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.