then Herman was selected to perform the unpleasant task

the general relevance-making relation, i.e., the occurrence (obtaining, existence) of p making the conditional probability that p equal to 1, is identical with an INFORMATION THAT relation.  (My ((probably non-existent)) reader will remember that the relevance-making relation is not always an INFORMATION THAT relation.)

And this (after having brought in a ruby example and a Herman’s getting selected example) concludes my working through of most of the peanut-under-a-shell examples.  I still have one more peanut and shell example to work through, namely,

If I turn over shell #4, I will see the peanut

which I will work through in a future post.

Today’s homage to Plato’s SYMPOSIUM is Channing Tatum, who has recently appeared in MAGIC MIKE II.

Channing Tatum is the very walking, talking, breathing, living definition of the words ‘age 35 and beautiful and sexy.’  One of these days I will get around to contemplating Plato’s Form of Beauty itself.  For now, though, I will rest content just contemplating the form of Channing Tatum.

July 18, 2015:  extensive revisions made in probably futile attempt to hide the vastness of the extent of my confusion.

July 21, 2015:  made one more revision in order to try to hide the lack of control I have over the subject matter.

August 02, 2015:  made yet another revision for the same dubious reasons as listed above.

## Shells, Peanuts, And Doorbells: Subjective Probability And The Relevance-Making Relation

So far then, we have the following:  following Relevant Logic, we can avoid Classical Logic’s paradoxes of Material Implication, according to which the following statements are true…

1) If Cliff lives in Houston, Texas, then the earth has just one moon.

2) If Cliff lives in Orange County, California, then Paris, Texas is the capital of France.

…by insisting that the antecedent p be relevant to the consequent q.  The question now is:  what is the relation that makes p relevant to q?  I propose that this relation (henceforth the ‘CONDITIONAL PROBABILITY IS 1 relation) can be stated as follows:  given p, the conditional probability of q, (under conditions c, and possibly given knowledge k) would be, or would become 1.

We will see that this relation involves a dependency on p of the value of the conditional probability of q; this dependency though is different from the dependencies I’ve discussed in the previous posts. This dependency is the relevance-making relation we are looking for in our quest to escape from the evil clutches of the Classical Logician.

There are two items in the way I have just stated the CONDITIONAL PROBABILITY IS 1 relation that cry out for discussion.  The first item is the distinction between subjective and objective probability.  (I am a bit surprised that I have not yet seen so far a discussion of this distinction by Dretske, though perhaps I have run across such a discussion but forgotten about it.) The second item is the phrase ‘given that.’

OBJECTIVE VS. SUBJECTIVE PROBABILITY:  In the doorbell examples given in the post below, the CONDITIONAL PROBABILITY IS 1 relation is in both cases objective. In the non-poltergeist example, were the doorbell ringing, the conditional probability would be 1 that someone or something is depressing the button outside. This probability would be 1 regardless of what anyone thinks, knows, or feels. The probability is objective. Likewise, in the poltergeist example, the conditional probability that the doorbell is ringing inside were I to press the button outside would be 1, regardless of what anyone thinks, knows, or feels. In both the poltergeist and the non-poltergeist examples, the CONDITIONAL PROBABILITY IS 1 relation is objective.

By contrast, when I first come across the four shells (in a situation in which I already know that there is a peanut located underneath one of the shells), the conditional probability that the peanut is underneath shell #4 would become 1 in three were shell #1 to prove to be empty; would then become 1 in 2 were shell #2 prove also to be empty, and finally would become 1 were shell #3 to turn out to be empty.  In each case, starting from the very beginning, the conditional probability hinges upon what I already know about the situation and changes with the alterations in my knowledge.  The CONDITIONAL PROBABILITY IS 1 relation in this case is subjective.

Henceforth I will use the phrase ‘would be’ to suggest that the CONDITIONAL PROBABILITY IS 1 relation is objective, and ‘would become’ to suggest that the relation is subjective.  ‘Would be’ suggests that the conditional probability is set from the very beginning and does not change with changes in a person’s knowledge of the situation; ‘would become’ suggests that the conditional probability is not fixed from the very beginning, and does change with increases (or decreases) in a person’s knowledge.

If we allow both objective and subjective probability and identify the relevance of p to q with the CONDITIONAL PROBABILITY IS 1 relation, we then get the result that IF-THEN statements are relative when the relevance relation is based on subjective probability.  In your situation, when you have first come upon the 4 shells (and you may not even know that there is a peanut is located underneath one of the shells!), the statement:

1)  If shell #3 turns out to be empty, Then a (the) peanut is located under shell #4

is false, because in your situation the Conditional Probability that a peanut is located under shell #4 would clearly not become 1 were shell #3 to turn out to be empty.  But in my situation, given what I know, that statement is true.  The Conditional Probability would definitely, in my situation, become 1 were shell #3 to prove to be empty.  So at least those IF-THEN statements belonging to a certain class — i.e., those whose relevance relation is based on subjective probability — display a relativity similar to the Galilean relativity of motion.

If one wants to avoid this (possibly, for some — at least for me –) counter-intuitive, paradoxical-seeming result, they may want to rule out subjective probability and base IF-THEN statements only on objective probability.  But what would ‘objective probability’ be in the case of the shell game?  I think it makes intuitive sense to claim something like:  ‘given that the peanut was located under shell #4 from the very beginning, chances were always 100% (the conditional probability was always 1) from the very beginning that the peanut was under shell #4.  (In other words, given p, the conditional probability of p is 1.  OMG — If p Then p!)   But let’s take a closer look at the phrase ‘given that’.

GIVEN THAT:  ‘Given that p, the conditional probability of q is 1′ means, I take it, that what the conditional probability of q is hinges upon, depends upon, p.  In the non-poltergeist doorbell example, that conditional probability of the button outside being pushed is 1 hinges upon the doorbell’s ringing.  If there is no ringing, the conditional probability of the button’s being depressed is not 1, but 1/100, or 1/100,000, or whatever.  (Remember that the conditions c of the doorbell’s defective wiring are such that 1% of the time the doorbell does not ring when the button outside is getting pushed.)  No ringing, no conditional probability equaling 1.   In the poltergeist doorbell example, that the conditional probability of the doorbell’s ringing inside is 1 and not 1/2, or 1/10,000, or whatever, hinges upon my pressing the button outside.  (Remember that in this example the conditions c of the doorbell’s defective wiring are such that 1% of the time the doorbell rings even when no one or nothing is depressing the button, creating the impression that a poltergeist must be dwelling inside the doorbell apparatus.)  No pressing of the button, no conditional probability equaling 1.

Note that this is a case of the value of the conditional probability of q hinging upon p.  This is to be distinguished from, for example, the ringing’s causally depending upon the button’s getting depressed, or the fact that I am about to see the peanut causally depends upon my lifting shell #4 (plus other factors).

Now if we do not allow subjective probability, the only GIVEN THAT relation that holds in the case of the shell game example is ‘given that the peanut is under shell #4, the conditional probability of the peanut’s being under shell #4 is 1’.  This is the only case that does not depend upon what a person already knows.  So statements 1 through 3 below would all be false for exactly the reason that 4) is false:  there is no longer any relation that would make p relevant to q by p‘s giving the conditional probability of q the value of 1:

1)  If shell #3 turns out to be empty, Then a (the) peanut is located under shell #4

2) If shell #1 turns out to be empty, Then a (the) peanut is located under shell #4

3) If shell #2 turns out to be empty, Then a (the) peanut is located under shell #4

4)  If Cliff lives in Houston, then a (the) peanut is located under shell #4

But there are situations in which statements 1 through 3 are true — situations in which my knowledge and yours vary.  I submit then that the price of jettisoning subjective probability is one that is too high to pay.  We need to keep subjective probability, and along with it the Galilean-like relativity of those IF-THEN statements whose relevance-making CONDITIONAL PROBABILITY is 1 relation is an instance of subjective probability.

Let me see what I will make of all of this in the morning, when I am sober.

Today’s homage to Plato’s SYMPOSIUM comprises Sal Mineo and the guy he crushed on, James Dean.

Beauty so wonderful, so fleeting.