what is the difference between logical necessity and metaphysical necessity?

I think of logical necessity as (predictably enough) the necessity imposed by the laws of logic. So, for example, it's logically necessary that no proposition and its negation are both true, a necessity imposed by the law of noncontradiction. But one might regard logical necessity as broader than that, since one might say that it also includes conceptual necessities such as "Whatever is red is colored." Metaphysical necessity is a bit harder to nail down. Every proposition that's logically or conceptually necessary is also metaphysically necessary, but there may be metaphysical necessities that are neither logically nor conceptually necessary, such as "Whatever is water is H2O" or "Whatever is (elemental) gold has atomic number 79." Nothing in logic or in the concepts involved makes those propositions necessary, but many philosophers say that those propositions are nevertheless "true in every possible world," which is the root idea of metaphysical necessity. Even if some proposition P isn't...

The notion of something being a "fake" seems linguistically odd. Normally, if you have an adjective and a noun, the noun notes what the thing being talked about is, and the adjective describes some quality of the thing in question. A "fake plant", however, doesn't seem to fit that pattern at all, because a fake plant isn't a plant to begin with; the noun seems to be violating its intended function. Is "fake" something other than an adjective, then, perhaps analogous to "not a"? Or is a "fake plant" actually a "fakeplant", i.e. the fake is a part of the noun rather than an adjective, despite its apparent form? Doesn't the adjective "fake" somehow undermine the purpose of nouns?

I'm having trouble confirming it online at the moment, but I believe that linguists have a category for words such as fake , artificial , would-be , and the like: I think they're called "cancelling modifiers" or "cancelling adjectives." These words are well-known exceptions to the rule that, given an adjective A and a noun N, any AN is an N. I don't think they "undermine the purpose" of nouns or adjectives; instead, they perform a special and useful adjectival function in language. Anyway, you might search for information on the linguistics of cancelling modifiers or cancelling adjectives. I hope you find the clarification you're seeking.

Is logic "universal"? For example, when we say that X is logically impossible, we mean to say that in no possible world is X actually possible. But doesn't this mean that we have to prove that in all possible worlds logic actually applies? In other words, don't we have to demonstrate that no world can exist in which the laws of logic don't apply or in which some other logic applies? If logic is not "universal" in this sense, that it applies in all possible words, and we've not shown that it absolutely does apply in all worlds, how can we justify saying that what is logically impossible means the not possible in any possible world, including our actual world?

I don't understand the question, because I don't understand the phrase 'a world in which the laws of logic don't apply'. I don't think any sense can be attached to that phrase. Is a world in which the laws of logic don't apply also a world in which they do apply? If no, why not? If yes, is that same world also a world in which the laws of logic neither apply nor don't apply? If no, why not? It's as if the questioner had asked, "Don't we have to demonstrate that no world can exist in which @#$%^&*?"

Is it possible for there to be a world that logic does not apply? That is, can't a "married bachelor" actually exist in some world that there is no logic or that there is a different logic that applies? And if so, then isn't it the case that we merely assume the first principles of logic (noncontradiction, identity, excluded middle, etc...) because we observe them in our actual world, which is 1 of many possible worlds? And if it is mere assumption, then can't we be wrong about them when we say they can/should apply to other possible worlds?

I don't think this question can be answered. I think no one -- including the questioner -- understands the question being asked. In asking "Is it possible for there to be a world where logic doesn't apply?" is the questioner asking (a) "Is it possible for there to be a world where logic doesn't apply?" or (b) "Is it possible for there to be a world where logic does and doesn't apply?" or (c) "Is it possible for there to be a world where logic neither applies nor doesn't apply?" or (d) "Is it possible for there to be a world where logic does apply?" If logic doesn't apply in a world, then...then what? In a world where logic doesn't apply, does logic also apply? If not, why not? Unless logic applies in every world, how can we tell which, if any, of (a)-(d) is the question that the questioner is asking?

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.

Can someone be an atheist and do good work in the philosophy of religion? what sorts of issues would attract such a person?

Most certainly. To give just four of many living examples: William L. Rowe , J. L. Schellenberg , Graham Oppy , and Erik Wielenberg . To see which issues they find interesting, start by following those links. One needn't believe that God exists in order to find questions in philosophy of religion worth pursuing, especially since so many people at home and abroad do believe that God exists (or tell pollsters that they do) and allow that belief to guide their behavior. Atheists regard theistic belief as false, but they needn't thereby regard it as unimportant.

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...

Does a proposition which is always false such as 'one plus one equals seven' have false truth conditions or no truth conditions?

I can't see how it could have no truth conditions if it's always false : if it's always false, mustn't it have truth conditions of a particular kind, namely, truth conditions that are never fulfilled? I wouldn't call those "false truth conditions," however; I'd call them unfulfilled truth conditions or, in the case of "One plus one equals seven," unfulfillable truth conditions.