is it logically impossible for there to be an infinite regress? A lot of people make an argument and then if it leads to an infinite regress, the argument is taken to be faulty. Something like the first cause argument where the conclusion that an infinite regress occurred is to be avoided. Why is this the case? I don't see how we couldn't have an infinite regress.

There's more than one issue here, I think. It is logically possible -- near as I can tell -- for there to be an infinite regresses of causes. Someone might find it unsatisfying that A is caused by B, which is caused by C, which is caused by D... with no end. BUt there's no contradiction or incoherence here, various proponents of various First Cause arguments notwithstanding. But sometimes what's at issue is justification. If I justify my belief that A by appeal to B, and justify my belief that C, then if B and C are equally as much in need of justification as A, then I've made no progress. And if I can show that something inherent in the style of justification I've adopted is bound to generate this sort of regress of justification, then the approach I've adopted isn't going to work. The problem isn't whether or not there could be an infinite regress. The probem is that the notion of justification doesn't allow for something to be justified by way of an infinite regress, if each step in the regress...

If I am correct, the opposite of 'A' is not 'B', 'C', 'D', etc., but rather, the opposite of 'A' is 'not-A.' Likewise, the opposite of 'Green' is not 'Blue', 'Orange', etc., but rather, the opposite of 'Green' is 'not-Green.' And the opposite of 'Dog,' is not 'Cat' or 'Whale,' but rather, the opposite of 'Dog' is 'not-Dog.' And so on. However, each letter 'B' through 'Z' is not 'A' (after all, it seems, 'B' is not 'A', 'C' is not 'A'. and so on). Does that mean that 'not-A' is, or can be, or includes 'B' through 'Z'? Thus, does that mean that the opposite of 'A' is or can be 'B', 'C', etc.? Logically, I suppose, letters can stand for anything -- so perhaps 'A' is or can be equal to, say, 'B' and, therefore, 'not-A' would be equal to 'not-B,' so the opposite of 'A' might be 'not-B'. But what about objects that are not logical symbols? Cats and whales or not dogs. So, if the opposite of 'dog' is 'not-dog', and if cats and whales are not dogs, then are cats and whales the opposite of dogs? Am I missing...

The idea of an "opposite" isn't really well-defined. What you're calling the opposite (e.g., "not-dog" as the opposite of "dog") is what a logician might call the contradictory . But even though "opposite"' doesn't have a precise meaning, it's clear from the way that people use the term that it doesn't just mean the contradictory. If we want to figure out what a term means, we're well-advised to attend to how competent speakers use it. Ask any competent speaker for the opposite of "white" and she'll say "black." Ask any competent speaker for the opposite of "tall" and he'll say "short." But what can we gather from this? First, that a term and its opposite can't both apply to the same thing. Opposites are contraries . A bit more precisely, if the term "Y" is the opposite of a term "X," then "a is an X" and "a is a Y" can't both be true. However, in typical cases of opposites, they could both be false. (My pen is neither white nor black, for instance.) Still, that isn't enough. After all, ...

Consider the argument: I am more than six feet tall. Therefore, I am over five feet tall. Is this a sound argument? Is it circular? Tautologous?

Let me muddy the waters in hopes that Peter will say more. According to at least some philosophers, it is simply impossible that something should contain water without containing H 2 O. If they are right, then given the notion of validity presupposed by Peter's (1), this is a valid argument: The plastic jug in my refrigerator contains water. Therefore, the plastic jug in my refrigerator contains H 2 O. But this doesn't strike most of us as a valid argument, and it doesn't help to invoke standard notions of meaning, since "water" and "H 2 O" aren't connected by meaning . One reply would be to invoke a notion of validity of the following sort: an argument is valid if there is no argument with the same logical form whose premises are true and whose conclusion is false. On that account, the little argument about water isn't valid. Needless to say, this raises tricky questions about the notion of "logical form," but it lets us honor the intuition that the water/H 2 O...

Following along from http://www.askphilosophers.org/question/2039: "Does the law of bivalence demand that a proposition IS either true or false today? What if the truth or falsity of this proposition is a correspondence to a future event that has yet to occur?" What's problematical about saying "yes, it's either true or false, but I don't happen to know which"? Is that substantively different from saying the same thing about an open problem in science or mathematics, to which the answer is presumably knowable but happens not yet to be known? The questioner seems to be demanding both that there be an answer, which may be a reasonable thing to want, and to be able to know what the answer is, which isn't necessarily reasonable. Is it reasonable always to expect somebody (other than deity) to know the answer to a question?

The issue about so-called "future contingent" propositions isn't just about whether we're in a position to know whether they're true, but whether there are any facts for them to pick out. And that issue arises from a tempting but controversial metaphysical picture: reality as it were "unfolds" in time. Reality consists at least of what's so now, and perhaps as well of what's already taken place, but on this picture there simply are no definite facts about future events. This may seem odd at first, but a couple of examples might make it seem less so. Suppose you think that people make choices that are free in the sense of being not just uncoerced but undetermined. Mary is a juror in the penalty phase of a trial. Tomorrow she will decide whether to vote that the defendant should be executed. If you think that there is nothing that fixes her decision before it's made, you might wonder what it would mean for there to be a correct answer to the question "What will she decide?" before she actually decides...

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....