Logic

Consider the implication (A&B=>R)=>(A=>R | B=>R). If we were to assign statements to these variables (as most Discrete Structures teachers will do to try to make logic more intuitive), we might be inclined to disagree that this statement is valid (i.e. a tautology). After all, if we let statement A be "over 18" and statement B be "male" and R be "draftable", it says that "over 18" and "male" together are what makes a person draftable, but not either of those alone.

If we look at the proof, though, or even just a truth table, we find that the statement above is valid. What does this mean? It means that if A&B is sufficient evidence for R, then either A alone or B alone is sufficient evidence for R, but we cannot really tell which it is. Note that the converse does not hold.

So how does this work? Suppose we have evidence for A&B=>R (if we do not, then the statement is vacuously true anyway). Then suppose we have no evidence for A&B. Then we have no evidence for one of A or B, and that means one of A=>R, B=>R is vacuously true. Note that at this point, we encounter the problem of not knowing which of A or B fails, just that one does. Then suppose we have evidence for A&B. Since we have evidence for A&B=>R and for A&B, modus ponens tells us we can conclude R.

Here's the tricky part, where intuition breaks down. Since we have evidence for A&B, we must have evidence for both A and B separately. Our statement says knowing this evidence for both A and B separately is sufficent to conclude that R is a necessary conclusion from just one. Or does it?

This statement might just be the result of the non-discrimitive classical form of implies, which is logically equivalent to not A | B. Taking this into account, the antecedent (A&B=>R) is a limitation on the relation between A, B, and R. If A and B are true but R is false, the whole statement is vacuously true. If one of A or B is false, the whole statement is vacuously true. Thus the only information this statement has is when A, B, and R are true. But then the implications hold by grace of T=>T is T. What this makes is more of a relation than a strict necessary/sufficient pairing, a coincidence of construction rather than a statement with actual meaning.