I studied philosophy in university and I recall that one of my tutors for symbolic logic was trying to walk me through a problem by saying that if you have a large enough set of premises, two of them will inevitably contradict one another. I've always had trouble understanding (and consequently, accepting) this proposition because: if one conceives of reality as a set of claims (e.g., I am right-handed, electron X is in position Y, 2 + 2 = 4, etc.) there are an infinite number of "premises" to the "argument" that is reality and consequently reality is self-contradictory. Am I missing something here? Can you explain which of us is right about this and in which sense? I should mention that I don't necessarily have a problem with reality being self-contradictory, but that really throws symbolic logic out the window (and doesn't throw it out the window at the same time)! Thanks to all respondents for their time. -JAK

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.

One can create axioms that make statements like "all bachelors are married" true. What is wrong with calling these truths analytic as a shorthand for the type of truth it is based on the type of axiom it is derived from, much in the way we use the adjectives arithmetic, set-theoretic, or logical to denote those types of formal truths? I feel like one could decide whether a truth is analytic by seeing which (kinds of) axioms need to involved in making it true.

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.")

Dear sirs and madams, I recently met my cousin, who is a very bright biologist. When she learned that I studied political science and philosophy at university, she asked respectfully me why I would study a self-perpetuating field. I know what my reasons are, but I would be interested in reading what some of the professionals have to say: Why study philosophy? Moreover, why study it since there is an impracticality associated with it? Have you ever gotten any flack from loved ones for philosophizing? Thank you for your time, -Justin

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...

Is a computer conceivable that would cut down on Philosophers' work by immediately identifying logic mistakes in arguments? For example: you enter "The Ontological argument for God" or "David Hume's argument against Inductive Reasoning" (or, for that matter, scan in the entire text of Plato's Republic) into the machine, and it immediately uses its programming (which tells it to watch out for contradiction, and all those other logic laws, etc.) and spits out the mistakes in reasoning. Is the problem with this that it would be too difficult to program, or that the laws of logic are under respectable attack?

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.

Can philosophy of mathematics influence mathematics, or it is just an abstraction of what actually works?

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.

I believe that Kant defended the "law of cause and effect" by stating this argument: (P) If we didn't understand or acknowledge the law of cause and effect, we couldn't have any knowledge. (Q) We have knowledge. Therefore: (P) we acknowledge the law of cause and effect. Isn't this line of reasoning a fallacy? P implies Q, Q, : P

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.

Can you help me evaluate Judah HaLevi’s “Kuzari” argument for the authenticity of the Jewish Tradition? If you’ve not heard of it, I am happy to offer an imperfect synopsis, but you’re better off consulting some more reliable sources (see below). The Kuzari, in a nutshell: If public miracles (e.g., manna of Exodus 16) had occurred, they would have left behind a huge amount of accessible evidence. Therefore, had the miracles not occurred, an entire generation of Jews (millions of people) would never have been duped into believing that they did. Therefore, since virtually the entire Jewish people (along with the Christians and Moslems, presumably) *do* believe those miracles occurred, the only explanation is that they must have occurred.

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.