Are first principles or the axioms of logic (such as identity, non-contradiction) provable? If not, then isn't just an intuitive assumption that they are true? Is it possible for example, to prove that a 4-sided triangle or a married bachelor cannot exist? Or must we stop at the point where we say "No, it is a contradiction" and end there with only the assumption that contradictions are the "end point" of our needing to support their non-existence or impossibility?

In any "complete" logical system, such as standard first-order predicate logic with identity, you can prove any logical truth. So you can prove the law of identity and the law of noncontradiction in such systems, because those laws are logical truths in those systems. But I don't think that answers the question you're really asking: Can we prove (for example) the law of noncontradiction using premises and inferences that are even more basic , even more trustworthy than the law of noncontradiction itself? No, or at least I can't see how we could. In that sense, then, the law of noncontradiction is bedrock. Pragmatically, we can explain the law of noncontradiction in terms of related notions such as inconsistency and impossibility, but I don't think we thereby "support" the law of noncontradiction by invoking something more basic than it.

Is mathematics grounded in logic or is logic grounded in mathematics?

I leave it to the experts on the Panel (and there are several) to give you a proper answer, but I would certainly reject the second of your alternatives: I can't see how logic could be grounded in mathematics. It's a more controversial issue whether mathematics is grounded in logic and, if it is, what that grounding amounts to.

Is it possible for something that is said to be logically impossible, to be physically possible? That is, what is the "proof" that logical impossibilities cannot actually exist (if there is any such 'proof')?

By "X is logically possible," I think most philosophers mean something like "X could exist (or could have existed) or could obtain (or could have obtained) in the broadest sense of 'could', i.e., 'could' without restriction or qualification." This sense of 'could' is supposed to be compatible with 'does', so the claim that you do exist is compatible with the claim that you could exist. In fact (to get to your question), the first claim obviously implies the second claim: any X exists (or obtains) only if X could exist (or obtain). It just makes no sense to say that something is true that couldn't have been true. That's the best "proof" I think I can give. Now, some analytic philosophers calling themselves " dialetheists " say that some logical contradictions -- some propositions of the form P & not-P -- are true. But they're not properly described as saying that some logical impossibilities are true or could be true; rather, they say that not all...

What is the definition of validity under possible world semantics?

Please excuse my parentheses; I hope they don't obscure my answer. As I understand it, an inference is (deductively) valid if and only if there's no possible world in (or at) which the premise(s) is (are) true and the conclusion is false. So "Socrates is a man; therefore, there's at least one man" is a deductively valid inference, since there's no possible world in (or at) which the premise is true and the conclusion is false. Ditto for the inference "Socrates exists; Socrates doesn't exist; therefore, snow is green": Barring equivocation, there's no possible world in (or at) which the premises are both true, and so there's no possible world in (or at) which the premises are both true and the conclusion is false.

Why is C.I. Lewis' strict implication not taken seriously in this day and age? Clarence Irving Lewis was known for criticizing material implication and for instead proposing strict implication. Why is he, his criticisms, and his proposed strict implication not taken seriously today? Many contemporary logic, philosophy, and mathematical texts refer to material implication rather than strict implication.

I'd say that C. I. Lewis's strict implication is very much alive in contemporary philosophy, although often called by different names, such as "logical entailment" or "logical implication." Philosophers frequently claim (or deny) that some proposition "entails" another, by which they very often seem to mean "strictly implies." Material implication, unlike strict implication, is a truth-functional relation between propositions: given only the classical truth-values of two propositions, you can tell which one materially implies the other (material implication will run in at least one direction between them, if not both). By contrast, strict implication isn't truth-functional: it requires asking about the truth-values that propositions take in worlds other than the actual world, which invites philosophical controversy. As a result, strict implication is a less clear-cut relation than material implication. So despite its unintuitive features (which, as you say, Lewis criticized), material...

Can tautology be defined as "unnecessary repetition of information"? In other words, does tautology have the same sense as repetition? Thanks.

In my experience, not every philosopher treats repetition as essential to a tautology. Sometimes I've seen "tautology" used to denote any logical or conceptual truth, even one that doesn't contain repetition, such as "All bachelors are unmarried." But I think most would agree that any statement that's logically true at least partly because of repetition, such as "All bachelors are bachelors," counts as a tautology. Whether the repetition is "unnecessary" requires asking, "Unnecessary for what?" If you start a statement with "All bachelors are...", there are plenty of ways to finish it that won't produce a truth, but repetition will. Yet repetition isn't necessary in order to produce a truth. Furthermore, repetition isn't sufficient for truth: "2+2=5. I repeat: 2+2=5."

It has long seemed to me that philosophers do not seem concerned with illusion, i.e., the appearance of reasoning that SEEMS valid but is at least questionable if not illusory. The Greek philosophers that I read in school seemed particularly questionable. My impression was that much of their argumentation was illusory, i.e., based on claims that are unidentified assumptions. An example of illusion is the argument that since everything has a cause, there must be a FIRST cause. This SOUNDS sound but of course is not. Causality is not simple and is not a matter of logic. Causality has to do with nature and we know very little about nature. For all we know the universe has been going on forever, i.e., had no beginning. Moreover, if EVERYTHING has a cause, then there cannot be a FIRST cause which is exempt from having a cause. Are there philosophers who are concerned with this problem of illusory or unfounded philosophical reasoning? I would love to read their ideas. Please note that I'm not calling...

You wrote, "It has long seemed to me that philosophers do not seem concerned with...reasoning that SEEMS valid but is at least questionable if not illusory." I must say I find that surprising, since philosophers devote a great deal of their time (and some of them virtually all of their time) to exposing hidden assumptions, faulty inferences, equivocations, etc., in the arguments of other philosophers. Indeed, much of the progress in philosophy comes from exactly this activity. The First-Cause Argument that you mentioned is a great example. Its many versions have been subjected to detailed and powerful philosophical criticism for centuries. You'll find a helpful summary of that criticism here . Among the important objections is one that you raised: Who says the universe had a beginning? There are quasi-scientific arguments that it did in fact have a beginning (based on Big Bang cosmology) and philosophical arguments that it must have had a beginning (based on the alleged impossibility of...

What are some of the most common mistakes of reasoning or logic that you have experienced being made by non-philosophers? What are some aspects of reasoning schools should particularly focus on?

In my experience, maybe the most common mistake in reasoning committed by non-philosophers (and certainly among the most exasperating) is the one that philosopher Paul Boghossian complains about here: "Pinning a precise philosophical position on someone, especially a non-philosopher, is always tricky, because people tend to give non-equivalent formulations of what they take to be the same view " (my italics). Boghossian's complaint in this case stems from the "defense" of moral relativism offered by the literary critic Stanley Fish: "Fish, for example, after saying that his view is that 'there can be no independent standards for determining which of many rival interpretations of an event is the true one,' which sounds appropriately relativistic, ends up claiming that all he means to defend is 'the practice of putting yourself in your adversary's shoes, not in order to wear them as your own but in order to have some understanding (far short of approval) of why someone else might want to wear them.' ...

I seem to remember the "heap paradox" being a very old one (given a heap and repeatedly removing a single grain of sand, when does it stop being a heap?). Yet I don't ever recall hearing a solution to it. No doubt there are different views of things, but is there at least a generally accepted solution to this paradox?

You asked, "Is there at least a generally accepted solution to this paradox?" Not by a long shot! The paradox of the heap (and its cousins that use other vague concepts) is in my opinion one of the greatest unsolved intellectual problems. It has generated a huge philosophical literature, and it's very much a topic of current philosophical debate, but I have yet to see a proposed solution that even comes close to being satisfactory. For starters, you might take a look at these entries from the Stanford Encyclopedia of Philosophy: SEP, " Sorites Paradox " SEP, " Vagueness " Best wishes as you work your way through this daunting -- but inescapable -- problem! I think you'll find it repays your careful thought even if you don't end up much closer to a satisfying solution.