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 t0 occurs 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?
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.)