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.

In trying to understand why Quine or others would not countenance antinomies, or real paradoxes, perhaps it would help to add that the conception of paradox in play here is that of an argument , a collection of premises that entails a conclusion. The arguments that appear to be paradoxes are ones whose premises seem obviously true, whose reasoning seems impeccable, and whose conclusion seems obviously false. So, on this conception what would a real paradox be? An argument that leads from true premises via correct reasoning to a false conclusion. Now, why be confident that there couldn't be such a thing? Because what we mean by "correct reasoning" is just reasoning that leads from true premises to a true conclusion. To judge something to be a real paradox would then be akin to judging something both to be an apple and not to be an apple. That is not a possible judgment. So, the judgment that there are no real paradoxes doesn't stem from an optimistic confidence in the powers of the human...

Each true claim can be paired with a unique false one, namely its negation (i.e., the result of prefixing the original claim with "It is not the case that ..."). And each false claim can be paired with a unique true one (again, its negation). So, there are exactly as many true claims as there are false ones.

(a) sounds a bit awkward and one might wonder whether it's ambiguous. Does it mean: (a1) You should make it be the case that (it is not the case that (you do not buy that book)), or (a2) It is not the case that (you should make it be the case that (you do not buy that book)). Using "S" to stand for "You should make it be the case that" and "N" for "It is not the case that", (a1) has the form: S(N(N( p ))). But (a2) has the form: N(S(N( p ))). (a2) does not mean the same as (b), which has the form: S( p ). But (a1) is arguably identical in meaning to (b). That's because "N(N( p ))" means the same as " p ". A cautionary note: In general, most people would agree that "N(N( p ))" means the same as " p ", that is, that a statement means the same as its double negation. What would a world look like, you might wonder, in which those statements differ in their truth or falsity? We can't even coherently describe it. I say "most people," though, because some have...

Thomas has given some examples of situations that are conceivable but not possible (in that they conflict with, say, laws of nature): for instance, in some sense one can imagine a puddle's turning into a human being, though such a transformation flies in the face of what we believe is physically possible. But there are also circumstances which are physically possible of which we can form no picture. For instance, it's physically possible for there to be a chunk of quartz with a thousand facets, though I cannot imagine such a thing. The question presupposes that we use the terms "conceivable" and "possible" in just one way, which is doubtful. For a little more, see Question 71 .

It's true that logic doesn't tell us how to do certain things, like dance or play badminton. Philosophers often distinguish between knowing how to do something and knowing that something is the case. The latter kind of knowledge is often termed propositional knowledge , because what we know is that a particular proposition holds. For instance, our knowledge that London is in England is an instance of propositional knowledge; the proposition we know is proposition that London is in England . Now let's return to the question of how to justify logical inferences. Can we hope to do so by pointing to any abilities (knowings-how) that we possess (abilities which, I grant you, are not given to us simply in virtue of our appreciating logical entailment relations)? I don't think so. Abilities, knowings-how, aren't really routes to knowledge. (They might be prerequisites for our acquisition of knowledge—for instance, if I don't have the ability to walk to the window, I might not be able...

Many kinds of considerations convince people. Everyone, not just philosophers, naturally sorts those considerations into "good" reasons and "bad" ones. People might sometimes disagree on where to draw the line, but most everyone agrees there's a line to be drawn. The good reasons are considerations that are relevant to the truth of the claim in question; the bad ones, irrelevant. Relevant in this sense: the truth of the considerations demands, or at least makes more likely, the truth of the claim being argued for. Turns out that we've made a lot of progress in understanding this relation of relevance. Logic studies one corner of it: that which concerns entailment relations between claims, that is, when the truth of one proposition forces the truth of another. With this knowledge in hand, we can see that people are often convinced by arguments that do not provide good reasons for their conclusion. And also, that they sometimes fail to be convinced by good arguments. Arguments that...

Also, if I were to tie your hand behind your back and then ask you whether you can touch your nose with it, that would be a peculiar question. And something similar is going on when one's asked whether one can defend all one's principles of reasoning. The whole practice of defending something assumes that principles of reasoning are in place. In fact, a cogito -like situation is indeed present: a state of affairs holds (thinking, defending) which demands presuppositions (existing, acceptance of rules) that make a certain doubt (about existence, about logic) self-stultifying.

I would say that a well-taught first course in formal logic would presuppose no mathematical knowledge and no mathematical sophistication. The material is technical in nature and freely employs symbols and terms drawn from the vernacular of mathematics -- but all these should be explained by your instructor. From teaching this material many times, I know that some "math phobic" students freeze up at the sight of parentheses and greek letters. But I always tell them that these symbols are their friends! They serve to make our lives easier by allowing us to say just what we want to say very briefly and perspicuously. We could in principle do without them -- but that would in fact make our lives a lot harder.

In the most straightforward case, that of deductive inference, reasons support a conclusion in this sense: if the reasons are true then the conclusion must be as well. Once one moves beyond deductive inference, the truth of a good argument's premises makes the truth of its conclusion more or less probable. If you're asking the question "How does it come to be that the truth of some claims makes the truth of others either necessary or highly probable," that's a much contested issue that haunts the philosophy of logic, the philosophy of language, and the philosophy of science.