# I have a small question about logic. In my text, "3 is less than or equal to pi" is translated as PvQ, where P is "3 is less than pi" and Q is "3 is equal to pi." Seems simple enough. But why isn't the statement better translated as (PvQ)&~(P&Q)? Of course, if you know what "less than" and "equal to" really mean, you'll understand that P&Q is precluded; but it bothers me that this is not explicitly stated in the translation. Someone who understands logic but not English might infer from PvQ that 3 may be simultaneously "less than" and "equal to" pi, and this strikes me as problematic.

Just to be sure I'm addressing your worry: it's often said that there are two senses of "or": an inclusive sense, where "P or Q" means "At least one of the statements 'P' and 'Q' is true, and an exclusive sense, where "P or Q" means "exactly one of the statements 'P' and 'Q' is true." Let's suppose I'm the sort of person who makes it a practice of always using "or" in the inclusive sense. Someone who knows this hears me say: "Mary is in San Francisco or in New York City." The logic of my statement doesn't rule out all by itself the possibility that Mary is in both places. What rules that possibility out are the facts of geography and of how people fit into space and time. (It's been claimed that some saints were capable of bilocation, but we'll assume that Mary is, at least in that respect, no saint.) Could someone who knew that I'm an inclusive "or" sort of guy but didn't know much about geography and the relationship between people and space correctly infer that if my statement is true, then Mary...

# As a young philosophy fanatic attempting to get to grips with the incumbent philosophical zeitgeist's obsession with logic as the source and answer to all its 'problems', I am having trouble finding any substantial reason for the unwavering authority and importance with which this analytic and logical character is treated within the whole of philosophical academia. Where is the incontestible evidence for such an incontestible reverence of such fundamental logical principles as the law of non-contradiction, other than within human intuition and common sense?

Before getting to your question, just an observation: all the philosophers I know believe that they should reason well and steer clear of contradiction, but I don't know any who think that logic is either the source of or the answer to all our philosophical problems. In any case, I'm not sure what would do the trick here. If I'm going to give you "evidence" for the law of non-contradiction, then presumably I'm going to have to reason from the evidence. And I don't know how to reason to the conclusion that one thing is so rather than another unless I take it for granted that contradictions can't be true. Unless you already assume the law of non-contradiction, you could reply to any argument I give for it by saying "I agree it's also true that sometimes a statement and its denial both hold. And in particular, even though the law of contradiction is true, it's also false." I don't really know what would count as "evidence" that the law of non-contradiction is true -- especially if I'm not allowed...

# My question is following: can we estimate how many validities (formulas that are always true) are there among all formulas of propositional logic? Is there a method of doing it?

As it turns out, the answer is easy: there are aleph-null tautologies (formulas true in every row of a truth table) in any standard system of propositional logic -- for sort, in SC (sentential calculus). Here "aleph-null" is the number of integers. Here's a sketch of a proof. First, how many formulas are there in SC? Infinitely many, of course. But it's possible to set up a function that pairs each formula in SC with a unique positive integer. (There are many ways to do this, in fact.) So there can't be any more formulas than there are integers; no more than aleph-null. But some standard arguments tell us that every infinite subset of the integers has the same number of members (aleph-null) as the set of integers itself. In the usual terminology, every infinite subset of a countable set is itself countable. (I recommend as an exercise thinking about how that might be proved.) So, we know that there are aleph-null formulas of SC. But we also know that there are infinitely many tautologies....