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 ...

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...

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...

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...

I'll assume, just for simplicity, that by a "thought" you mean a belief and by "something impossible" you mean a proposition that cannot possibly be true . I hope my assumptions aren't off the mark. (I'm not a neuroscientist, so I'll say nothing about how neurons work.) If my assumptions are correct, then your question becomes "How can anyone believe a proposition that cannot possibly be true?" One answer is this: "Easy! For example, many people down through the ages believed that they had accomplished the famous geometric construction known as squaring the circle . But the proposition they believed cannot possibly be true, because squaring the circle is impossible, as was finally proven in 1882. Those who believed the proposition obviously didn't see the impossibility of the construction." An opposing answer is this: "They can't! Indeed, we can understand the behavior of those misguided geometers only if we attribute to them a false belief that could have been true, such as the belief that a...

In Aristotle's syllogistic logic (including in his square of opposition), "All men are mortal" implies "Some men are mortal." But in the standard logic of the past 100 or so years, that implication doesn't hold. This failure of implication arises because modern standard logic construes "All men are mortal" as a universal quantification over a conditional statement: "For anything at all, if it's a man then it's mortal." Intuitively, I think we can see why the universally quantified statement can be true even if no men exist. Compare "For anything at all, if it's a unicorn then it's a unicorn," which seems clearly true despite the fact that (let's assume) no unicorns exist. In modern standard logic, then, "All men are mortal," "No men are mortal, " and "All men are immortal" come out true if in fact no men exist. Importantly, "Some men are immortal" does not come out true in those circumstances. A similar lesson applies in set theory, in which "All of the members of the empty set are even" and...

Thanks for the chance to clarify my answer to Question 25338 . I can see how my answer might have given the wrong impression. I didn't mean to suggest that, whatever one's desired conclusion, one can always find less controversial premises that imply it. That is, one may fail in the attempt to improve one's argument, no matter how hard one tries. One's conclusion may just not follow from less controversial premises. My point was simply that a logically valid argument from less controversial premises to conclusion X is better than a logically valid argument from more controversial premises to conclusion X. It's something one should strive to find, even though there's no guarantee of success, and failing to find it may be a good reason to reconsider one's conclusion.

I'm intrigued by your suggestion that there may be a paradox here, but I'm having trouble reconstructing your reasoning. As far as I can tell, your reasoning relies on the premise that any definition must contain whatever it defines . But I'm not sure that premise is plausible. Merriam-Webster.com defines "walrus" as follows: "a large gregarious marine mammal ( Odobenus rosmarus of the family Odobenidae) of arctic waters that is related to the seals and has long ivory tusks, a tough wrinkled hide, and stiff whiskers and that feeds mainly on bivalve mollusks." That definition (more precisely, the definiens ) doesn't contain the word ("walrus") being defined, which is good: otherwise the definition would be unhelpful. Nor does the definition contain a walrus; we'd need a cage to do that. Instead, the definition gives us a string of words meant to pick out the walrus from among other things. If it's a genuine definition, then the principle on which it seems you're relying is false. But I may...

I think both arguments can be analyzed as inductive arguments and still distinguished in terms of their quality. The book argument is a stronger inductive argument than the coin-toss argument for a simple reason: the probability that Smith's book C is great isn't independent of whether Smith's books A and B are great. That is, Smith's having written great books A and B makes the probability that Smith's book C is great higher than it would be had Smith not already written two great books. Important: higher than it would be otherwise, which needn't mean higher than one-half. Even though Smith's track-record raises the probability that book C is great, the track-record needn't make it more probable than not that book C is great. By contrast, the probability of tails on any given toss of a fair coin is independent of whether the coin came up tails twice already: that history of tosses neither increases nor decreases the probability of tails on a third toss.

Good question, and as fundamental a question as anyone could ask. I think that the laws of logic must be not only independent of human minds but independent of any minds, including God's mind if such exists. At any rate, I don't think anyone can see how it could be otherwise. To say that the laws of logic depend on human or divine minds is to imply that the following conditional statement is nontrivially true: If (1) human or divine minds had been different enough, then (2) all of the laws of logic would be different from what they are . (By "nontrivially true," I mean that the statement is true not merely on the ground that (1), its antecedent, is logically impossible. If (1) is logically impossible, then the conditional statement is trivially true, even if (2), its consequent, is also logically impossible.) We can't make sense of the italicized statement without presupposing that (2) is false . If the italicized statement means anything, then it doesn't mean this: If (1) human or...