Category Archives: Probability

The Monty Hall Paradox And Borges’ GARDEN OF FORKING PATHS

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.) And 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. 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.

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 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. 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 it 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 version 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 (although this is not relevant to the example, the name 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. But I digress) sets up on the table a shell game with three shells and a single peanut.  The shells are labelled in order 1, 2, and 3. Employing a randomizing device of some sort (say, he throws a die), Elizarraraz places the peanut under the shell selected by his randomizer. Naturally, he knows under which shell the peanut is hidden.

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.

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 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 ‘non-Monty-Hall shell game. Again, using a randomizer, Smith selects one of the shells, and turns it over. Naturally, either there is a peanut showing up, or there is not. I think it would be uncontroversial to say that the probability there is a peanut there is 1/3, and the probability that there is not is 2/3.

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. But he knows 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. Given that the original sample space [*] of three has now been reduced to two, surely the probability is now 50/50! But hold that thought for a few more paragraphs [1] while I discuss for a bit the notion of ‘a possibility’.

At this point, Smith confronts two possibilities. 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.

Now Morgenstern arrives on the scene.

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. 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.

Shell #1 Shell #2 Shell #3 If Same If Different Which Means That
peanut, initial selection empty, not initial selection empty, not initial selection Smith wins Smith loses either shell #2 was turned over, leaving shell #3 to be select should Smith opt to change his selection; or shell #3 was turned over, leaving shell #2 to be selected should Smith opt to change … in either case, Smith loses if he opts to change his selection
empty, initial selection peanut, not initial selection empty, not initial selection Smith loses Smith wins shell #3 is the only shell eligible to be turned over, which means that Smith will choose shell #2, and win, if he opts to change his selection
empty, initial selection empty, initial selection peanut, initial selection Smith loses Smith wins shell # 2 is the only shell eligible to be turned over, which means that Smith will choose shell #3, and win, if he opts to change his selection
peanut, not initial selection empty, initial selection empty, not initial selection Smith loses Smith wins shell # 3 is the only shell eligible to be turned over, which means that Smith will choose shell #1, and win, should he opt to change his selection
empty, not initial selection peanut, initial selection empty, not initial selection Smith wins Smith loses either shell #1 was turned over, leaving shell #3 to be selected should Smith opt to change his selection; or shell #3 was turned over, leaving shell #1 to be selected should Smith opt to change. In either case, Smith loses if he opts to change his selection
empty, not initial selection empty, initial selection peanut, not initial selection Smith loses Smith wins shell #1 is the only shell eligible to be turned over, which means that Smith will choose shell #3, and win, if he opts to change his selection
peanut, not initial selection empty, not initial selection empty, initial selection Smith loses Smith wins shell #2 is the only shell eligible to be turned over, which means that Smith will choose shell #1, and win, if he opts to change his selection
empty, not initial selection peanut, not initial selection empty, initial selection Smith loses Smith wins shell #1 is the only shell eligible to be turned over, which means that Smith will choose shell #3, and win, if he opts to change his selection
empty, initial selection empty, initial selection peanut, initial selection Smith wins Smith loses either shell #1 was turned over, leaving shell #2 to be select should Smith opt to change his selection; or shell #2 was turned over, leaving shell #1 to be selected should Smith opt to change. In either case, Smith loses if he opts to change his selection

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.

Fortunately, Borges’ story, THE GARDEN OF FORKING PATHS, gives us a picture, another way of showing the 1/3 and 2/3 probabilities without the burden of this complication. We can picture the Monty Hall shell game as a series of forking paths. Doing so will nail down the 1/3 and 2/3 probabilities quite conclusively. Picturing the game this way will also provide at least one reason 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 (“You are so smart”, says my colleague Daryl Kwong, though I suspect he meant 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.]

Monty-Hall Shell Game Forest of Forking Paths Starting with E’s Choosing to Hide the Peanut under Shell #1

In the chart shown above, 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 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 (darker viridian green) or stick to his initial choice (lighter viridian green). The winning choice (Smith gets the peanut) is shown by the thick purple arrow.

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 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.

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.

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

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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.  ūüôā

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 ő© — as follows:

The Sample Space ő© =

ő© = { 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 ‚ą© 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 ő© 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 ‚ą© F ) / P(E)

P( E ‚ą© F ) = |E ‚ą© F| / |ő©| = 8/16 = 1/2

P(E) = |E| / |ő©| = 8/16 = 1/2

So:

P( E ‚ą© 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 ‚ą© 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 ‚ą© 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.

Boxer_XML_OnlyComputerBookBoughtJustForTheCover_

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)

peanutshell_03

So one more time — but this time with feeling: ¬†following Relevant Logic, we can avoid Classical Logic‚Äôs paradoxes of Material Implication, according to which the following statements are true‚Ķ

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

‚Ķby 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.

shell_02

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.

SmallShell

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

ChanningTatum_02png

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.

SmallShell

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.

Continue reading


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.

             shell_02shell_02shell_02shell_02

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.

James_Dean_SalMineo_4

Beauty so wonderful, so fleeting.