# Category Archives: Plato

## 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:

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:

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.

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.

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.

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

## Semantic Arguments vs. Adjuncts (Revised)

This is a version of the post below, revised so as to try to eliminate a number of confusions.

The Wikipedia article Argument (linguistics) starts its discussion of the argument/adjunct distinction by asserting that an argument is what is demanded by a predicate to complete its meaning, while an adjunct is not so demanded.  For example, if someone asks me “What is Joe eating?” my answer would be drastically incomplete if I replied “eats.”  My answer would still be drastically incomplete if I supplied just one argument, ‘Joe’, to say ‘Joe eats.’  Only when I supply a second argument, say, ‘a fried egg’, would my reply not create a sense of a question ludicrously left hanging and an answer simply not given.  The predicate _eats_ has two parameters ( shown here as ‘_’) demanding two arguments, such as  ‘Joe’ and ‘a fried egg’ for my reply to make any sense.

( This example, of course, is my own; I am offering it (maybe tendentiously?) in order to make drawing certain conclusions more natural. )

‘[I]n the kitchen’, however, is an adjunct, since nothing would be left ludicrously left hanging in the air were I to leave that phrase out of the proposition “Joe eats a fried egg in the kitchen.”  The predicate eats does not have a parameter demanding something like ‘in the kitchen’ as an argument.

This criterion — i.e., what is demanded by a predicate to complete its meaning … henceforth I will call this the ‘demands criterion’ — runs into trouble when one notices that sometimes eats demands two arguments, but sometimes demands just one.  One might say:  “Joe goes into the kitchen.  Joe is ravenous.  Joe sees food.  Joe eats.”  ( Imagine a novelist or short-story writer working in a certain style.)  The argument ‘a fried egg’ is not demanded in this particular piece of discourse.

But if ‘a fried egg’ is an argument, not an adjunct to eats, it would seem one would  have to abandon the ‘what is demanded by a predicate to complete its meaning’ criterion and find another criterion for what is to count as an argument and what is to count as an adjunct.  This a contributor (doubtlessly not the same person who put forward the ‘demands’ criterion) to the Wikipedia article cited above tries to do.

But if one wants to retain the demands criterion, they (I am intentionally using ‘they’ as a genderless singular pronoun) can assert that two different predicates, each with a different number of parameters, may get invoked when someone utters  ‘eats’ in a stretch of discourse.  Sometimes the one-place predicate _ eats is invoked, sometimes the two-place predicate _eats_.   Which predicate one uses is optional, depending upon what they feel is called for by the situation and what they want to do with the predicate.  Sometimes the context forces one to use, for example, the two-placed predicate (for example, in answer to the question ‘Joe is eating what?’; sometimes which predicate one invokes is purely a matter of choice.

If all of the predicates demand a certain argument (for example, ‘Joe’ in ‘Joe eats’), what is so demanded is an argument that is not also an adjunct.  If not all of the predicates demand a given argument (‘fried egg’, ‘in the kitchen’), that argument is an adjunct.  In this way, the demands criterion is rescued.

I picture the relations formed by these predicates as follows:

One-place relation formed by _eats:

EATS
PERSON_EATING
PERSON( NAME(‘Joe’) )
PERSON( NAME(‘Juan’) )
PERSON( NAME(‘Kha’) )
PERSON( NAME(‘Cliff’) )

Here the key is, of course, PERSON_EATING.  The ellipses ‘…’ indicate all the further tuples needed to make this relation satisfy the Closed World Assumption.  (The Closed World Assumption states that a relation contains all and only those tuples expressing the true propositions generated by completing the predicate with the relevant argument(s).)

Two-place relation formed by _eats_:

EATS
PERSON_EATING FOOD_ITEM_BEING_EATEN
PERSON( NAME(‘Joe’) ) FOOD_ITEM( NAME(‘This fried egg’) )
PERSON( NAME(‘Khadija’) ) FOOD_ITEM( NAME(‘This souffle’) )
PERSON( NAME(‘Juan’) ) FOOD_ITEM( NAME(‘This fajita’) )
PERSON( NAME(‘Kha’) ) FOOD_ITEM( NAME(‘This bowl of Pho’) )
PERSON( NAME(‘Cliff’) ) FOOD_ITEM( NAME(‘This plate of Thai food with a 5-star Thai-spicy rating’) )

Here the relation formed by _eats_ is a subtype of the supertype formed by _eats.  That is to say, PERSON_EATING is a unique key in this relation, but it is also a foreign key to the PERSON_EATING attribute of the relation formed by _eats.

This means of, course, that in each tuple there is just one thing that the person is eating.  This constraint would be natural enough if one restricts the now of the present tense eats enough so that only one thing could possibly be getting eaten, for example, the egg one piece of which Joe is now bringing to his mouth via a spoon.  But, of course, if one stretches out this now enough so that our hypothetical author could write:   “Joe goes into the kitchen.  Joe is ravenous.  Joe eats a fried egg, an apple, and a salad,” one could not treat the one-place relation as a subtype of the two-place relation.  I think the solution in this case would be to treat what gets eaten as a meal, a meal comprising one or more items.  The meal then could be treated relationally the way an order and its order-items get treated, the orders going into one relation, and orders and order-items going into another, with the orders and order-items together comprising a unique key.

The predicate _eats_ _ (as in ‘Joe eats the fried egg in the kitchen’) can be treated the same way.  And so on for any number of possible adjuncts that a predicate might accept.

If I can get away with this move, then, an adjunct would be any argument that is 1) accepted by a predicate in which the corresponding relation is a subtype of another relation, and 2) the parameter which takes that argument corresponds to an attribute in the subtype relation which is not a foreign key of the supertype relation.  An adjunct then is one kind of argument.  Non-adjunct arguments (arguments that are just arguments, arguments simpliciter) correspond to a unique key in a supertype relation; adjuncts in turn are arguments not corresponding to any attributes in the subtype relations that are foreign keys to that unique key in the supertype relation.

Notice how this treatment of arguments vs. adjuncts (that is to say, arguments that are just arguments and arguments that are also adjuncts) corresponds to the way “optional (nullable) columns” in SQL tables get turned into actual relations, which cannot contain “null values”:

SQL Table (what is eaten is an optional or “nullable value”):

EATS
PERSON_EATING FOOD_ITEM_BEING_EATEN
Joe  Fried egg
Juan
Kha Bowl of Pho
Cliff
…

Here PERSON_EATING is a not-null column, and FOOD_ITEM_BEING_EATEN is a “nullable” column.

This looks like a single relation with an optional parameter (FOOD_ITEM_BEING_EATEN).  So if one both accepts the demands criterion and takes the  SQL table as their cue, PERSON_EATING would be an argument because it is not optional, i.e., always demanded and FOOD_ITEM_BEING_EATEN would be an adjunct because it is optional.  But then one has no way of accounting for when FOOD_ITEM_BEING_EATEN isn’t optional — for example in answering the question ‘what is Joe eating’?  (Compare with the COMMISSION column in the EMP table of Oracle’s sample SCOTT schema when the employee is a salesman.)  One would either have to try to explain away — an impossible task? — the times when eats surely seems to demand not one, but two arguments, or they would have to give up the demands criterion as the way to distinguish between arguments and adjuncts.

But of course SQL is confused.  The SQL table above is mushing together two different relations, the relation formed by _eats and the relation formed by _eats_.  Disentangle the two relations, and you get a two-fer.  You get rid of the nulls, and you also rescue the demands criterion for distinguishing between arguments simpliciter and arguments that are adjuncts.

When you disentangle the relations, you can see that what is optional, when one is talking about adjuncts, is not the attribute value (e.g., fried egg), but which predicate one invokes when they say eats.  To put it a different way, the attribute value is optional only because the predicate is.

I submit, then, that treating a verb as invoking different predicates whose corresponding relations are involved in subtype/supertype relationships does away with the confusing situation that challenges the demands criterion:  i.e., the initially confusing fact that sometimes an argument seems to be demanded for the verb, and sometimes it seems not to be.

Today’s homage to Plato’s SYMPOSIUM is Channing Tatum (aka Magic Mike) again, as in the previous post.

How can anyone get anything done with such beauty walking the earth?

The Wikipedia article Argument (linguistics) starts its discussion of the argument/adjunct distinction by asserting that an argument is what is demanded by a predicate to complete its meaning, while an adjunct is not so demanded.  For example, if someone asks me “What is Joe eating?” my answer would be drastically incomplete if I replied “eats.”  My answer would still be drastically incomplete if I supplied just one argument, ‘Joe’, to say ‘Joe eats.’  Only when I supply a second argument, say, ‘a fried egg’, would my reply not create a sense of a question ludicrously left hanging and an answer simply not given.  The predicate _eats_ demands two arguments, such as  ‘Joe’ and ‘a fried egg’ for my reply to make any sense.

( This example, of course, is my own; I am offering it (maybe tendentiously?) in order to make drawing certain conclusions more natural. )

‘[I]n the kitchen’, however, is an adjunct, since nothing would be left ludicrously left hanging in the air were I to leave that argument out of the proposition “Joe eats a fried egg in the kitchen.”  The predicate eats does not demand that argument.

This criterion — i.e., what is demanded by a predicate to complete its meaning … henceforth I will call this the ‘demands criterion’ — runs into trouble when one notices that sometimes eats demands two predicates, but sometimes demands just one.  One might say:  “Joe goes into the kitchen.  Joe eats.”  ( Imagine a novelist or short-story writer working in a certain style.)  Although one could just as well say “Joe goes into the kitchen.  Joe eats a fried egg”, the argument ‘a fried egg’ is not demanded in this particular piece of discourse.

So if one wants to maintain that the predicate eats takes two arguments, they would  have to abandon the ‘what is demanded by a predicate to complete its meaning’ criterion and find another criterion for what is to count as an argument and what is to count as an adjunct.  This a contributor (doubtlessly not the same person who put forward the ‘demands’ criterion) to the Wikipedia article cited above tries to do.

But if one wants to retain the demands criterion, they can assert that two different predicates may get invoked, depending upon the context, depending upon the circumstances, when someone utters the word ‘eats’ in a stretch of discourse.  ( I am not clearly distinguishing between predicate and word here; perhaps I don’t necessarily need to just right here.)  When one invokes the predicate in order to answer the question ‘What is Joe eating?’, invoking the predicate creates a proposition, or tuple, in a 2-place relation.  In circumstances in which nothing is left ludicrously hanging in the air when one says ‘Joe eats’, the predicate creates a proposition, or tuple, in a 1-place relation.  There are two different predicates that may get invoked when one utters ‘eats’.  And depending upon which predicate gets invoked, ‘a fried egg’ is either an argument or an adjunct.

Two-place relation (demands what is eaten to complete the meaning):

EATS
PERSON_EATING FOOD_ITEM_BEING_EATEN
PERSON( NAME(‘Joe’) ) FOOD_ITEM( NAME(‘This fried egg’) )
PERSON( NAME(‘Khadija’) ) FOOD_ITEM( NAME(‘This souffle’) )
PERSON( NAME(‘Juan’) ) FOOD_ITEM( NAME(‘This fajita’) )
PERSON( NAME(‘Kha’) ) FOOD_ITEM( NAME(‘This bowl of Pho’) )
PERSON( NAME(‘Cliff’) ) FOOD_ITEM( NAME(‘This plate of Thai food with a 5-star Thai-spicy rating’) )
PERSON( NAME(‘Cliff’) ) FOOD_ITEM( NAME(‘This strip of bacon’) )

Here the key is composite, comprising both PERSON_EATING and FOOD_ITEM_BEING_EATEN, since we would may want to answer the question “What is Cliff eating?’ with “Cliff eats a fried egg and Cliff eats a strip of bacon.”

One-place relation (does not demand what is eaten to complete the meaning):

EATS
PERSON_EATING
PERSON( NAME(‘Joe’) )
PERSON( NAME(‘Juan’) )
PERSON( NAME(‘Kha’) )
PERSON( NAME(‘Cliff’) )

Here the key is, of course, PERSON_EATING.

Sometimes what Joe eats is a ‘core element of the situation’, sometimes it is not.  In a possible world there exists a tribe for whom the amount of  energy pounded into the ground by John’s running is a core element of the situation runs, such that something is left ludicrously hanging in the air when one simply says ‘John runs’ and not (to invent a new syntactic marker, ‘tha’, which expresses ‘the energy absorbed by the ground when John runs”’, just as ‘to’ expresses ‘the place to which John ran’ ) ‘John runs tha 1,000 <<some unit of energy>>’.

When what is eaten is an adjunct, not an argument, one can, I think, treat the attribute PERSON_EATING in the two-place relation as a foreign key dependent upon the  PERSON_EATING attribute in the one-place relation.   would be both a unique key in that relation and a foreign key to the one-place relation.  This kind of design is, of course, how one would avoids “nulls” or “optional values” in a SQL table like the following:

SQL Table (what is eaten is an optional or “nullable value”):

EATS
PERSON_EATING FOOD_ITEM_BEING_EATEN
Joe  Fried egg
Juan
Kha Bowl of Pho
Cliff
Cliff

Yes — there is a certain oddness, a certain ugliness, to having Cliff suffer from two “null values”.  Maybe there is something fishy about the SQL idea of a “null value”?  But the SQL table does convey the idea that an adjunct is an optional value, while an argument is required.  After conveying this idea, we can get rid of the SQL table with its dubious nulls and replace it with the two-place relation EATS whose PERSON_EATING attribute is a foreign key to the one-place relation.

EATS
PERSON_EATING FOOD_ITEM_BEING_EATEN IN ORDER TO
PERSON( NAME(‘Joe’) ) FOOD_ITEM( NAME(‘This fried egg’) ) REASON( NAME(‘Gain Nutrition’) )
PERSON( NAME(‘Khadija’) ) FOOD_ITEM( NAME(‘This souffle’) ) REASON( NAME(‘Gain Nutrition’) )
PERSON( NAME(‘Juan’) ) FOOD_ITEM( NAME(‘This fajita’) ) REASON( NAME(‘Gain Nutrition’) )
PERSON( NAME(‘Kha’) ) FOOD_ITEM( NAME(‘This bowl of Pho’) ) REASON( NAME(‘Gain Nutrition’) )
PERSON( NAME(‘Cliff’) ) FOOD_ITEM( NAME(‘This plate of Thai food with a 5-star Thai-spicy rating’) ) REASON( NAME(‘Show how macho he is’) )
PERSON( NAME(‘Cliff’) ) FOOD_ITEM( NAME(‘This plate of Thai food with a 5-star Thai-spicy rating’) ) REASON( NAME(‘Show how much pain and suffering he can endure’) )
PERSON( NAME(‘Cliff’) ) FOOD_ITEM( NAME(‘This strip of bacon’) ) REASON( NAME(‘Indulge in a guilty pleasure’) )

Here of course, the key is PERSON_EATING, FOOD_ITEM_BEING_EATEN, and IN_ORDER_TO.

This is the way of treating the argument/adjunct distinction that I prefer at the moment, possibly with no good argument for preferring this way to the alternative. The alternative that is at the back of my mind as I write this is something like the following:  there is only one predicate eats, which is a two-place relation.  Or rather, there is only one primary, non-derived predicate eats.  In those cases in which the what-is-eaten argument is optional (so we are giving up the demands criterion for what is to count as an argument), we are projecting on the relation EATS on the PERSON_EATING attribute, to generate propositions such as “Joe eats something.”

EATS(1)
PERSON_EATING SOME ATTRIBUTE
PERSON( NAME(‘Joe’) ) Some thing or things
PERSON( NAME(‘Khadija’) ) Some thing or things
PERSON( NAME(‘Juan’) ) Some thing or things
PERSON( NAME(‘Kha’) ) Some thing or things
PERSON( NAME(‘Cliff’) ) Some thing or things

Here I envisage the demi-urge performing the needed projection by ignoring the FOOD_ITEM_EATEN attribute (perhaps even forgetting there is such an attribute in the relation), then, in order to avoid duplicates (we don’t want our demi-urge to be seeing double!), collapsing what had been two appearances of Cliff into just a single appearance.

The picture of relations above may be pretty (forget the picture of the SQL table — that is definitely not pretty…nothing connected to SQL ever is), but even prettier is  Channing Tatum aka Magic Mike, who is today’s homage to Plato’s SYMPOSIUM:

Notwithstanding all of my rapturous sighs at the moment, my sole interest in Magic Mike is, of course, as a stepping stone first, to the Relational Algebra, and then, ultimately, to the Platonic Form of Beauty.

## Paul Vincent Spade On Motivating The Mediaeval Problem Of Universals

“It is well known that the problem of universals was widely discussed in mediaeval philosophy — indeed, some would say it was discussed then with a level of insight and rigor it has never enjoyed since.” What follows is an extremely good motivation of the medieval problem of universals, offered by Paul Vincent Spade in the introduction to FIVE TEXTS ON THE MEDIAEVAL PROBLEM OF UNIVERSALS.

“It is easy to motivate the problem of universals. Consider these two capital letters: A A. Ignore everything else about them and for now observe only that they are of the same color: they are both black.

As you look at the two letters, how many colors do you see?  Two different answers are plausible.  You may want to say  you see only one color here, blackness.  You see it twice, once in each of the two capitals, but it is the same color in both cases.  After all, did I not just say the two letters were “of the same color“?  Isn’t that obvious by just looking at them?  This single blackness is the kind of entity that is repeatable, found intact in both letters at the same time; it is what philosophers call a “universal.”  If this is your answer, then you believe in the reality of at least one universal, and are in that sense a “realist” on the question.

But now reset your mental apparatus and look at the two letters again.  On second glance, isn’t it obvious that you see two colors here, two blacknesses:  the blackness of the first A, this blackness, and then the blackness of the second A, that blackness?  The two colors look exactly alike, yes, but aren’t they visually as distinct as the two letters themselves?  If this is your answer, then you do not believe in the reality of universals (at least not in this case) and are a “nominalist” on the question.  The problem of universals is in effect the problem of deciding between these answers. ”

Of course, my eros for the mediaeval problem of universals is just a stepping stone on the path to eros for the platonic form BEAUTY.  And the stepping stone previous to eros for the mediaeval problem of universals is, in the grand tradition of platonic philosophy, eros for gorgeous young men such as this one, who is today’s homage to Plato’s SYMPOSIUM.

If Plato can include a bad boy like Alkibiades in his SYMPOSIUM, I can include a bad boy like Josh in my post.

## Logical Pairings

In previous posts I’ve tried to interpret the canonical Tagalog sentence (e.g., maganda si Taylor Lautner) in terms of an equality relation, GORGEOUS_EQUALS_GORGEOUS.  Conceptually, the relation is formed by logically pairing each member of the set GORGEOUS (MAGANDA) to each of the members, then taking a subset of the set that results from this logical paring.  That subset comprises those logical pairings in which each member of the pair is identical with the other.

What do I mean by ‘logical pairing’?  In the real world, to pair one thing with another is to bring the two things together in some way.  One may pair, for example, some particular matte board, with its particular color, with the painting one is getting framed.  Here, the matte board and painting are getting physically paired.  Or one may pair John with Bill by picturing them in the mind’s eye as together as a couple.  Or one may pair John with John by first seeing him double (i.e., seeing him twice but simultaneously), then by realizing the two Johns are in fact one.

To get a logical pairing, abstract from any concrete form of pairing, that is, ignore any particular way in which the bringing together is done.  Ignore in fact everything about them except that they go under the heading ‘bringing together’ (since maybe that is the only single thing they all have in common.)   Then be content with the fact that, while each member of the set MAGANDA can potentially be brought together with every member of that set,  any actual pairings will be performed just every now and then, and only for a few members.  (For example, in a particular article, Dan Savage pictures Ashton Kutcher and Matt Damon together.)  A logical pairing is a bringing together in which all concrete details of the bringing together (how it is done, in what sense the things are brought together?  Physically?  In the imagination only?  By already knowing that the “objects” of one’s double vision are in fact one and the same?) are ignored.  One salient detail in particular is ignored:  is the pairing actually being done in any given instance, or is it just something that could be done?

If one does not want to rest content with each member of the set being brought together just potentially with every other member of the set, they (plural third person intentionally being used here as a neutral singular third person) are free to imagine a Demiurge ala Plato or a God ala the medievals whose cognitive capacities are sufficiently large as to simultaneously bring together in its mind’s eye every member of the set MAGANDA with every member of that set, so large, in fact, as to be able to see Matt Damon twice with the mind’s eye but already know that Matt Damon is, well, Matt Damon.

I will end by confessing that I like to think of projection as the Demiurge’s ignoring one or more attributes of a relation, and of restriction as the Demiurge’s ignoring one or more tuples in the relation.

Today, my homage to Plato’s SYMPOSIUM (first, gorgeous guys, then the Relational Algebra, then the form Beauty itself) will take the form of a concrete (not just a logical) pairing of Matt Damon and Ashton Kutcher:

Sigh.  There is too much beauty in the world.

## A Doubtlessly Lame Attempt At Explaining The Awkwardness

Caution:  The following belongs to the category of ‘let’s see how long I can get away with this before it convincingly gets shot down’.  Either that, or to the category ‘This is so obvious and has been stated so many times in the past that it is a puzzle why you bother mentioning it.’

The motivation for the following blather:  In a previous post I was bothered by the (I think true) assertion that one can reduce propositions to states of affairs…my botheration arising from the fact that while propositions are always either true or false, it seems awkward to say things like “Don’s standing to the right of Genghis Khan is a true state of affairs.”

The blather itself:  Let’s suppose that we could describe a state of affairs as either true state of affairs, or a false state of affairs, using the ‘state of affairs’ vocabulary, only if any state of affairs could be so described.  Not every state of affairs can be described as either true or false:  for example, “Guile riding his bicycle.”  This is a state of affairs that occurs at any given moment, when Guile is riding his bicycle at that moment, or that fails to occur at that moment; but it is neither true nor false.

So if we try to describe “Guile rode his bicycle a moment ago in Angeles” as a true (or false) state of affairs, we will be beating our head against the wall, for we will be using a vocabulary that cannot be applied to any state of affairs, but only to some of them.  So instead of beating our head against a brick wall, we will invent a new kind of thing:  propositions, which are either true or false.  And we will reduce propositions to states of affairs by saying propositions are nothing but a subset of the set of states of affairs, namely, those that always occur (or fail to occur).  (“Guile rode his bicycle a moment ago, i.e., at time t_1, in Angeles” is a state of affairs that will always occur if Guile did ride his bicycle during that time and at that place, or it is a state of affairs that will never occur.)

Conclusion:  This way we can reduce propositions to a subset of states of affairs without having to talk about true or false states of affairs.

My homage to Plato’s SYMPOSIUM for this post is Brad Pitt again:

How can anyone get anything done with beauty like that walking the earth?

## Oh My God, I Get Tons Of Likes When…

Oh my god, I get tons of likes when I put Ashton Kutcher in relation to a Cavafy poem, but none at all when I talk about the Relational Algebra.  🙂

Male Pilipino total gorgeousness:

This is a high-minded homage to Plato’s SYMPOSIUM, of course.