Category Archives: Information Theory

What Is Relevance Anyhow?

But What Does ‘Relevance’ Mean? If (at the time of this writing) one googles for a definition of the word ‘relevance’, the gist of what they will get will be something like:  a state of affairs1 p is relevant to a state of affairs q when p is connected to q in some way and that connection is important to us in some way.  The connection matters.

Any given state of affairs will of course bear a very large (perhaps indefinitely large) number of connections to any other state of affairs.  I am trivially connected for example to all people in the world whose last name begins with ‘W’ (I bear a W connection to each of them); and I am trivially connected to everyone else in the world whose last name does not begin with ‘W’ (I bear a non-W connection to each of them).

But some connections matter to us, perhaps in relation to some particular goal, or in relation to some highly pervasive desire.  The importance of the connection selects out those cases in which p is relevant to q.

The Ice Example:  Warning — I Intend To Use This As A Metaphor For Implication:  For example, the thickness/thinness (or even complete absence) of the ice covering a river (state of affairs p)  is connected to my reaching the other bank of the river (state of affairs q) by way of enabling/hindering/rendering-impossible my reaching that other bank.  This connection matters to me when I have the goal of reaching the other side alive, or at least in some reasonable approximation thereto.  (And I have this goal because of something else that matters to me.  I need, say, to evade the secret police on this side, or the only food there is exists only on the other side.)  The importance of this connection, the place it has in the web of my goals, renders p relevant to q.

So when the Relevant Logician insists that p be relevant to q in propositions of the form IF p THEN q, they can plausibly be construed as asserting that there is some connection between p and q, and this connection is important to us.  What this connection is and why it is important to us may be suggested by the following examples.  The first example to follow (Madame Olensky) does not quite get us to this connection, but it is suggestive enough to put us on the right track leading to it (The Doorbell).

The Matter Regarding Madame Olensky And Professor Plum:  When Madame Olensky is caught standing over the body of Professor Plum with a smoking gun in her hand, this state of affairs (p) bears a definite connection to another (quite) possible state of affairs, namely, that Madame Olensky murdered Professor Plum (q). This connection consists in the fact that p‘s obtaining/being true increases the probability (in this case drastically) that q obtains/is true.  That probability is now somewhere greater than 0 but equal to or less than 1.  The connection matters to us whenever we are concerned enough to ask (say, out of a desire for justice, I should hope, or at least out of a general desire to get things right):  Did Madame Olensky murder Professor Plum?  Because this increases-the-probability connection matters to us, it renders Madame Olensky’s standing over the body of Professor Plum (whose last twitches ceased just one second ago) with a smoking gun relevant to the possible state of affairs comprising Madame Olensky’s just having murdered Professor Plum.

But the Relevant Logician will want something a bit stronger for the connection between p and q that will make p relevant to q in propositions of the form IF p THEN q.  For in propositions of that form, the obtaining/being true of q is guaranteed should p obtain/be-true.  In other words, the probability of q, given p, needs to be 1.  Not 0.86, not 0.9999, but 1.  Implication needs to be completely reliable.

In other words, the ice needs to be so solid that the chances of falling through, of losing one’s footing and plunging into deep cold water while trying to cross to the consequent q are zero.

Although Madame Olensky’s standing over the body of Professor Plum with a smoking gun definitely increases the probability that she is the murderer of Professor Plum beyond 0, that probability is doubtlessly not 1.  For a sufficiently competent writer of mystery novels can invent a scenario just barely within the realm of possibility in which, despite the bald fact that Madame Olensky is standing over the body of Professor Plum with a smoking gun in her hand, she is in fact not the actual murderer of Professor Plum.  The probability is, say, a mere 0.99999999999.

In the matter regarding Madame Olensky and Professor Plum, there is a minuscule, but real chance that one might fall through the ice, lose their footing, plunge into the deep cold swift water while crossing to the other bank of the river.

So the statement

1) IF Madame Olensky is standing over the body of Professor Plum with a smoking gun, THEN Madame Olensky is the murderer of Professor Plum

is false.  It is false because, although the state of affairs comprising Madame Olensky’s standing over the body of Professor Plum with a smoking gun is definitely relevant to the possible state of affairs comprising Madame Olensky’s being the murderer of Professor Plum, the connection which generates this relevance is not the right relevance-making connection.

The Doorbell (In Perfect Working Order):  The right relevance connection does exist, I think, taking a cue from Fred Dretske, in the case of a doorbell whose wiring is in perfect condition.  Given the condition of the wiring, the probability, when the doorbell is ringing (p), that someone outside is pushing the doorbell button, or that, at least, something is depressing that button (q), is 1.  The constraint created by the perfect condition of the wiring makes p a completely reliable indicator of q.  So this IF THEN statement:

2) IF the doorbell is ringing THEN someone or something is depressing the button outside

is true.  That someone or something outside is depressing the doorbell button is guaranteed by the doorbell’s ringing inside.

This particular increases-the-probability (to 1) connection between the doorbell’s ringing and someone-or-something’s depressing the button outside matters to (most of) us because there is, I should think, a pervasive desire to get things right, to know how things actually stand outside the room, to know what is actually the case among the things that are not immediately present to us, to be able to tell what is happening.  This mattering selects out this particular connection as a relevance-making connection between p and q.  Because of this relevance of p to q, 2) above is true.

The doorbell’s ringing (when the condition of the wiring is perfect) is, of course, the classic example of Information That, of informational content.  The ringing (r, for reception) is information that the button outside is getting depressed (s, for source), if we follow Dretske’s definition of informational content:

Informational content:  A signal r carries the information that s is F = The conditional probability of s‘s being F, given r (and k), is 1 (but, given k alone, less than 1)

Fred Dretske, KNOWLEDGE AND THE FLOW OF INFORMATION, Stanford, CSLI Publications, 1999, p. 65

I will dwell on the knowledge k part of this definition in some detail later.

That the conditional probability of the button outside’s getting depressed increases to 1 when the doorbell rings is both what makes the ringing a signal, information that the button outside is getting depressed and what makes p relevant to q in 2) above.  Therefore, it is tempting to identity the relevance-making relation between p an q with the information-that relation.   Implication, it is tempting to say, is always information that.  The following:

3) IF Cliff lives in Houston THEN the earth has just one moon

fails to be a true implication because Cliff’s living in Houston is not information that the earth has just one moon.  I will be returning to this point later.2

To revert back to the river ice metaphor, the antecedent in 3) is ice that never formed in the first place.  There is no chance one can cross to the consequent q on the basis of p.  One cannot even lose their footing here, because there was only ever swift cold water to plunge into.

However, there are of course a number of rather severe challenges to the notion that implication is always information.  I will consider some of these in the snippets that follow.

At the time of this writing, I am suffering under the delusion that once all the challenges that I have considered so far have been dealt with, one ends up with the concept of relevant implication as always to be made sense of in terms of the concept of information — sometimes as full-blooded information, sometimes as degenerate or denatured information, and sometimes as the radical absence of information.  Whichever is the case, there is always the reference to the concept of information.  We will see if I end up having to eat crow on this point.

Some Housekeeping:  First, however, I want to do some housekeeping.  The careful reader will notice that I keep shifting back and forth between talking about p and q as states of affairs and as propositions.  I will continue to shift back and forth because I will be following Roderick Chisholm in treating propositions as a subspecies of states of affairs.3  The state of affairs comprising this cat, Munti sitting on this Persian mat can obtain or not obtain at different times.  The state of affairs comprising ‘Munti is sitting on on this Persian mat on October 31 at 12:00 am’ either always obtains or never obtains according as it was true or not true October 31 at 12:00 am that Munti was sitting on the Persian mat.  The latter is a state of affairs (obtaining or not obtaining) that is also a proposition (true or false); the former is a state of affairs (obtaining or not obtaining) that is not also a proposition.

Propositions are true or false; a proposition can follow from another or fail to follow from it.  Implication, therefore, is a relation between sets of states of affairs obtaining/failing to obtain being true/failing to be true at particular times (the doorbell is ringing at times t0, t1, t2, t… tn) and the button outside is getting pushed at times t0, t1, t2, t… tn).

One Final Point:  I have defined relevance in terms of mattering.  Since in Relevant Logic p has to be relevant to q in implication propositions in order for those implications to be true, does this mean that no implication statement was true before any sentient creature existed to whom anything could matter?  (I don’t think so, but this still needs to be shown, of course.)  If so, is this a weirdness that is off-putting enough to make one prefer Classical Logic to Relevant Logic?

 

1 I will leave ‘state of affairs’ as an undefined primitive.
2 One reason p is not information that q here is, of course, that the earth has just one moon is “old information” and therefore not information at all. But the more important reason is that even if this were not “old information”, Cliff’s living in Houston would still not be information that the earth has just one moon because the former, by itself, leaves the probability of the latter at 0. This ‘even if’ is pertinent to my claim that implication is to be understood in terms of information even if a particular example of an implication proposition is not an instance of information that.
3 Roderick Chisholm, THEORY OF KNOWLEDGE SECOND EDITION, Englewood Cliffs, Prentice-Hall, Inc., 1977, pp. 87-88.

 

 

Back To Main
Back To The Problem

 

 

 

 

 

 

 

 

Edit Log:  June 04, 2017:  Made some fairly minor edits in an always-ongoing and never-fully-accomplished effort to avoid complete and total embarrassment.

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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’s paradoxes of Material Implication, according to which the following statements are true…

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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.

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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.


My Attempt To Identify The IF-THEN Relation With The INFORMATION-THAT Relation Ignominiously Bites The Dust

Here is yet another challenge to the idea that ‘If p Then q’ is true when the occurrence of p is information that q.  Unfortunately, I think this challenge nails the matter. Consider Dretske’s shell game example.  The peanut is under shell #4.  So the following statement is true (given that my visual faculties are in sufficiently good working order, and that I am looking in the proper direction with my eyes open):

If I turn shell #4 over now (t0), I will see a peanut at time t1

(t1 being one millisecond or whatever later than t0.)  Is my turning shell #4 over at time t0 information that I see a peanut at t1? Certainly the situation largely fits Dretske’s definition of ‘information that’:

Informational content:  A signal r carries the information that s is F = The conditional probability of s‘s being F, given r (and k), is 1 (but, given k alone, less than 1)

Fred Dretske, KNOWLEDGE AND THE FLOW OF INFORMATION, Stanford, CSLI Publications, 1999, p. 65

(k represents what the receiver already knows about the source.)  The conditional probability of my seeing the peanut at t1 is certainly 1 given my turning the shell over at t0 (and given the other conditions mentioned).  So the IF-THEN statement above certainly fits that part of the definition of informational content.

But is my turning the shell over at time t0 a signal that at time t1 that I see the peanut?  A signal is  “…any event, condition, or state of affairs the existence (occurrence) of which may depend on s‘s being F.”  (Dretske, p. 65.)  Does my turning the shell over now depend upon my seeing the peanut one millisecond in the future?  How can a present event depend upon a future event?  Clearly not.

A signal cannot occur before the event or thing or state of affairs the occurrence (existence, obtaining) of which it signals.  The smoke does not occur before the fire (or the smoldering).  The doorbell does not ring before the button is pushed.  The deer tracks in the snow do not appear before the deer show up.  Were the watchman in Aeschylus’ play AGAMEMNON in the ORESTEIA trilogy to light his fire before he spots Agamemnon’s ships, his fire would not be a signal informing Clytemnestra of the appearance of those ships on the scene:  Clytemnestra would be receiving false information.  Something cannot be announced before it occurs (exists, obtains).

“But the dark clouds signal the rain that is about to fall; the sports official signals the race that is about to start in one millisecond by firing the pistol into the air.”  Someone may object in this way to my (seemingly obvious) claim that a signal cannot occur before the thing it signals.  Yet, although we can doubtlessly “round up” the dark clouds and the firing of the pistol to the status of signals, they are not so in the very strictest sense of ‘signal’ that I intend to use here.  For the conditional probability that, given the dark clouds, rain will fall is perhaps only 99%, while the probability that the race actually will start given the firing of the pistol is perhaps only 99.9999999999% (the supernova that will hit us eventually may choose that exact millisecond to intervene by making its presence glaringly, searingly obvious, or a huge earthquake might strike at that very moment….).

A signal is  “…any event, condition, or state of affairs the existence (occurrence) of which may depend on s‘s being F” and therefore cannot occur before the occurrence (existence, obtaining) of s‘s coming to be F.   The examples I’ve just given are not signals because they occur after what they “signal”, and — surely not coincidentally — they do not depend upon what they “signal.”  Let me dwell a moment, perhaps a bit obsessively/compulsively, on this notion of dependence.  Let me say that an event, object, or state of affairs p depends upon an event, object, or state of affairs q when, given a condition c,  p would occur (exists, obtain) only because q occurs (exists, obtains).

Consider, for example, a doorbell whose wiring is defective in such a way that, 99% of the time when the button outside is getting depressed by someone or something, the doorbell rings.  But 1% of the time the doorbell does not ring when the button outside is getting depressed. (I state the example this way to make it mirror the fact that p does not follow from If p Then q; q.)  Also, there is no poltergeist inside the wiring that sometimes generates the ringing sound even when no one or nothing is pressing the button outside; likewise, there is never, ever any freak burst of electricity ultimately caused by a butterfly flapping its wings in the Amazon that generates a buttonless ringing sound.  Nor (somewhat more plausibly) is there any defect in the wiring that would ever cause a buttonless ringing sound to occur. Let c be the condition of the defective wiring as just described (including the absence of ring-generating poltergeists).  Given c (which I will call the non-poltergeist condition), the doorbell would ring only because the button outside is getting depressed (even though the button’s getting depressed does not necessarily result in the doorbell’s ringing)*.  Given c, the doorbell’s ringing depends upon someone or something’s depressing the button outside and is therefore a signal.  (A signal, moreover, carrying the information that someone or something is depressing the button outside, because the conditional probability of this is 1 given the doorbell’s ringing under condition c.  Another way to put this is to make the perhaps obvious/tautologous point that to be a signal is to carry information.)

Consider another example, one which is perhaps belongs more to the realm of probability than to causality.  One has turned over shells #1 and # 2 and verified that both are empty. They have information that the peanut is located in one of the four shells.  So c is now the condition that either the peanut is located under shell #3 or under shell #4.  Given c, shell #3 would be empty only because it is shell #4 that is covering the peanut.  It is, in fact, difficult to come up with any clear idea of anything else that could be the reason why shell #3 is empty.  Shell #3’s being empty therefore depends upon the peanut’s being located under shell #4, and the former would be a signal carrying information that the latter.  (Conversely, given that there is only 1 peanut at play in the game and given the rest of c, shell #4’s turning out to have the peanut would be a signal carrying information that shell #3 is empty.  Shell #4 would have the peanut only because shell #3 is empty. )

Now consider again the turning over shell #4 example given above as an instance of an event, object, or state of affairs that very definitely is not a signal carrying information.  It would be difficult to give any meaning to the assertion:

my turning shell #4 over at time toccurs only because I will see a peanut at time t1

Such an assertion would not, I submit, make any clear sense, since the dependency aka only because relationship is a vector traveling forward (to speak metaphorically) in time.

Also consider yet one more doorbell example:  suppose that the doorbell’s wiring is screwy in such a way that every now and then little bursts of electricity get generated which produce the ringing sound even when no one or no thing is depressing the button outside.  (Or, if you prefer, there is a poltergeist residing inside the wiring that every now and then gets agitated by a freak burst of air pressure inside the contraption that is ultimately caused by a butterfly flapping its wings in the Amazon….)  Nonetheless, the condition of the wiring is such that the doorbell always rings when the button is getting pushed.  100 percent of the time the doorbell rings when the button outside gets pushed, but 1% of the time the doorbell is ringing buttonlessly. (I state the example this way to make it mirror the fact that q does not follow from If q Then p; p.  And I am making it mirror this because, of course, the whole point of these interminable disquisitions is to dig into the nature of IF-THEN statements.)  Let me call this condition of the wiring c, as usual.  (In a moment I will be calling it the ‘poltergeist condition.>)  Given c, it would be difficult to give any sense to the following assertion:

My pressing the button outside occurs only because the doorbell is ringing.

Clearly, my pressing the button outside does not depend upon, and is not a signal for, the doorbell’s ringing.  Again, the pressing of the button does not depend upon the doorbell ringing because the dependency aka only because relationship is a vector traveling forward, not backward, in time.

“Feel free to come to the point when you finally have one,” my (possibly non-existent) reader may want to say.  Well, the point of all of the above is the following.  Given their respective condition c’s, each of the following IF-THEN statements is true:

1) If I turn shell #4 over now (t0), then I will see a peanut at time t1

2) If I press the button outside [given the poltergeist condition], then the doorbell will ring.

3) If shell #3 is empty, then the peanut is located under shell #4.

4) If the doorbell is ringing [given the non-poltergeist condition], then someone or something is depressing the button outside.

Although the antecedent p is a signal carrying the information that q in the last two examples, it is not such a signal in the first two examples.

These examples, I think, nail it:  IF-THEN statements cannot be identified with an information relation.  My attempt to identify the IF-THEN relation with the INFORMATION-THAT relation has ignominiously bitten the dust.  (Sob, sob.) Does this mean, then, that we are stuck after all with Classical Logic’s paradoxes of Material Implication, whereby both of the following statements are true?

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

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

(Please God, please God, please don’t let these statements be true.)  Well, maybe we aren’t forced to accept these horribly ugly statements as true after all.  For in each of the 4 numbered examples given above, the conditional probability of the consequent (given the antecedent plus the relevant condition c ((plus the relevant knowledge k))) remains 1.  It is just that in the first two examples the antecedent does not depend upon the consequent, and therefore is not a signal carrying the information that the consequent.  It is not a p only because q relationship.  Perhaps, then, we can identify the IF-THEN relation with a different (but similar) relation, which I will call ‘the conditional probability is 1‘ relation. If so, it would remain true that in examples 3 and 4 above, the antecedent p is a signal carrying information that q.  So whenever p does depend upon q in such a way as to be a signal for q the corresponding IF-THEN statements would, possibly, have the (at least to me) weird properties mentioned in a previous post:

Third, the informational relation is both intentional and relative, as described by Fred Dretske in his KNOWLEDGE AND THE FLOW OF INFORMATION.  Treating If p Then q as an information relation would make implication both intentional and relative.  The very same If p Then q statement would be true inside some frameworks and false inside others.  Rather than accept this, some would perhaps rather accept Classical Logic’s paradoxes of Material Implication.

(Sidenote:  Dretske’s measles example displays the intentional character of information.  By pure chance, all of Herman’s children happen to have the measles; moreover, one does not know this.  So when one discovers that a particular person is a child of Herman’s, they do not have information that this person has the measles.) Or are we truly stuck with this weirdness? Can we find a way to make implication non-relative and non-intentional even in those cases in which p happens to be a signal carrying the information that q?

Today’s homage to Plato’s SYMPOSIUM is this gorgeous young Asian Man: GorgeousAsianGuy

It is hard to understand how anyone can get any work done at all with Beauty like this walking the earth, but somehow we do. How sleek all those black, white, and gray tones are!

Post Updated on June 27, 2015 to make the temporal vector nature of the dependency/only because relation clearer. (Or, if my reader is particularly suspicious, they are free to think I made the update in order to cover up some totally obvious mistakes, not simply to make a somewhat muddy post slightly clearer.)


Measles, Wormy Red Apples, And God (And Peanuts)

In his Knowledge and the Flow of Information, Dretske argues that what information a signal carries is relative to what the receiver already knows about the possibilities at the source:

To illustrate, suppose that there are four shells and a peanut is located under one of them.  In attempting to find under which shell the peanut is located, I turn over shells 1 and 2 and discover them to be empty.  At this point, you arrive on the scene and join the investigation.  You are not told about my previous discoveries.  We turn over shell 3 and find it empty.  How much information do you receive from this observation?  How much do I receive?  Do I receive information that you do not receive?  … [Dretske goes on to argue that the answer is ‘yes’ because the amount of information and what information is received depends upon the reduction in possibilities achieved in each case.  Information is all about reduction in possibilities.] … This constitutes a relativization of the information contained in a signal because how much information a signal contains, and hence what information it carries, depends on what the potential receiver already knows about the various possibilities that exist at the source.

Fred Dretske, KNOWLEDGE AND THE FLOW OF INFORMATION, Stanford, CSLI Publications, 1999, pp. 78-79

The third shell’s proving to be empty when it is turned over is, for me, information that the peanut is hidden under shell 4.  But for you, it is not information that the peanut is hidden under shell 4.  What information a signal carries (here the signal is the third shell’s proving to be empty when turned over) is relative to what one already knows.

Let’s apply this conclusion to the measles and wormy read apple examples.

Suppose that one has received information that all of Herman’s children have the measles.  Should one then discover (say, a friend tells them this) that this layabout in front of one’s shop is a child of Herman’s, that this person is a child of Herman’s is now, all of a sudden, information that this person has the measles.  Before one knew that all of Herman’s children have the measles, that this person is a child of Herman’s was not information that the person has the measles.

The same reasoning applies mutatis mutandis to the wormy red apple example.  If one has information (say, received from a person who has previously examined all of the apples in the pile)  that all of the red apples in the pile are wormy, then that the apple in one’s hand drawn from this pile is red is information that the apple is wormy.  Before one has received the information that all of the red apples in the pile are wormy, a signal that the apple in one’s hand is red is not information that it is wormy.  In both the measles and the wormy red apples examples, what information a signal carries depends upon, is relative to, what one already knows.

So if one claims that If p Then q is true only when the occurrence of p is information that q, then the truth of these sentences (henceforth the ‘measles’ and  ‘wormy red apple’ statements)…

If this layabout loitering about on the front of my shop is a child of Herman’s, then this person has the measles.

and

If this apple (drawn from this particular pile) in my hand is red, then it is wormy

…is relative to what one already knows.  They will be true relative to the person who already knows that all of Herman’s children have the measles (without necessarily knowing that this particular person in front of their shop is a child of Herman’s) and that all of the red apples in this pile happen to be wormy.  They will be false relative to the person who does not already know these things.

In previous posts, I noted as an autobiographical fact that I had the strong intuition that both statements above are true, regardless of what one already knows.  But perhaps this intuition, in spite of its being my intuition, should not be regarded as totally sacrosanct.  For I will venture that most people would not be bothered by the relativity of this statement (henceforth the ‘third shell proves empty’ statement):

If the third shell proves to be empty, then the peanut is located under the fourth shell

Clearly (although I say ‘clearly’ with some trepidation, in the spirit of ‘let me throw this piece of spaghetti onto the wall, and see if it sticks,’ or, alternatively, ‘let me see if I can get away with this statement without too many screams of protest’), this statement would be true in the situation occupied by the person who already knows that the first and second shells are empty, and false in the situation occupied by the person who does not already know these things.

What can be learned from, inferred from, concluded from the third shell’s being empty, the apple’s being red, the layabout’s being a child of Herman’s, depends upon the situation one is in that is defined by what one already knows.  There isn’t, I think, anything controversial or counter-intuitive about this.  IF-THEN statements have everything to do with what can be learned from, inferred from, concluded from a given situation.  So the truth/falsity of the corresponding If p Then q statements is also relative to the situation one is in as defined by what one already knows.

And if one is still bothered by this, would one rather return to the paradoxes of Material Implication?

(Begin aside:  Remember that what is motivating this entire attempt to argue that If p Then q is true only when p is information that q is to escape from the paradoxes of Material Implication, which would count both of the following statements as true:

If Calypso music originated in Wisconsin, then the earth has two moons

and

If Paris is the capital of France, then the earth has one moon

To escape these paradoxes, we need to find a way to make p relevant to q in some way.  And the most plausible way to do this, I assert, is to insist that p be information that q.  End Of Aside.)

To undermine my initial intuition further, suppose that one has obtained information that all of the apples in the pile — both yellow and red — are wormy.  In that case, should one (blindfolded) handle each apple in turn and say ‘If this apple is red then it is wormy’, his statement would be (I venture) false.  For the redness of the apple is, in this situation, no longer what excludes the possibility that it is not wormy, or, put another way, is no longer the factor that renders as 1 the probability that the apple is wormy.  That factor is now the fact that the apple is from this pile, not that it is red.  Since the apple’s being red is no longer relevant to its being wormy (is no longer what makes the probability the apple is wormy 1), one cannot learn from, conclude from, infer from its being red that it is wormy. The apple’s being wormy no longer hinges on its being red. The statement is now false for exactly the same reason that ‘If Paris is the capital of France then the earth has one moon’ is false.

One might try to preserve a version of the intuition that the measles and wormy red apple statements are true regardless of anyone’s knowledge by proposing that these are true independently of what any finite intelligence knows or doesn’t know.  What if there were an infinite intelligence — a God who knows everything in general, and the measles status of Herman’s children, the worminess status of the red apples in the pile, and the location of the peanut under the fourth shell in particular.  One could then accurately say the ‘measles’, ‘wormy red apples’, and ‘the third shell proves empty’ statements are true objectively, that is to say, sub specie aeternitatis, even if they are true or false as the case may be, from the subjective standpoints of this or that finite intelligence.

The analogy would be with Galilean motion studied in High School physics.  An object may be moving at 10 miles per hour given one reference frame and 60 miles an hour given another reference frame; nonetheless, there was to be some absolute reference frame embracing all of them which would let one give an absolute, non-relative value to the object’s speed.

But the intuition cannot be rescued this way.  For clearly, nothing could ever be a signal, could be information-that, for an infinite intelligence that knew everything.  Such an intelligence with its penetrating x-ray vision would already know, for example, that the peanut was located under the fourth shell.  Given this knowledge, the third shell’s proving empty would not reduce to 1 for this intelligence the number of possibilities regarding the location of the shell.  For the number of such possibilities was already 1 for this intelligence.  Likewise, for this all-knowing intelligence, that this particular layabout is a child of Herman’s would do nothing to reduce to 1 the probability that this person has the measles.  Nor would the fact that this particular apple is red reduce for this intelligence the number of possibilities regarding the worminess status of the apple from 2 (the apple is wormy or non-wormy) to 1 (the apple is wormy).  With no reduction of possibilities, there is no signal carrying information-that in any of these cases.

God’s knowledge cannot serve as the equivalent in logic of the Galilean absolute reference frame.

Not only is information-that relative to what one already knows, it also requires finitude.  No limitation on one’s knowledge — no hiddenness — no information-that.  And if the truth of If p Then q statements requires that the occurrence of p be information that q, the truth of these statements also require finitude.

One final note:  how can one account for the illusion (if it is that) that both the measles and the wormy red apply statements are true regardless of what one already knows?  I think the answer lies in the fact that, after completely talking through one’s hat at time 1 with the statement “If this apple is red, then it is wormy,” one were later at time 2 to examine all of the red apples and discovered they were all wormy (and that just some of the yellow apples were), it would seem that, since the statement is true at time 2, it would have to have been true at time 1.  The truth value of a statement like this can’t change, can it?  Maybe we would prefer to accept the paradoxes of Material Implication after all.  But it seems to me that one should accept that, at least in the case of the ‘third shell proves empty’ statement, the truth value of that statement can change with time as one obtains more knowledge (you later get information that the first and second shells also proved to be empty).  So the truth value of the measles and wormy red apples statements changing over time should not prove to be an absolute obstacle.

     *****

The entire point of this exercise is not just to make grandiose metaphysically-existentialist-sounding statements such as ‘logical implication requires finitude’ (although I must admit this is one of my aims), but also to escape from Classical Logic’s paradoxes of Material Implication by insisting that there must be some relation between p and q that makes p relevant to q, and that this relation consists in p‘s being information that q.

In the previous post, I noted two apparent counterexamples (the measles and wormy red apple statements) that would seem to preclude identifying this hoped-for relevance-making relation with information-that.  These statements seem to be true even though in these cases p is not information that q.  Also, identifying this relation with information-that would make the truth of IF-THEN statements relative to what one already knows, an implication that may make one prefer the paradoxes of Classical Logic’s Material Implication.

In this post, I employ the ‘third shell proves empty’ statement, as well as the close connection (I claim) that IF-THEN statements have with what one can learn from, infer from, or conclude from a situation to remove whatever counter-intuitiveness might adhere to the notion that the truth of IF-THEN statements is relative to what one knows.  (Of course what one can learn, infer from, conclude from a situation depends upon what one already knows.  Of course the truth/falsity of ‘the third shell proves empty’ statement depends as well upon what one already knows.)  If one can accept the relativity of IF-THEN statements, they will be in a better position to accept the idea that those cases in which p is not information that q (the redness of the apple sometimes fails to be information that the apple is wormy; that this person is a child of Herman’s sometimes fails to be information that this person has the measles)  are also cases in which If p Then q is false.

This leaves the third difficulty mentioned in the previous post:  what to do about the statement If p Then p?  Is a channel of information supposed to exist between p and the self-same p?

Do I have a song and dance that will eliminate this difficulty?

*****

Today’s homage to Plato’s SYMPOSIUM is the soccer player James Rodriguez.

James_Rodriguez

From math teachers to soccer players…How can anyone get anything at all done with beauty like this walking the earth?


Measles And Wormy Red Apples: IF-THEN Statements And INFORMATION THAT (An Apparent Counter-Example)

It would seem that there are some clear counterexamples to the idea that If p Then q is true when p is information that q.

Consider the following (somewhat gruesome, in the light of the irresponsibility of our contemporary anti-vaxxers) measles example from Fred Dretske.  Dretske, by the way, does not discuss this example in the light of IF-THEN statements.

…an exceptionless uniformity … is not sufficient for the purposes of transmitting information.  Correlations, even pervasive correlations, are not to be confused with informational relations.  Even if the properties F and G are perfectly correlated (whatever is F is G and vice versa), this does not mean that there is information in s’s being F about s‘s being G (or vice versa).  It does not mean that a signal carrying the information that s is F also carries the information that s is G.  For the correlation between F and G may be the sheerest coincidence, a correlation whose persistence is not assured by any law of nature or principle of logic.  All Fs can be G without the probability of s‘s being G, given that it is F, being 1.

To illustrate this point, suppose that all Herman’s children have the measles.  Despite the “correlation,” a signal might well carry the information that Alice is one of Herman’s children without carrying the information that Alice has the measles.  Presumably the fact that all Herman’s children (living in different parts of the country) happened to contract the measles at the same time does not make the probability of their having the measles, given their common parentage, 1.  Since this is so, a signal can carry the information that Alice is one of Herman’s children without carrying the information that she has the measles despite the fact that all Herman’s children have the measles.  It is this fact about information that helps to explain (as we will see in Part II) why we are sometimes in a position to see that (hence, know that) s is F without being able to tell whether s is G despite the fact that every F is G.  Recognizing Alice as one of Herman’s children is not good enough for a medical diagnosis no matter what happens to be true of Herman’s children.  It is diagnostically significant only if the correlation is a manifestation of a nomic (e.g., genetic) regularity between being one of Herman’s children and having the measles.

Fred Dretske, KNOWLEDGE AND THE FLOW OF INFORMATION, Stanford, CSLI Publications, 1999, pp. 73-74

Myself, I would rather choose a less gruesome (given the sometimes horrific consequences of measles), even if still somewhat gross, example.  Suppose that there is a pile comprising red and yellow apples in my grandfather’s orchard.  By pure chance, some of the yellow apples happen to be wormy, while all of the red apples are so.  Given his measles example, Dretske would surely claim that just the fact that a given apple from the pile is red would not constitute information that the apple is wormy.  But suppose that, blindfolded, I handle each apple in the pile one by one, saying each time:

If this apple is red, Then it is wormy.

In my mind’s inner ear, my intuition is shouting to me:  “This is TRUE TRUE TRUE TRUE TRUE!!!!!!”

Likewise, surely the following statement is also true:

If this person loitering here in front of my shop among all these other disreputable-looking lay-abouts is a child of Herman’s, Then she has measles.

This statement would be true, it (strongly) seems to me, even if the person uttering it is talking completely through their hat, even randomly, and has absolutely no evidence that ‘this person’ has the measles, or that she is a child of Herman’s, or that there is any connection at all, even an accidental one, between Herman’s children and the measles.

Therefore, there would seem to be clear cases in which an If p Then q statement is true even when the occurrence of p is not information that q.

Nonetheless, I (at least as of this writing) think I can show in a later post that Dretske’s discussion of the relativity of information drastically undercuts what he thinks his measles example shows.  (I am also thoroughly confident, by the way, that if my doubts are valid, they have already been discussed a thousand times already by everyone and their uncle.)  So the idea that what makes p relevant to q in any true If p Then q statement is an informational relation . . . this idea might find a rescuer after all.

 *****

I hope that today’s homage to Plato’s SYMPOSIUM has never suffered from the measles.  This gorgeous hunk is a math teacher in Great Britain (perhaps hailing ultimately from Italy) who moonlights as a model.

pietronew

I am confident that this math teach will inspire many of his students, both male and female, to start the ascending the platonic ladder whose lowest rung consists in the contemplation of the Beauty of Gorgeous Guys, whose next rungs consist in the contemplation of the Beauty of Math and Logic, and which finally leads to the contemplation of the Form of Beauty Itself.

For now, however, I will linger a bit at the lowest rung, the Contemplation of the Beauty of Gorgeous Guys.  I will get to the Form of Beauty Itself sometime.


IF-THEN Treated As INFORMATION THAT

Relevant Logic tries to resolve the following paradoxes of Classical Logic’s Material Implication by insisting that for any If p Then q statement, p must be relevant to q:

If Cliff Wirt resides in Houston, Texas, Then the earth has just one moon.

If Calypso music originated in Wisconsin, Then the earth has two moons.

According to Classical Logic, both of the above statements are true because they fulfil the truth-functional requirements of true IF-THEN statements.  (T T and F F.  According to Classical Logic, F T also yields a true IF-THEN statement; the only truth-table combination that yields a false IF-THEN statement is T F.)  Nonetheless, one may be excused if they think that regarding the two statements as true is a bit paradoxical, to put it mildly.  One cannot conclude, infer, or learn from Cliff Wirt’s residing in Houston that the earth has just one moon.  Even less can one conclude, infer, or learn from the “false fact” that Calypso music originated in Wisconsin the equally “false fact” that the earth has two moons.  One would think that both IF-THEN statements are false because in both, the antecedent, p, is irrelevant to the consequent, q.

So the truth-functional account of the IF-THEN statement has to go, I am thoroughly persuaded, because it can take into account only the truth or falsity of the antecedent and consequent, leaving completely out of view the relevance of the antecedent to the consequent.

What, then, would make the antecedent relevant to the consequent?  What is the relation between p and q when we say If p Then q?  I am partial to the hypothesis that the relation is informational.  If p Then q is true when the occurrence of p is information that q.  If the doorbell is ringing, then someone or something outside has depressed the button; that the doorbell is ringing would be information that someone or something outside has depressed the button.  The first is information that the second because there is a channel of information extending from the button to the ringing sound, such that, when that channel is in good working order, the probability that the button is being depressed is 100% when the ringing sound occurs.

Because this informational relation exists between the ringing sound and the button’s being depressed, one can conclude from, infer from learn from the doorbell’s ringing that someone or something is depressing the button outside.  So — oh my god! — there is a close affinity between If p Then q and p’s being information that q.

There are, however, several obstacles in the way of treating the IF-THEN statement as an informational relation.

First, how would one deal with If p then p?  Is there somehow supposed to be a channel of information between p and itself?

Second, there are (seemingly) clear cases in which If p Then q is true when p is most definitely not information that q.

Third, the informational relation is both intentional and relative, as described by Fred Dretske in his KNOWLEDGE AND THE FLOW OF INFORMATION.  Treating If p Then q as an information relation would make implication both intentional and relative.  The very same If p Then q statement would be true inside some frameworks and false inside others.  Rather than accept this, some would perhaps rather accept Classical Logic’s paradoxes of Material Implication.

*****

Today’s homage to Plato’s SYMPOSIUM takes the form of a very kalos Bruno Mars.  According to Plato, one ascends a ladder whose first rung consists in the beauty of gorgeous young men, whose middle rungs consist in the beauty of things like Classical and Relevant logic, and whose final rung consists in the Form of Beauty Itself.

bruno-mars-promo

I will get to adoring the Form of Beauty Itself eventually.  For now, I will content myself with adoring the Form of Bruno Mars.