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.
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 contradictions are logical impossibilities in the first place.
Again, logical possibility is usually taken to be the broadest or weakest kind of possibility, with physical possibility being a narrower or stronger kind of possibility: Everything physically possible is logically possible, but the converse needn't hold.
Fallacies are forms of reasoning that fail to provide support for the conclusions reached via that reasoning. In other words, the premises could all be true, but the conclusion still false. Just because something is a fallacy does not make the conclusion of such reasoning false, however.
For example, here is a (deductive, logical) fallacy with a true conclusion:
- If my name is Nicholas D. Smith, then I have a very common last name.
- I do have a very common last name.
- Hence, my name is Nicholas D. Smith.
All the premises are true, and so is the conclusion, but the reasoning is fallacious (called "affirming the consequent"), because the truth of the premises does not in any way support or ensure the truth of the conclusion. To see this, consider another example of the same sort of inference (affirming the consequent):
- If I am swimming, then I am wet.
- I am wet.
- Hence, I am swimming.
Nah! I live in Portland, Oregon--folks here are wet most of the year from the rain. So even if the first two premises of the little argument above are true, notice that it doesn't "follow" that the conclusion is true. Hence, the truth-value of the conclusion is in no way supported or assured by the premises. That's what it means for something to be a fallacy.
To answer your question now, we attempt to avoid fallacies because we care about what is true and we want to believe what is true and not what is false (at least when we are being reasonable). So we want to avoid reasoning that does not help us (and may actually hinder us) from our pursuit of truth.
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.
Without disagreeing with anything Alex has said, let me just add one more thing: There are logicians who sympathize with this sort of question, and so who would deny that an argument with inconsistent premises is always valid. There are logics, that is to say, that do NOT validate all inferences of the form: A & ~A, therefore B, for arbitrary B. Such logics are called "paraconsistent, and if you'd like to read about them I'd recommend the Stanford Encyclopedia article as a start.
I was intrigued that you take human knowledge to be very fragile. The reason you gave was that there's no way for all conclusions to be backed by premises, which I take to be a way of saying that not all of the things we take ourselves to be know can be based on reasoning from other things we take ourselves to know- at least, not if we rule out infinite regresses and circles. But why should that fact of logic (for that's what it seems to be) amount to a reason to think that knowledge is fragile?
Most of us - including most philosophers and even most epistemologists - take it for granted that we know a great deal. I know that I just ate lunch; you know that there are people who write answers to questions on askphilosophers.org. More or less all of us know that there are trees and rocks and that 1+1 = 2 and that cheap wine can give you a headache. Some of the things we know call for complicated justifications; others don't call for anything other than what we see when we open our eyes or (as in the case of things like 1 =1) understanding what we've been told.
This sort of reply is likely to prompt someone to ask "But how do you know that you know all those things?" That question will make some people fret, but here's a perfectly good answer: I don't know how I know all those things. Coming up with a good theory of knowledge is hard work and tends to produce controversial answers. But knowing things doesn't call for a theory of how we know things. People knew things for centuries before anyone got around to asking what exactly knowledge is and how it works.
A few things do seem clear, however. One is that not everything we know comes from syllogistic or any other sort of reasoning. Another is that we can use parts of what we know to evaluate the usefulness of other possible ways of knowing things. For example: by careful investigation, we've learned a lot about the unreliability of eyewitness testimony and memory (though we haven't learned that they're never reliable.)
But the most important thing is that there's no good reason to follow Descartes and thinking that knowledge must be based on foundations that are beyond all possible doubt. That's a premise eminently worthy of doubting, not least because it does such a lousy job of accounting for something that seems much less open to doubt: that we really do know a great deal about a great many things.
You're quite right: ordinary moral intuitions aren't infallible. However, the sort of criticisms you have in mind doesn't really suppose that they are.
Start with an extreme case. Suppose someone came up with a moral theory with the consequence that most of our common moral beliefs were wrong. Now ask yourself: what sort of reason could we have to believe this moral theory? The point is that there's no possible way of making sense of this; perhaps there is. But if I'm told that my ordinary moral judgments are massively wrong, there would be a real problem about what sort of reason we could have to accept the very unintuitive theory from which that consequence flowed.
Or take a more concrete example. Suppose some moral theory had the consequence that wanton cruelty toward innocent people was a good thing. I don't know about you, but I find it hard to imagine what could possibly make this moral theory more plausible than my ordinary moral belief that wanton cruelty is very wrong indeed.
Here's another way of getting at the point: if I don't give any weight to my ordinary moral judgments, then it's not clear what basis I could have for giving weight to a theory whose output was supposed to replace those judgments. If I cold be so massively wrong about ordinary moral matters, what hope would I have for picking the correct Big Picture of morality?
On the one hand, what's been offered so far is essentially a string of rhetorical questions. But the point of the questions is to make vivid that there is a close connection between our judgments about ordinary moral questions and larger theoretical questions about morality.
One way this is sometimes put is by saying that ordinary moral judgments play an evidential role to play in evaluating moral theories; the ability of a moral theory to make broad sense of our considered moral judgments is a point in its favor; the failure of a theory to do that job is a serious strike against it. This doesn't mean that ordinary judgments get the final say; sometimes we give up our intuitions in the face of compelling general arguments or principles. But an "ethical theory" that gave no weight to first-order moral judgments would have a hard time making the case that we should accept its deliverances.
When someone does use a highly selective set of examples to support their conclusion (Wittgenstein referred to this as a matter of relying on too narrow a diet of examples) a person might be begging the question --which, technically, is assuming the very thesis you are seeking to support or prove. But probably the informal fallacy you may be looking for is simply called a hasty generalization: e.g. reaching a conclusion inductively on the basis of too few cases, as when I might observe a dozen white swans and draw the conclusion that 'All swains are white,' notwithstanding the fact that some swans are black.
As an aside, I think that the term 'begging the question' is now used (at least by most of my students) not in its technical, prior use (here is the St. Martin's Dictionary of Philosophy definition: "The procedure of taking for granted in a statement or argument, precisely what is in dispute"). Many students seem to use it to mean that an event / statement / argument calls for questioning, as in: 'Wittgenstein's remarks beg many questions' meaning 'Wittgenstein's remarks need to be questioned / investigated / challenged.' 30 years ago, a writing class instructor would correct that usage, but today I am not so sure.
Good wishes in your writing, professional and otherwise!
It should also be said that there is nowadays a lot of formal, logical work that is devoted to various forms of implication, like strict implication. Part of this is done within so-called "modal" logic; part of it is done in theories of conditionals generally; some of it concerns non-classical logics like relevant logic.
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."