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.

Hi, what an awesome website! I have another free will related question to add to the heap! I saw an interview with Steven Weinberg, the Nobel Prize winning physicist and, I don't remember the precise phrasing, but he said something like 'I know of no law saying that nature is here to make physicists happy.' He wasn't referring to free will, but it got me thinking about something... From what I've read and heard in papers and talks (which is certainly not nearly exhaustive), it seems that their is a tendency for those who chime in on the free will issue (even professional philosophers) to approach it from the perspective that the challenge is to show that free will does not or cannot exist. What I mean is that there seems to be a tacit presumption that since we "feel" free, the burden of proof is on those who contend we are in reality not free. I understand this perspective (and it is not unique to the free will debate), but it seems to presuppose some kind of rule that says that our feelings...

As a panelist, I'm glad you like the website. Spread the word! I don't think we need to invoke feelings in order to assign the burden of proof to those who challenge the existence of free will. In the literal sense, feelings have nothing to do with it: I don't think we have reason to believe, for instance, that causally undetermined actions have a characteristic "feel" to them that causally predetermined actions lack. But neither do we have reason to believe that they don't have a characteristic feel to them. So you're right that it's only fair to leave the phenomenology -- the literal feel of our actions -- out of it. There's also the metaphorical use of "feel," as in "I just feel that human beings sometimes act freely." Feelings in that sense too are irrelevant, I think. But the reason that those who challenge the existence of free will bear the burden of proof is that pre-philosophically -- i.e., before examining the issue philosophically -- we start with a widespread set of...

In one famous trolley case, it seems clear that the driver should divert the trolley to the spur, killing one while saving five. In another, it seems clear that a bystnader should not push the fat man off the bridge, again killing one to save five in the trolley's path. But what is the justification for my intuition? Do you see any relevant, principled difference between the two cases that would explain why I should divert the trolley yet refrain from pushing the fat man?

I think it should be noted that Professor Pogge's reply invokes, or alludes to, the controversial Doctrine of Double Effect (or else something close to that doctrine). For an account of the doctrine and of some of the controversy surrounding it, see this SEP article . One problem for the doctrine emerges when we consider a reply that B2 might make: "I didn't intend the trolley to hurt the fat man. I intended only that he stop the trolley (although of course I foresaw that his being harmed would be a side-effect of his stopping the trolley). So I didn't intend harm any more than B1 did." B2's reply seems lame, no?

It has long been recognized that free will appears to be incompatible with the causality observed in the rest of the universe. There is now evidence from neuroscience that free will does not really exist. Does the fact that many people - including many philosophers - find this conclusion absolutely unacceptable constitute a manifestation of the limits of the human mind's abilities?

My diagnosis is different: I think the arguments claiming that causality (specifically, deterministic causation) rules out free will, or claiming that neuroscientific results cast doubt on the existence of free will, are bad arguments. This issue came up recently here: see the discussion at Question 4792 , particularly the reply by Professor Nahmias and the book review he links to.

I am a 16 year old and i have been asking myself the same question for a very long time but only recently was able to finally word the question.. Isnt it true that there can not be a certainty of anything outside a person's current observed world?  It still sounds very wierd but if i am sitting in a room in a building that i walked into myself, i saw all of my surrounding as i entered the building and the room. The door is closed and there is no way for me to observe anything on the outside of the room.   I can say that i know exactly what is outside that door because i saw it as i came in the room, but in reality i have zero way of being competely certain of anything i cant see or hear outside the room. I could, potentially, be in a room floating in space and have no way of knowing, givin there isnt any ovservable evidence of my location.  It may sound strange, but i believe it could be related to particle physics, etc.  The fact is that i have no certainty of anything outside my personal observed ...

I congratulate you on your interest in philosophy at the age of 16 -- in this case, your interest in epistemology (the theory of knowledge) and, within epistemology, the issue of skepticism . You posed your question in terms of certainty. I said in response to Question 4721 that the term 'certain' works in a way that allows you to be certain that some proposition P is true even though it's logically possible that P is false. But you raised a harder question: Can you be certain that P is true when the evidence you now have is the same evidence you'd have if P were false? This question is hotly debated: there are plausible grounds for answering "No" and plausible grounds for answering "Yes." You might start your investigation by reading Peter Klein's SEP article on skepticism . It's long and challenging, but I think it will reward your patience. I recommend paying particular attention to Klein's discussion of the concept of evidence. (By the way, I don't think particle...

Recently I read a newly published very short book criticisng the concept of Free Will. I thought the book made some good points and some not-so- good points, but what really disturbed me is that the author didn't ever carefully define what he meant by Free Will. Is the definition of Free Will so obvious and clear that there is no need to define it in a book intended for lay readers?

You're right to be disturbed if the author never defines 'free will' in a book criticizing the concept of free will! One frequent obstacle to progress in discussing free will is that different parties to the discussion rely on different (and often unstated) conceptions of free will. Some think of free will as whatever is necessary and sufficient on the part of an agent to make the agent morally responsible for his/her action. Some think of it as requiring the ability to have acted otherwise than one in fact acted. Some think of it as requiring the ability to have acted otherwise than one in fact acted even under the same circumstances that preceded one's action. Those conceptions of free will aren't the same -- or aren't obviously the same. It's a good question which of those conceptions, or which other conception, best fits the way we use the concept of free will (assuming we use the concept in a stable way). It's another good question how well those conceptions stand up to scrutiny. In answering...

Eddy: Nice article! I'm glad they tapped you to review Harris's book. (I too suspect that it's the book the questioner is referring to.) --Steve