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If there is a category "Empty Set" it has to have the property "nothingness". Thus it is not propertyless - contradiction?

As far as I can see, the definitive property of the empty set is not nothingness but instead emptiness : It's the one and only set having (containing, possessing) no members at all. The empty set can be empty, in that sense, without itself being nothing. So I see no threat of contradiction here. Indeed, the empty set can belong to a non-empty set, such as the set { { } } , which couldn't happen if the empty set were nothing.

Does logic rule out the possibility that someone could travel into the past and affect events so that they turn out otherwise than we remember them?

Does logic rule out the possibility that someone could travel into the past and affect events so that they turn out otherwise than we remember them? No, because our memory of those events could be mistaken. But: Does logic rule out the possibility that someone could travel into the past and affect events so that they turn out otherwise than they in fact did? Yes, so far as I can see.

As all logical arguments must make the assumption that the rules of logic work, is there any way to derive the laws of logic?

As you suggest, all logical arguments (and hence all derivations) depend at least implicitly on laws of logic. So I can't see any way of deriving any law of logic without relying on other laws of logic. Nevertheless, we can derive every law of logic, provided we're allowed to use other laws of logic in our derivation. We needn't fret about our inability to derive a law of logic while relying on no laws of logic, because the demand that we do so is simply incoherent.

Do these two sentences mean the same thing?- a) If I feel better tomorrow, I'll go out. b) Unless I feel better tomorrow, I won't go out.

I'd say that they have different meanings. I interpret (a) as implying that your feeling better tomorrow is a sufficient condition (all else equal, presumably) for your going out, whereas (b) implies that your feeling better tomorrow is a necessary but maybe not sufficient condition for your going out. That is, (b) seems more cautious, more hedged: (b) allows that you may not go out even if you do feel better tomorrow. Compare: (c) If you feed your pet goldfish, it will flourish; (d) Unless you feed your pet goldfish, it won't flourish. Given how easy it is to overfeed a pet goldfish, (c) is doubtful: your pet goldfish may not flourish even if you feed it. Given that pet goldfish depend on being fed, (d) isn't at all doubtful.

If it's possible for a cat to be alive and dead at the same time, or for a particle to be in two places at the same time, would that show there are at least some things about which one couldn't rely on "Either P or not P" as a sound step in reasoning?

Your question concerns the classical law of excluded middle (LEM): For any proposition P, either P or not P. Because logic is absolutely fundamental, ceasing to rely on LEM will have ramifications that are both widespread and deep. In classical logic, we can derive LEM from the law of noncontradiction (LNC), so to give up LEM is to give up LNC or the equally obvious laws that allow us to derive LEM from LNC. We should be very reluctant to do that. In my view, the alleged possibilities that you cite from physics are not enough to overcome that reluctance. First, they are possibilities only according to some, not all, interpretations of quantum mechanics. Second, even if we accept them as possibilities, rejecting LEM or LNC is more costly than (1) reconceiving "being dead" and "being alive" so that they name logically compatible conditions and (2) reconceiving "being here at time t " and "being elsewhere at time t " so that they name logically compatible conditions. It's less costly to mess with the...

Why is the sorites problem a "paradox"? Isn't it fundamentally a problem of definition?

The sorites problem is a paradox for the reason that any problem is a paradox: it's an argument that leads from apparently true premises to an apparently false conclusion by means of apparently valid inferences. I don't think it's fundamentally a problem of definition, because the concepts that generate sorites paradoxes would be useless to us if they were redefined precisely enough to avoid sorites paradoxes. Take the concept tall man . In order to make that concept immune to the sorites, we'd have to define it in terms that are precise to no more than 1 millimeter of height, because a sorites argument for tall man exists that involves men who differ in height by only 1 millimeter. But defining a tall man as (say) a man at least 1850 millimeters in height would mean that in many cases we couldn't tell whether a man is tall without measuring his height in millimeters. Given the impracticality of taking such precise measurements in the typical case, we'd likely stop classifying men as "tall" and ...

Is it possible to employ a truth predicate or truth set (set of all true propositions) in ordinary first order logic?

To my knowledge, no. Ordinary first-order logic quantifies only over individuals (none of which are literally true) rather than over truth-valued things such as sentences or propositions. Thus there's nothing in first-order logic to which the predicate "is true" can apply. For that you need higher-order logic, which is a topic of controversy in its own right. By "set of all true propositions," I take it you mean "a set of all the true propositions there are," i.e., the extension of the predicate "is a true proposition." A Cantorian argument due to Patrick Grim concludes that no such set is possible. It works by reductio . Let T be any set containing all of the true propositions. If T exists, then it has infinitely many members, but that doesn't affect the argument. Now consider the power set of T -- P(T) -- which is the set whose members are all of the subsets of T. It's provable that any set has more subsets than it has members. With respect to each of those subsets in P(T), there is a true...

Mustn't there be a counterexample to any statement that's a generalisation, because if there weren't a counterexample the statement would be a matter of fact and not a generalisation?

It may help to distinguish between universal generalizations and statistical generalizations. An example of a universal generalization is "All swans are white," and a statistical generalization might be "Swans tend to be white." As it happens, that universal generalization is false, because some swans are not white, and the statistical generalization will be true or false depending on the context (in some parts of Australia, it may be a false statistical generalization). When people say things like "That's just a generalization," I take it they're talking about a statistical generalization -- a claim about how things tend to be, a claim about how things are substantially more often than not. The reason that we expect statistical generalizations to have exceptions, I think, is that if the person asserting a statistical generalization were in a position to assert a universal generalization -- a logically stronger claim -- then he or she would. That's why the universal generalization "All triangles have...

When you ask why people believe in logic, it seems to me that the commonest answer is, "It works." But that answer seems problematic to me; how do you know it won't stop working? I guess what I'm asking is -- are logical laws nothing more than empirical regularities, models of how things behave? Are logical laws any different from empirical laws? Is there any stronger reason to have faith in logic apart from the fact that it works and has always worked?

Yes: As I see it, logical laws are different from empirical regularities. Many of our empirical predictions come true, but some of them don't, and in any case it's not hard to imagine any particular empirical prediction turning out false. I predict that the chair I'm now sitting in won't levitate before I finish answering your question, but it's easy for me to imagine being wrong in that prediction. Indeed, I can even imagine that universal gravitation stops working in the way we've become used to. But what would it be to suppose that the laws of logic stop working? Would it be to suppose that the laws of logic stop working and continue to work exactly as they always have? If yes, why? If no, why not? (Presumably not because the laws of logic would prevent it!) So I'm not sure it's possible to entertain the supposition that the laws of logic stop working. Indeed, I'm not sure that there's any such supposition in the first place. In my view, the question "What makes us so confident that it will never...