I am reading a by book by the great logician Raymond Smullyan. In this book he says that any statement of the form, "All As are Bs" are true if there are no "As". That is, these statements are vacuously true. He gives the following example, "All Unicorns have 5 legs" is true since there are no unicorns. So is "All unicorns have 6 legs", and "All unicorns are purple", etc. But this strikes me as obviously false. For example, "All unicorns have two horns" and "All unicorns are necessarily existing" are false statements. The first is false in virtue of the fact that unicorns are by definition one-horned. The second is false in virtue by the fact that it is impossible for something to be both necessarily existing and nonexistent. Am I missing something here or misreading Smullyan? Or are these counterexamples sufficient in refuting the claim that any statement of the form "All As are Bs" is vacuously true if there are no "As"? For reference the book is, "Logical Labyrinths" from pages 99-101. Thanks...

I don't know that book in particular, but I can give you a standard explanation that at least makes sense of the view you find puzzling. In Aristotle's logic, any statement of the form "All S are P" implies that at least one S is P, so the statement comes out false (rather than vacuously true) if nothing is S. By contrast, in contemporary logic, "All S are P" is interpreted as saying "For anything at all, if it is S, then it is P": it is interpreted as a universal quantification applied to a conditional statement. Crucially, the conditional statement "If it is S, then it is P" is standardly treated as a truth-functional conditional that is equivalent to the disjunction "It is not S, or it is P." Now suppose that nothing is S, so that "It is not S" is true of everything. Then the disjunction "It is not S, or it is P" will come out true no matter what we substitute for "it," because a true disjunction needs only one true disjunct. In that case, the truth-functional conditionals "If it is S, then it...

Recently I read a comment on an online debating site where someone stated “ Every deductive statement regarding the real world relies on induction” to me that does not sound correct am I missing something?

One reason it doesn't sound right to me is that I don't know what could be meant by a "deductive statement." I know what a deductive argument is, but it always contains more than one token statement. Did the site say, instead, "every declarative statement" (i.e., every declarative sentence)? In any case, consider the statement "There are no colorless red cars." It's a declarative statement. Does it regard the real world? Arguably, yes: it's at least partly about cars. But knowing its truth doesn't require induction -- it's analytically true. On the other hand, maybe despite appearances it's not a statement even partly about cars but only about the logic of the concepts red and color . We'd need an agreed-on criterion of "aboutness" in order to decide.

Can you coherently consistently imagine a universe where laws of thoughts are false?

If by "laws of thoughts" you mean laws of logic, then no. No coherent (that is, self-consistent) situation can violate any law of logic. Even philosophers, such as Graham Priest, who claim to be able to imagine situations that violate the law of non-contradiction concede that those situations are not self-consistent.

Is the Sorites paradox really a paradox, or is it more properly considered to be a logical fallacy? By definition, the term "heap" is indeterminate. Yet the Sorites paradox tries to force a specific definition on what is by design an indeterminate concept: the very idea of defining the term "heap" as a specific number of grains of sand is fallacious, is it not? I don't see a paradox here as much as I see confusion about how terms are defined. How many grapes are in a bunch of grapes? How many leaves are in a head of lettuce? How many grains are in an ear of corn? How many chips are in a bag of potato chips? in each of the above questions, the answer will vary from one example to the next, the exact number is not particularly germane to the concept. So what makes a heap different from a bunch or any of the other examples?

I see the sorites paradox as a very serious problem, not a logical fallacy that's easy to diagnose and fix. The paradox arises whenever we have clear cases at the extremes but no known line separating the cases where a concept applies from the cases where the concept doesn't apply. Clearly, 1 grape isn't enough to compose a bunch of grapes. Just as clearly, 100 grapes is enough to compose a bunch of grapes. So which number between 2 and 100 is the smallest number of grapes sufficient to compose a bunch of grapes? If there's no correct answer, then the sorites paradox shows that the concept enough grapes to compose bunch of grapes is an inconsistent concept. But inconsistent concepts, such as the concept colorless red object , necessarily never apply to anything, in which case it would be impossible for anything to be a bunch of grapes. One might reply, "Okay, fine. Necessarily there are no bunches of grapes. Life goes on." The problem, however, is that the sorites paradox applies to every vague...

For some reason, the sorites paradox seems quite a bit like the supposed paradox of Achilles and the turtle with a head start: every time Achilles reaches where the turtle had been, the turtle moves a little bit forward, and so by that line of reasoning, Achilles will never be able to reach the turtle. Yet, when we watch Achilles chase the turtle in real life, he catches it and passes it with ease. By shifting the level of perspective from the molecular to the macro level, so to speak, we move beyond the paradox into a practical solution. If we try to define "heap" by specifying the exact number of grains of sand it takes to differentiate between "x grains of sand" and "a heap of sand," aren't we merely perpetuating the same fallacy, albeit in a different way, by saying that every time Achilles reaches where the turtle had been, the turtle has moved on from there? If not, how are the two situations qualitatively different? Thanks.

In my opinion, the reasoning that generates the paradox of Achilles and the tortoise isn't nearly as compelling as the reasoning that generates the sorites paradox. The Achilles reasoning overlooks the simple fact that Achilles and the tortoise are travelling at different speeds : if you graph the motion of each of them, with one axis for distance and the other axis for elapsed time, the two curves will eventually cross and then diverge as Achilles pulls farther and farther ahead of the tortoise. All of this is compatible with the fact that, for any point along the path that's within the tortoise's head start, the tortoise will have moved on by the time Achilles reaches that point: that's just what it means for the tortoise to have a head start. It's not that the Achilles reasoning is good at the micro level but bad at the macro level. It's just bad. By contrast, the only thing overlooked by the sorites reasoning is the principle that a small quantitative change (e.g., the loss of one grain of...

Given a particular conclusion, we can, normally, trace it back to the very basic premises that constitute it. The entire process of reaching such a conclusion(or stripping it to its basic constituents) is based on logic(reason). So, however primitive a premise may be, we don't seem to reach the "root" of a conclusion. Do you believe that goes on to show that we are not to ever acquire "pure knowledge"? That is, do you think there is a way around perceiving truths through a, so to say, prism of reasoning, in which case, nothing is to be trusted?

It's not clear to me what you're asking, but I'll do my best. Given a particular conclusion, we can, normally, trace it back to the very basic premises that constitute it. I doubt we can do that without seeing the conclusion in the context of the actual premises used to derive it. The conclusion Socrates is mortal follows from the premises All men are mortal and Socrates is a man , but it also follows from the premises All primates are mortal and Socrates is a primate . So which pair of premises are "the very basic premises" for that conclusion? Outside of the actual argument context, the question has no answer. I don't know what you mean by "the root of a conclusion," but you seem to be suggesting that any knowledge is impure if it depends on -- or if it was acquired using -- any reasoning at all. Perhaps the term inferential would be a better label for such knowledge. On this view, even if I have direct knowledge that I am in pain (when I am), I have only...

Logic is supposed to be an objective foundation of all knowledge. But if that's the case then why are there multiple systems of logic? For example there's 'dialetheism', which allows for true contradictions, and 'fuzzy logic' in which the law of excluded middle does not apply. If people can just re-write the rules to create their own system of logic, then doesn't that make logic subjective and arbitrary? It doesn't seem like arguments would have much weight if I could simply just choose whichever system best supports the conclusion I want.

You've asked a very good question, and your final sentence makes a good point. Those who defend one or another non-classical system of logic (paraconsistent, dialetheistic, intuitionistic, fuzzy, quantum, etc.) insist that they're not simply choosing a system of logic on a whim or merely out of convenience. Instead, they say, we're forced to accept non-classical logic because (a) it's an objective fact that arbitrary contradictions don't imply every proposition; because (b) some propositions are objectively both true and false; because (c) some propositions are objectively neither true nor false; because (d) some tautologies aren't completely true and some contradictions aren't completely false; because (e) the data gleaned from reliable experiments don't obey the classical laws of distribution, etc. Having looked into them, I find none of their arguments for (a)-(e) persuasive. But what's most interesting, as various philosophers have observed, is that the defenders of non-classical logic sooner or...

How is this argument valid? Either Oscar is an octopus or he is a whale. Oscar is a zebra. Therefore, Oscar is an octopus.

Validity in an argument comes down to one question: Is it possible for all the argument's premises to be true and its conclusion false? If no, then the argument is valid. So, assuming it is impossible for Oscar to be both a whale and a zebra, the argument is valid. Even so, the argument is not formally valid, because the following is not a valid form: Octopus(Oscar) or Whale(Oscar) Zebra(Oscar) Therefore: Octopus(Oscar) Not all valid arguments are formally valid. Furthermore, assuming that Oscar is not both an octopus and a zebra, the argument is unsound despite being valid, because in that case the second premise and the conclusion are not both true. The same holds for this argument (on similar assumptions): Oscar is an octopus, or Oscar is a whale. Oscar is a zebra. Therefore: Oscar is a whale. Valid but unsound. So neither argument establishes its conclusion.

Why is it important to study logic in philosophy? One answer might be that logic teaches you correct reasoning, but that is not something that is unique to philosophy, as it's important in other fields as well (e.g. history, economics, physics, etc.), and those usually do not include any explicit study of logic.

In my experience, philosophy courses take the explicit, self-conscious formulation and evaluation of arguments (i.e., reasoning) more seriously than any other courses of study, with the possible exception of those math courses that emphasize proofs. Moreover, the breadth and depth of philosophical problems exceed those encountered in math. Therein lie the advantages of philosophy courses as compared to, say, math or economics courses. If you pursue philosophy, I think you'll discover that the standards of argumentative rigor expected in philosophy courses surpass -- sometimes by far -- the standards of rigor expected in any courses outside of math, and again they're applied to a much more varied, and often deeper, set of questions.

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