Category Archives: Let Me See If I Can Get Away With This

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.)

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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?