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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. 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.

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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(Asymptotically) All Republicans Are Racist

In order to argue that (asymptotically) all Republicans are racist, you need to take time into account. Before actual Nazis and outright racists started getting NOMINATED, before Trump won* the presidency,before Republican politicians started issuing racist dog whistles, before Nixon put in place the Southern strategy, it was definitely possible to give any given Republican the benefit of the doubt. As time goes on, decent people leave the Republican party, disgusted by the (at first covert, now overt) racism. Now that the racism is overt, with actual Nazis getting nominated for office, it is morally incumbent upon any remaining Republican to either demand that the Nazis and overt racists leave the party, form a new party in which these people are not welcome, or join the democrats. As time goes on and they do none of these things, it becomes more and more apparent that these are not decent people deserving respect and deserving the benefit of the doubt. One starts to think that the same thing that draws the overt racists and the Nazis to the Republican party is the same thing that draws THEM to the party and keeps them there, i.e., racism, white tribalism, the desire to keep heterosexual white males on the top of the heap and to continue to be granted automatic deference.

All Republicans are racist.  Just to be clear, by ‘all’ I mean ‘asymptotically all’ — as time goes on the proportion of Republicans who cannot deny they are racists approaches 100% asymptotically.

The Republican party delenda est. The Republican party must be destroyed; salt must be plowed into its ruins.


The Time We Should Be Giving Any Republican The Benefit Of The Doubt Is Long Past

There are three questions that any Republican needs to ask themselves before they can be considered deserving of any respect at all. — And no, looking and acting avuncular does not entitle you to the presumption that you are a decent human being and deserving a minimum of respect for that reason.

These questions are inspired by this Washington Post article, and some of the wording is taken directly from that article.

1) Why is it that all these racists are so supportive of my party? Why is it that a bunch of actual Nazis won Republican nominations for elected offices this year, and our nominee for the Senate in Virginia is a neo-Confederate? Why is it that every white nationalist thinks they can find a home in the GOP?

2. What can I do to change that?

I and the writer of the article would be interested to hear their ideas. But so far, we’ve heard pretty much nothing.  In other words, Republicans are not especially interested in making their party unattractive to out and out racists and Nazis. Nor have we seen any effort to create a new, center-right party that does not draw overt racists and Nazis.

Given this, any honest Republican needs to ask themselves:

3) Especially given that I am not interested in making my party unattractive to racists and Nazis or forming a new party, how much of my own attraction to the Republican party stems from the same racism that attracts the Nazis? — the same racism, just not overtly expressed, and doubtlessly hidden even from themselves. (The human capacity for self-deception is practically infinite.)

Drawing on a certain informal principle of plausible reasoning, which can be stated as

 

Birds of a feather flock together.

or again as:

If you see a bunch of Nazi flesh flies feasting on a piece of rotting carrion along with a bunch of ostensibly non-Nazi flesh flies, all of them are probably drawn to the same stench.

I think the answer is a lot.

If you see members of a flock of birds perfectly content to associate with a bunch of birds with swastikas emblazoned on their wings, and if you observe them failing to form a new flock minus those members, this contentment renders more credible the conclusion that all of the birds feel a certain … affinity … with one another.

Likewise, the togetherness of the flesh flies renders more credible conclusion that both varieties of flesh flies share the same racism.

Among Republicans, this racism is usually not expressed overtly.  It is typically hidden from themselves by an immense amount of self-deception.  Nonetheless, given the usual vehemence with which they react to the charge, their racism is clearly a sore — though unacknowledged — wound for them.

The number of Republicans asking themselves the three questions posed above is vanishingly small. The number of Republicans deserving of any respect at all is vanishingly small. The time is long past that we should give any of them the benefit of the doubt.

Homework Assignment:  Relate the principle stated above to G. Polya’s PATTERNS OF PLAUSIBLE REASONING, especially to pages 111-116.


On Cruelty, And The Distinction Between Amorality And Immorality

This comes close to nailing it.

TomMorris_0

My two cents:

TomMorris

The context was attitudes toward cruelty in the ODYSSEY.

Tom Morris I like to think there is an objectivity to beauty, alongside the subjectivity of experience relevant to it and what delights us or attracts us. I would view cruelty the same way. Cruelty is first an inner state, an intent to harm without reason or beyond justification, to inflict pain, physical or emotional, for its own sake, outside of any other goal. Cruel acts I would define as acts that arise from that intention. That leaves space for a harmful act that was not intended to be such being viewed by the harmed person as cruel, even though it literally wasn’t. Cruelty is an inner state of the soul or mind and heart. I see it as distinct from sadism, which I could define the same except to add an element of pleasure to the mindset. I don’t see cruelty as demanding pleasure on the part of the cruel person. That just makes him a sadist. Some of the suitors were cruel, I think. Others were just selfish and oblivious to the max. Does that make sense to you? Great questions as always.
Cliff Wirt
Cliff Wirt Tom Morris I’ve been looking for a way to distinguish ‘amoral’ from ‘immoral,’ and this may give me a START. An unambiguously amoral action (putting lead in gasoline because that is the easiest way to make money; firing large numbers of employees because that is a way to temporarily bump up the stock price) is one in which the harm one inflicts is in service to a goal; one does not have the specific intention of inflicting harm even though they may know that harm will be inflicted; accomplishing the goal is more important to one than any harm one may inflict ; and a norm is violated — i.e., the goal SHOULDN’T be more important to one than the harm one inflicts. By contrast, a cruel action would be unambiguously immoral, i.e., an action in which one has a specific intention to do harm. (By ‘specific intention’ I mean one has not accomplished what they set out to accomplish unless harm was done, no matter what other goals one may have had in performing the act.) This is just a start.
Tom Morris
Tom Morris Cliff Wirt I like it, and even more than hitting ‘like’ would indicate.

The Quality Quest

[The following is a letter I wrote a while ago to the editor of Chicago’s NEW ART EXAMINER responding to an article by Betty Ann Brown.  Betty Ann Brown’s article is badly vitiated, if I may say so, by the sort of sloppy reasoning peculiar to postmodern political flimflam.  As might be expected from the low quality of Brown’s article, Brown’s only response was to engage in some perfunctory hand waving.]

Betty Ann Brown (“A community self-portrait,” NAE, December, 1990) would have us retire the word “quality” because she believes that the concept the word expresses has built into it standards which improperly and objectionably tend to exclude women and artists of color from museums, galleries and exhibitions.  (I will put “quality” in double quotes when I am talking about the word, and in single quotes when I am talking about the concept.)  That is to say, the concept is constructed along class/race/gender lines.  She seems to identify ‘quality’ with the concept of formalistic quality, i.e., a work’s excellence or lack of excellence considered as hinging on such factors as line quality, touch, handling, composition, spatial balance, relations between forms, relations between colors, and so on.  ‘Quality’ interpreted as ‘formalistic quality’ is the concept, she asserts, whose use excludes women and artists of color.  Instead of the word “quality,” she would have us use “worthy.”  According to Brown, a work is worthy when its content “…authentically [accurately?] reflects the artist’s social/historical/political moment.”  She prefers work that grates on her, reflects experiences beyond her own, and concerns issues of race, gender, and class.

I very much doubt whether Brown is really rejecting the concept of quality at all.  If she uses “worthy” in such a way that “This work is good or excellent” follows from “This work is worthy” (surely the word means nothing if this does not follow), then the concept of quality has not been done away with.  For if a work is high in quality, it is good or excellent, and if it is good or excellent, it is high in quality.  Thus I suspect Brown is really just advancing a different theory of what artistic quality (worth, merit, excellence, being good) consists in.  She thinks that a work’s quality hinges not on its formalistic values, but on its authentically reflecting an artists’s social/historical/political moment.

However, Brown’s theory of quality (or worth, merit, excellence, or whatever) is obviously false.  Consider all the dull, heavy-handed, poorly observed works stemming from the nineteenth century that use vicious stereotypes to depict African Americans, male and female.  Surely these works reflect their artists’ social/historical/political moment in the most authentic way possible.  They even grate on me, reflect on experiences beyond my own, and concern the issues of race, gender, and class that Brown holds so dear.  Brown is not about to value them as worthy.  If her theory of quality is true, however, there is no way one could escape the conclusion that they are worthy, their shoddiness and viciousness notwithstanding.  Brown could try to avoid this unappetizing conclusion by claiming that the content of  work must reflect the correct politics if it is to count as excellent, but such a move would be clearly ad hoc, if not laughable.  The only reason to make such a move would be to save Brown’s theory.

In the absence of any plausible alternative, one is left with the formalistic theories of quality.  Do these theories in fact have built into them standards that improperly and objectionably tend to exclude women and artists of color?  Consider the following theory, and see if it has any such standards built in.  I submit that the concept I describe below is the one operative in most critical discourse.

A work of art is a symbol that both expresses and sometimes denotes (to use Nelson Goodman’s terms) a content or subject matter.  The work’s excellence or lack of excellence is a function of both its formalistic values and what it expresses.  If what the work expresses is of low value, the work itself is of lesser value, even if (and in fact partly because) its formalistic values express its content perfectly.  Suppose, for example, that Jones, a critic, becomes convinced that Jackson Pollock’s drip paintings express the same types of feelings expressed by New Age music.  Since Jones holds those feelings in low esteem, she values the paintings less than they are usually valued.  Similarly, Smith, a curator at the Metropolitan Museum of Art, holds in low esteem what Anne Ryan’s collages express, namely, a sense of intimacy and pleasure (usually regarded as feminine) in materials and fabrics.  The fact that the formalistic values of the collages expresses those things perfectly hardly commends them to him.  He therefore places the works in storage.

Clearly, Smith’s application of the concept ‘quality’ has been guided by his gender attitudes.  He regards feminine stuff as minor and of lesser value.  I take it this is the sort of case Brown has in mind when she claims that ‘quality’ has built into it standards that improperly and objectionably tend to exclude women.  In what follows, I argue that the claim is nonetheless false.  The argument focuses on the expressive content of an artwork.

There are two possibilities concerning the value of what an artwork expresses.  1) Conventional, relativistic, folk wisdom is correct.  Conventional folk wisdom would like to relativize value the way Einstein relativizes motion.  In Einstein’s theory, of course, the speed of an object is relative to a frame of reference.  In one frame of reference, the speed is 60 mph, and in another it is 1 mph.  Folk wisdom treats Smith and Jones as one-person frames of reference.  In the Smith frame of reference, what Ryan’s work expresses has a low value, while in the Jones frame of reference, say, it has a high value.  Just as there is no absolute measure of speed, but only the speed in this frame of reference and the speed in that one, there is no absolute measure of value for what Ryan’s work expresses.  There is only its value for Smith, and its value for Jones.  2)  What an artwork expresses has a value that is not relative to particular individuals, and Smith and Jones can measure that value accurately or inaccurately, correctly or incorrectly.

Assume that 1) is right.  Suppose also that Jones is a feminist who wants to believe that Smith’s exclusion of Ryan (and the exclusion of other women artists on similar grounds) is improper and objectionable.  Jones, however, cannot cogently criticize or object to Smith’s exclusion of Ryan’s work.  For surely the following thesis is true:

A) If an artwork is of low value (is not good, excellent, worthy, etc.), excluding it (putting it into storage in a museum, not exhibiting it in a show, not buying it, and so on) is not objectionable or improper.

This is, I suspect, an intuition everyone shares.  Even Brown’s view commits her to it, since if a work is worthy, it is surely not low in value.  Now in the Smith frame of reference, Ryan’s collages are low in value.  It follows from A), then, that Smith’s putting her work into storage is not improper or objectionable.  The mere fact that in the Jones frame of reference the collages have a high value does not make the exclusion objectionable.  For disputing the exclusion on those grounds would be like disputing a measure of speed made in another frame of reference on the grounds that it does not match the measure one has made in his own frame of reference.

So if the relativism outlined in 1) is correct, Smith’s exclusion of Ryan’s work is not objectionable.  I assume, by the way, that Brown objects to ‘quality’ because it allegedly leads to cases of objectionable exclusion.

Assume now that 2) is right.  Smith has either correctly or incorrectly valued the expressive content of Ryan’s work.  If he has valued that content correctly, then Ryan’s work is of lesser quality and therefore of lesser value.  It follows from A), then, that Smith’s exclusion of Ryan’s work is not objectionable or improper.  Smith’s exclusion has not resulted from biases and prejudices that have prevented him from valuing the work correctly.  So the concept ‘quality’ is not open to criticism in this case because it has not led to an improper or objectionable exclusion.

Suppose now that Smith has valued the expressive content of Ryan’s work incorrectly (presumably because of gender biases).  He was wrong to put it in storage.  (This is, incidentally, the view I hold, and I suspect Brown would prefer to hold it as well.)  In this case, however, the fault does not lie with the concept ‘quality,’ but with a bad and misguided application of that concept to a particular case.  The application of ‘quality’ went afoul because prejudice prevented Smith from valuing correctly the expressive content of Ryan’s work.  In cases like these, then, the concept ‘quality does not have built into it standards that improperly and objectionably exclude women; rather, it is particular application of the concept that can objectionably exclude women (not all women, by the way) when the expressive content of a work gets wrongly valued.

In each case, then, either the concept ‘quality’ is not the culprit, or the exclusion in question is not objectionable.  Contrary to Brown, it turns out that ‘quality’ does not have built into it (through some kind of white male conspiracy) standards which improperly and objectionably exclude women.  If women are underrepresented in museums relative to their population, the fault lies not with ‘quality,’ but with other factors, including bad applications of the concept (assuming that relativism is false and that female concerns are incorrectly assigned a low value — if relativism is true and female concerns are correctly given a low value, cases of the sort discussed above, which I take to be bad applications of the concept, are in fact not objectionable), prejudice, and social discouragement.  The same analysis applies mutatis mutandis to artists of color.

Cliff Engle Wirt                                                                                                                                        Chicago, IL

Today’s homage to Plato’s SYMPOSIUM takes the form of James Dean and Sal Mineo.

James_Dean_SalMineo_4

‘Look at me the way I look at Natalie Wood,’ James Dean reportedly told Sal Mineo during the filming of REBEL WITHOUT A CAUSE.  Mineo, having a crush on Dean, needed very little prompting to heed this instruction.  Homoerotic expression is, I dare say, something that in the past has been given an incorrect valuation.


“How Can We Know What The Probability Is?” And Other Objections And Remarks


The following, of course, is not yet developed.


“Where did you get that 99% probability from?” someone may object.  “Did you pull it from your ass?”  Well, I did stipulate it.  But the general objection remains valid nonetheless:  it would seem that there is no way to come up with an objective evaluation of what the probability actually is unless it is 0 or 100%, these figures being based on the physical laws of the universe or the laws of probability.  Deal with this.  See if Dretske’s discussion of this works.


Inductive:  probability of less than 100% but greater than 0.  Deductive (or what supports deduction):  conditional probability is 100%.  Absolute reliability, absolute safety.  What makes the transmission a case of information is also what makes it something supporting deduction.


IF p THEN p — either a complete lack of transmission of information or the exact opposite — a complete surfeit of “transmission” (quote unquote) at the “zero point”.

Update: 09/08/2018: Graying this out because it is too revealing of my vast ignorance of subjective vs. objective probability.

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The Problem

The Problem:  What Does Relevance Consist In?  Following Relevant Logic, we can avoid Classical Logic’s paradoxes (or at least weirdnesses) 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.

But what is it that makes p relevant to q?  What is relevance anyhow?

 

 

 

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Next Snippet:  What Is Relevance Anyhow?

 

 
 
 
Edit Log: June 04, 2017: Made some minor changes.