This is an incredibly complicated question. One question we might want to ask is what's meant by "justifying" a law of logic. I'll do my best to ignore that question here. It's tempting to say that one can't justify anything without using the laws of logic, but that is arguably too strong. I think my belief that there is a computer in front of me is justified by my perception of it, and I doubt that the laws of logic have to be invoked there. Moreover, it is not obvious that the laws of deductive logic have always to be invoked even when justification is somehow "inferential". Often, they will, but, again, it's not clear they always must be. What is meant here by "laws of logic"? Do we mean such generalizations as that, if a conditional is true and its antecedent is also true, then its consequent is true? Or do we mean to count what we'd otherwise call instances of logical laws, such as "Either Dubya likes popcorn or Dubya does not like popcorn", as "laws of logic" for the purposes...

There's also this point: If one has shown that, if X is true, then Y is true, then one has proven, without making any assumptions, that, if X, then Y. One might say that, if one has given an argument, then one has assumed that it is legitimate to use whatever principles of argument one applied in the argument. But, as Lewis Carroll once observed in a famous paper titled "What the Tortoise Said to Achilles", that claim leads to the conclusion that argument itself is impossible. We have to distinguish between an argument's justifiably employing certain principles and its assuming that those principles may legitmately be employed. I take it, however, that the question was not concerned with whether we can prove this kind of claim but, perhaps, with whether we can prove anything that would actually be regarded as contentious. Can we prove, for example, that abortion is immoral? or, conversely, that it is morally permissible? In such cases, the answer may well be "No".

There are some syllogistic figures that at least some Aristotleansregarded as valid that are not treated as valid by modern logic. Anexample would be: All Fs are G; all Gs are H; therefore, some Fs are H.This is valid if ,but only if, one supposes that "univeraljudgements are existentially committed", as it is sometimes put, thatis, if one supposes that, if "All Fs are G" is to be true, there mustbe some Fs. That assumption is not usually made in modern logic, and sothe contemporary translation of this syllogism: ∀x(Fx → Gx); ∀x(Gx →Hx); therefore, ∃x(Fx ∧ Hx), is not valid. However, if one does thinkthat "All F are G" is existenally committal, one can perfectly welldefine a new quantifier, "∀ + x", that incorporates that assumption. Andthen the inference can be shown to be valid. Whether the English statement "All Fs are G" is existentially committed is not for a logician ( qua logician) to decide. That's an empirical question about natural language.

I'm not sure what you have in mind here by "Aristotelian principles of logic". I can think of a couple possibilities. I should say first, however, that some philosophers spend a lot of time thinking about Aristotelian logic, namely, historians of ancient philosophy. But I take it that your question concerns contemporary philosophy. One aspect of Aristotle's writings on logic is his theory of valid inference. This is not much discussed because it has been essentially supplanted by modern logic, which can explain the validity of the valid figures of the syllogism and do much that Aristotle's logic could not do. Famously, Aristotelian logic cannot explain the validity of the inference from "Every horse is an animal" to "Every horse's head is an animal's head". The validity of that inference can be explained by modern logic. Since such inferences occur throughout mathematics and ordinary reasoning, Aristotelian logic is simply far too limited in its scope. Another aspect of Aristotle's...

One might add that it is by now well established that people are, in general, terrible at probabilistic reasoning. So if there's anything hard-wired in that case, it probably doesn't conform to the laws of probability. It's a nice question why not, but it might be, for example, that reasoning according to certain heuristics that don't always work is faster or what have you than reasoning according to the rules that are actually valid. Or maybe there's some other reason.

There's a nice article on intuitionist logic in the Stanford encyclopedia.The differences between it and classical logic become more profound inconnection with quantifiers such as "all" and "some". For example, inintuitionistic logic, it can be true both that not everything has someproperty and that there is nothing that does not have it! Andin the intuitionistic theory of the real numbers, we can actually findsuch a property and prove such a statement: We can prove that not everynumber is either positive, negative, or equal to zero; we can alsoprove that there is no number that is neither positive, nor negative,nor equal to zero. Think about that for a while. Of course, we can only "prove" it in the sense that it follows fromthe principles of intuitionistic analysis. Whether it is true dependsupon whether those principles are true. Since you mentioned internal consistency, perhaps I'll mention something even stranger, so-called paraconsistent logics . These are systems in which...

On Dan's comment. The distinction between so-called weak counterexamples and strong ones is, of course, important. But it really is possible to prove, in intuitionistic analysis, the negation of the claim that every real is either negative, zero, or positive. The argument uses the so-called continuity principles for choice sequences. I don't have my copy of Dummett's Elements of Intuitionism here at home, but the argument can be found there. A short form of the argument, appealing to the uniform continuity theorem—which says that every total function on [0,1] is uniformly continuous—can be found in the Stanford Encyclopedia note on strong counterexamples . There is an important point here about the principle of bivalence, which says that every statement is either true or false. It's sometimes said that intuitionists do not, and cannot, deny the principle of bivalence but can only hold that we have no reason to affirm it. What's behind this claim is the fact that we can prove that we will not...

Most logicians would regard self-contradiction as a flaw, yes. Thereason is that a good argument is supposed to be one whose conclusionmust be true if its premises are true. If at some point in the argumenta contradiction appears, then either (i) the reasoning was bad or (ii)the premises cannot all be true. That said, however, one can usepossibilty (ii) to argue for something by what is called reductio ad absurdum :If you want to show that not-p, show that p (possibly together withother things that are agreed to be true) leads, via good reasoning, to contradiction. Then not all of the premises can be true. So if the ones other than p are, it isn't. Now,how do you tell if you have a contradiction when the argument in words?There's no magic bullet, I'm afraid. Being able to translate intosymbolic logic only helps so much, and in the really hard cases it'llbe controversial how to do the translation, anyway. So, to a firstapproximation, you have reached a contradiction if you have reached...

So far as I know, deontic logic has never entered mainstream work on moral philosophy. One of the key ideas of deontic logic is to allow for impossible (combinations of) obligations. My sense is that, while there have been proponents of the idea that there could be such things (notably, Bernard Williams), most have rejected the idea. The argument I have usually heard (this is going back to grad school, so it's been a while) is that the relevant notion of obligation is one of all things considered obligations, and these cannot conflict. Perhaps deontic logic would be of more interest, however, if regarded as a logic of prima facie obligation. But then the deep question is how conflicts between prima facie obligations are supposed to be resolved, and, so far as I know, that's not really the focus of work on deontic logic. Perhaps more recent work than is known to me has addressed that question.