You've asked about one version of an ancient paradox called the "Liar paradox" or the "Epimenides paradox." One good place to start looking, then, is the SEP entry on the Liar paradox, available here . Philosophers are all over the map on how to solve paradoxes of this kind, and their proposed solutions are sometimes awfully complicated! Best of luck.

What is the difference between something that is logically a contradiction and something that happens to not be instantiated? As I think you already suspect, it's the difference between (1) a concept whose instantiation is contrary to the laws of logic or contrary to the logical relations that obtain among concepts; and (2) a concept whose instantiation isn't contrary to logic but only contrary to fact. Examples of (1) include the concepts colorless red object and quadrilateral triangle . Examples of (2) include the concept child of Elizabeth I of England . Concepts of type (1) are unsatisfiable in the strongest sense; concepts of type (2) are merely unsatisfied. Could you explain how the concept of a ghost is not a contradiction? Good question. I'm not sure the concept isn't internally contradictory. Can ghosts, by their very nature, interact with matter? Some stories seem to want to answer yes and no . If I recall correctly (it's been a while) the movie ...

The topic is controversial (as I indicate below), but the inference rules of standard logic do allow you to derive any conclusion at all from any (formally) contradictory premise. Here's one way (let P and Q be any propositions at all): 1. P & Not-P [Premise: formal contradiction] 2. Therefore: P [From 1, by conjunction elimination] 3. Therefore: P or Q [From 2, by disjunction introduction] 4. Therefore: Not-P [From 1, by conjunction elimination] 5. Therefore: Q [From 3, 4, by disjunctive syllogism] Those who object to such derivations usually call themselves "paraconsistent" logicians; more at this SEP entry . They typically reject step 5 on the grounds that disjunctive syllogism "breaks down" in the presence of contradictions. I confess I've never found their line persuasive.

@William Rapaport: Unless disjunctive syllogism or one of the other two rules used in the derivation fails, the "irrelevance" of the conclusion to the premise is irrelevant to whether the conclusion follows from the premise. Relevance logic has to give up at least one of those rules, none of which is easy to give up.

Even asking "Is there a way to prove that logic works?" presupposes that logic does work at least at the level of its most basic laws, such as the Law of Noncontradiction, because the question itself has meaning only if the most basic laws of logic hold. To put it a bit differently: No sense at all can be attached to the notion that logic doesn't work (or even sometimes doesn't work). See also my reply to Question 4837 and Question 4884 . So we have what philosophers call a "transcendental" proof of the reliability of logic: If we can so much as ask whether logic is reliable (provably or otherwise), then it follows that the answer to our question is yes . You might say that this proof won't impress someone who doubts the most basic laws of logic in the first place. But I'd reply -- predictably -- that no sense can be attached to the notion of doubting the most basic laws of logic.

Yes. Completely. The tricky question is why . It's tempting to answer that necessarily everything is bound by the laws of logic because the alternative -- the claim that something isn't bound by the laws of logic -- is necessarily false. But, as I suggested in my reply to Question 4837 , no sense can be attached to the claim that something isn't bound by the laws of logic. So the claim can't be false , strictly speaking. Perhaps all we can assert is a wide-scope negation: it's not the case that something isn't bound by the laws of logic, just as it's not the case that @#$%^&*. Necessarily everything is bound by the laws of logic because the alternative is literally nonsense? I wish I had a better explanation!

To test a rule of inference, you can try to find counterexamples to it, cases in which the rule lets you derive a falsehood from true premises. Professor Vann McGee offered a well-known (and controversial) such attempt in this article . But there's no getting around rules of inference entirely. Even as you test one rule of inference you unavoidably rely on others. Because any attempt to answer the question "Why should we trust rules of inference at all?" will rely on reasoning, it will trust some rules of inference, whether or not those rules are made explicit in the reasoning. There's no way to get "outside" all rules of inference and see how they measure up against something more trustworthy than they are.

Notice that the same question arises in math, where the laws also hold no matter what. Arithmetic contains commutative laws of addition and of multiplication, associative laws of addition and multiplication, a distributive law of multiplication over addition, etc. Are those laws different? Their representations on the page certainly look different. I take it that you're asking, at bottom, how truths that hold in all possible worlds could count as distinct truths. The answer depends on how propositions are to be individuated , and here philosophers give various answers. On some theories, there's only one proposition that's true in all possible worlds, although there are indefinitely many sentences (some logical, some mathematical, some metaphysical) that express this single proposition. Other theories give a more fine-grained way of individuating propositions that allows for the existence of multiple propositions that are true in all possible worlds. You'll find more...

If I understand your first question, the answer is no (unless the existential premise is superfluous). By an "existential claim," I take it you mean an existential generalization such as "There exists an x such that F x ," rather than a claim of the form "F a ," which implies an existential generalization. But you might wish to look into the rule of Existential Instantiation (or Existential Elimination in natural deduction systems); you'll find a brief summary of it here . I'm not sure I understand your second question. There are two ways of interpreting the universal and existential quantifiers: the objectual way and the substitutional way. I can't find a handy link to recommend, but if you search for discussions of those terms, you may find something relevant to your third question.

In this context, by "normative claims" I take it you mean claims that one ought to (or ought not to) do some particular thing. Can we get such claims out of principles of deductively valid inference? I think so. If you accept P, and you recognize that P implies Q, then there's a sense in which you ought to accept Q: you're logically and rationally committed to Q by propositions that you accept and recognize. If you accept Q, and you recognize that P implies Q, there's a sense in which you ought not to deduce P from those propositions alone: doing so would be fallacious. Now, you might say that the ought and ought not in those cases is only hypothetical: " If you want your deductive reasoning to be reliable, then you ought (or ought not)...." But I think the antecedent of that conditional (the "if" part) is easy to discharge. Plenty of people do want their deductive reasoning to be reliable, and so there's a sense in which such people really ought to use ...

I think of logical necessity as (predictably enough) the necessity imposed by the laws of logic. So, for example, it's logically necessary that no proposition and its negation are both true, a necessity imposed by the law of noncontradiction. But one might regard logical necessity as broader than that, since one might say that it also includes conceptual necessities such as "Whatever is red is colored." Metaphysical necessity is a bit harder to nail down. Every proposition that's logically or conceptually necessary is also metaphysically necessary, but there may be metaphysical necessities that are neither logically nor conceptually necessary, such as "Whatever is water is H2O" or "Whatever is (elemental) gold has atomic number 79." Nothing in logic or in the concepts involved makes those propositions necessary, but many philosophers say that those propositions are nevertheless "true in every possible world," which is the root idea of metaphysical necessity. Even if some proposition P isn't...