Several pertinent posts can be found in Question 362 and Question 2156 .

There are a few similar questions already on this site, e.g., see Question 2110 .

I'm not sure what your tutor was getting at either. If your tutor meant that one can always enlarge a set of premises to make it an inconsistent set, that's obviously true: simply add the negation of one of the premises already in the set. If he meant that any axiom system with infinitely many premises (say, one that employs an axiom schema) is inconsistent, then there's no reason to believe that.

There is nothing stopping you from defining an analytic theorem of a formal system to be one whose derivation requires appeal to at least one member of a designated subset of axioms. But on what basis are you deciding to single out that particular subset of axioms? If you say you're being guided by the fact that those particular axioms express truths about meanings, whereas other axioms express substantive truths about the world, then you owe an explanation of what that distinction amounts to -- and arguably, that will be no easier to give than an outright analysis of "analytic". (You might also look at W.V. Quine's discussion of Semantic Postulates in his paper "Two Dogmas of Empiricism.")

I wonder whether there is no question that tires philosophers more than this one -- if it's not "What's your philosophy?" or "If a tree falls in a forest ...". The assumption made in the question, that "there is an impracticality" to philosophy, is false. It's not the common perception of employers or graduate schools, and it's not the case. A recent article in The New York Times spoke to this. The article was unfortunate, in my view, because a reader might think that the main reason students do, or should, study philosophy is instrumental: it sharpens various skills which will be of value throughout one's life, regardless of its particular shape. That might be true for the occasional student but in general skill-sharpening is not a strong enough motivator to keep curious people studying a subject. The real reason is that issues in philosophy are central to our lives as thinking creatures, and the specific form these issues take in the questions, answers, and arguments of the great...

Philosophy would be much easier if we could program such a machine -- and boring too. But it's not going to happen. For one thing, there's your interesting point that philosophical disputes can go very deep, so deep as to include disagreement about what the laws of logic, of correct inference, ought to be. Secondly, even for first-order classical logic, there simply is no computer that can decide whether any given inference is correct. (This is known as Church's Theorem and was proved by Alonzo Church in 1936.) Finally, there's the fact that evaluating the logical cogency of arguments is only a (small) part of the business of figuring out what to think about someone's argument in philosophy: at the very least, one must also understand and evaluate the assumptions to which the logical reasoning is applied.

As Peter Smith's examples make clear, sometimes "the philosophy of mathematics" appears in other than philosophy journals and is done by other than people in philosophy departments. Another instance of this is the long current of constructivism in mathematics. The development by mathematicians of constructivist mathematics (most notably, intuitionistic mathematics) is often motivated by their -- for want of a better term -- philosophical reflection on the nature of mathematics.

See Question 442 .

It seems to me you haven't reported the inference accurately. The conclusion, "We acknowledge the law of cause and effect," is the negation of the antecedent of (P) and not, as you report, (P). (That is, your premise (P) is of the form: if not-X, then not-Y. And the conclusion of your argument is X.) So, the argument really has the form "If not-X, then not-Y" and "Y", therefore "X". This is a correct form of inference in classical logic. You're right that "If X, then Y" and "Y" do not imply "X"; that is indeed a fallacy. But this argument is rather of the form: "If X, then Y" and "not-Y", therefore "not-X". And that is a correct inference.

History and our own personal experiences tell us that people -- even very large numbers of people -- can be mistaken or even bamboozled. So it cannot be right to say that "the only explanation" for those people's beliefs is that the alleged miraculous events occurred. But might it be the best explanation for their reports? The locus classicus is David Hume's "Of Miracles," which appears as Section X of his Enquiry Concerning Human Understanding . He argues there that the answer to the last question is a resounding No. If you want a powerful assessment of the argument you've presented, I'd start there.