I'd answer your three questions as follows. (1) Very important. (2) No: There are lively disagreements in logic concerning particular issues, and there may be few if any issues in logic on which everyone agrees. (3) Some philosophers say that different situations call for different kinds of logic. For what it's worth, I disagree: I'm not persuaded that there are any situations to which standard (or "classical") logic doesn't apply.

The last two of your three questions suggest this: We can't properly regard some law P as true (rather than merely as a guideline for thought) unless there's some more fundamental law Q that enables us to see that P is true. But presumably Q must also be something we properly regard as true, in which case your suggestion implies an infinite regress: there must be some more fundamental law R that enables us to see that Q is true. Likewise for R, and so on. This infinite regress may be a good reason to reject your suggestion. Why must our properly regarding P as true depend on there being some more fundamental law? In any case, I can't see how there could be any law more fundamental than the law of non-contradiction (LNC). Let F be any such law. If the claim "F is more fundamental than LNC " is meaningful (and it may not be), then it conflicts with the claim "F isn't more fundamental than LNC " -- but that reasoning, of course, depends on LNC.

I share your skepticism about Heidegger and his work. But, to give him a fair shake, I'd recommend reading the long and detailed SEP entry on him, available at this link . It appeared in the SEP in 2011, which is surprisingly late given Heidegger's fame and influence. (By comparison, Derrida's entry appeared in 2006, Rorty's in 2001.) Anyway, the job of the entry-writer is not only to explicate the philosopher's major ideas but also to make a case for the interest and importance of those ideas. If, after reading the entry, you're not satisfied by the explication and persuaded that the ideas are interesting and important, then I'd recommend moving on to something else in philosophy. There's plenty of good stuff to be found elsewhere.

Two replies: 1. I'd caution against equating natural with non-human , let alone with non-living . Many living beings are in their natural state despite being alive, and many (if not all) human beings are in their natural state despite being human. A hermit crab in its borrowed shell is in its natural state, and so is a human being fully clothed inside his/her house. We human beings naturally clothe and shelter ourselves. 2. Your question seems a bit like the old chestnut 'If a tree falls in the forest, and nothing is around to hear it, does it make a sound?' If a given object reflects light in the visible spectrum, but nothing is around to detect that reflected light, is the object visible? Both questions turn on how we define terms, in particular 'sound' and 'visible'. If sound is simply vibrations that would be detected by a sound-sensor were one present, then surely the falling tree makes a sound. Likewise, if 'visible' means ' would be seen by a normal human observer looking...

I can't see how it could be. Beating up someone for making sexist or misogynist comments is using physical violence to punish the commenter. That seems like literally the wrong type of reaction to merely verbal misconduct. (Notice that we don't punish slander or libel that way.) It's something like a category mistake , in addition to being a moral mistake.

I would caution against inferring from 'The principle of noncontradiction is an abstraction' to 'The principle of noncontradiction doesn't exist naturally or non-naturally'. A number of philosophers, and maybe an even larger number of mathematicians, think that at least some abstract objects must exist -- and exist non-naturally. It may be that the principle of noncontradiction is among those abstract objects. You may find this SEP entry on the topic helpful.

Having read this question and Question 5466, I think I may see what you're saying. If your opponents deny that there's a dichotomy between whatever has no consciousness at all and whatever has at least some consciousness, then they're mistaken. Maybe nothing occupies the first of those categories, but it's still a genuine dichotomy. On the other hand, if they're claiming merely that consciousness comes in degrees , then their claim is compatible with the existence of the dichotomy. Compare the real numbers, which also come in degrees (of size): -1 is smaller than 0; 0 is smaller than pi; etc. Yet there's still a dichotomy between the negative and the non-negative real numbers: no real number is both; no real number is neither. Because the real numbers are densely ordered, either there's a largest negative real number or there's a smallest non-negative real number, but not both. (In fact, it's the second option: 0 is the smallest non-negative real number, and there's no largest negative real number....

You've asked an interesting question, one related to what's often called the "Benacerraf problem" in the philosophy of mathematics (see section 3.4 of this SEP entry ). I'm not sure that the problem is peculiar to mathematics. Imagine that the philosopher tried to impress his queen by creating a colorless red object. Was his failure caused by the fact that colorless red objects are impossible? If facts about color and facts about redness in particular can have causal power, can the fact that colorless red objects are impossible have causal power? Part of the problem may be that these questions assume that we have a better philosophical grasp of the concept of fact and the concept of cause than we actually do. Given our currently poor grasp of those concepts, I don't think we should be confident that mathematical explanations or mathematical knowledge must depend on the causal power of mathematical facts.

You've asked a venerable question in epistemology, the area of philosophy that investigates knowledge and related concepts. My short answer would be "Why not?" In ordinary life, we confidently take ourselves to know things. I'm confident that I know I have hands. I'm confident that I know I'm now awake and typing at a keyboard. I'm confident that I know the surname of the current U.S. president. And on and on. What reason do I have to abandon that confidence? Now, over the centuries, various skeptical arguments have been offered that challenge the claim that we know the things we take ourselves to know. Those arguments are well worth investigating. You might start with the SEP entry on skepticism, available here .

You asked, "If the physical processes that give rise to thought are rational, how can a human being have an irrational thought?" You might be misinterpreting the claim that "the physical processes...are rational." Presumably what's meant by the claim is that the physical processes can be discovered and understood by rational means , such as empirical investigation and logical reasoning. The claim doesn't mean to attribute rationality to the physical processes themselves: the processes don't literally investigate or reason, either well or badly. So the fact that the physical processes can be discovered and understood rationally doesn't imply that irrational thoughts can't result from those processes. Furthermore, we can rationally investigate the physical causes of irrational thoughts, even if science isn't very far down that road at present. In any case, we should resist the suggestion that the universe sometimes violates the laws of logic: that suggestion is either impossible or not even...