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.