Advanced Search

Suppose P is true and Q is true, then it follows logically that P --> Q, that Q --> P and therefore that P Q. Now, suppose that P is 'George W. Bush is the 43rd President of the US' and Q is 'Bertrand Russell invented the ramified theory of types', both propositions are true, and therefore the truth of both guarantees the truth the aforementioned propositions. But it seems bizarre to say that Russell's invention of the theory of types entails that Bush is the 43rd president, as well as the other logical consequences. After all we can conceive of a scenario where Russell invents the ramified theory of types, but Bush becomes a plumber (say), if that is a possible scenario, it would seem that the proposition "If Russell invents the ramified theory of types then Bush is the 43rd President of the US" is false given the definition of 'if then'. But after all, does it make sense to say that a proposition entails another only in the actual world? (That doesn't seem to have as much generality as we...

Briefly, yeah. I think I see what you're getting at. When P and Q are true (which I think is what you mean by 'P&Q'), then P->Q, Q->P, and P is materially equivalent to Q. But note that this was the case for your earlier puzzle, too. Keep in mind that in standard first-order propositional logic, P->Q is a matter of only "material" implication, and P equivalent to Q is a matter only of "material" equivalence. That is, all 'P->Q' says is that when P is true Q is also true. All the equivalence says is that they have the same truth values. It doesn't say that there's some reason for the truth values being as they are, that there's any other connection between the statements, or that P and Q will be true in every possible or imaginary world. In our world P and Q happen both to be true, and that's enough. The issue you're pointing to is addressed by what's come to be called "Relevance Logic" and it is sometimes used to mark a difference in the use of the terms "implication" and "entailment." In relevance...

I have a question about the identity of a certain kind of fallacy, namely: A = C B = C therefore A = C Confusingly, I have read that the above syllogism is valid; and yet consider this argument I've heard recently: Obama = Good speaker Hitler = Good speaker therefore Obama = Hitler Clearly the latter is a fallacy. So, I have two questions, really: 1) What is the name of this fallacy? 2) How can it be a fallacy if the first syllogism (A = C, B = C, therefore A = C), whose form it follows, is considered to be valid . . . or am I wrong about it being valid?

Well, yes and no. What you've got here is a tangle, just the sort of tangle that actually does lead to serious philosophical problems. You see, what you've got in the first place isn't exactly a syllogism. So, it's neither a valid nor an invalid syllogism. It looks a lot like the following syllogistic form (which is invalid): "All P are M. All S are M. Therefore, All S are P." You can see that this invalid by plugging in the following terms. "All Pigs are Mammals. All Siberian Huskies are Mammals. Therefore, all Siberian Huskies are Pigs." While the two premises are true, the conclusion is clearly false--and that doesn't happen in valid arguments. This invalid form doesn't have a specific name, really, but it does commit the fallacy of "undistributed middle." I think one reason you may have lost your way here is because you use equal signs in your presentation. If you intend to use the equal sign as short hand for the verb "to be" ("are") in the same what as I have used "to be" ("are") in my example,...

Tautology is popularly defined two main ways: 1) An argument that derives its conclusion from one of its premises, or 2) logical statements that are necessarily true, as in (A∨~A). How are these two definitions reconciled? The second definition is only a statement; it has no premises or conclusions.

You've definitely put your finger on a problem. I'd say that for most purposes the two definitions aren't reconcilable because they belong to different discourses or contexts. The first usage is more colloquial and rhetorical. The second is a technical definition. The term "valid" is used in similarly different ways. In common discourse, one can make a "valid point"; but in technical terms only arguments or inferences, not points, can be valid. There is, however, at least one way to make the two definitions consistent: assume in the first that the premise from which the conclusion is derived is the same as the conclusion. From two premises A and B, the conclusion A follows. This, of course, becomes a variant of begging the question. Conversely, I suppose, one could argue that a tautology follows from itself, making the second definition applicable to an argument. (Note that the way you've phrased the first definition is a bit odd, since all good arguments derive their conclusions from their...

I believe that Kant defended the "law of cause and effect" by stating this argument: (P) If we didn't understand or acknowledge the law of cause and effect, we couldn't have any knowledge. (Q) We have knowledge. Therefore: (P) we acknowledge the law of cause and effect. Isn't this line of reasoning a fallacy? P implies Q, Q, : P

You have certainly put your finger on a complex issue. One might say you've got a dragon by the tail. First, I should call your attention to the fact that you've rendered his argument in two logically different ways. The first rendering is actually a valid form of deductive inference, not a fallacy. Philosophers, in their pretentious way, call it a modus tollens. The terms in which you've put it allow for this rendering: 1. If Not-P, then Not-Q. 2. Q. 3. Therefore, P. And, by the way, that first rendering can also be restated in another valid form called a modus ponens: 1. If we have knowledge (Q), then we understand or acknowledge the law of cause and effect (P). 2. We have knowledge (Q). 3. Therefore, we understand or acknowledge the law of cause and effect (P). There's a rather large issue lurking here, too, as to what "understanding" and "acknowledging" mean, how they're similar, how they're different. (See, for example, Stanley Cavell's, "Knowing and...

How do we get better at reasoning, and what would such an ‘improvement’ be exactly? What sort of benefits would be gained that would distinguish reasoning from some other sort of guide to the truth (whatever that might be)?

There are broadly speaking four ways we get better at reasoning (narrowly speaking there are countless). 1. We learn to apply existing logical principles more skillfuly, using them in new contexts and using them more effectively in old contexts. 2. We invent or discover new logical principles. 3. We learn to apply existing error theories better so that they help us better understand how and why we go wrong in reasoning. 4. We invent or discover new error theories. Reasoning might be understood as a set of discursive procedures or rules that make it possible for us to preserve or secure truth. By this I mean that if we begin with a set of truths (even a single truth) reason allows us to proceed to new truths with a significant degree of assurance, perhaps even certainty. Reasoning well means doing this skillfullly in lots of different contexts, in lots of different ways, with lots of different forms of language and thought. It also means understanding how people go wrong so that even...

Can someone please explain to me the difference between induction and deduction? I think I get it, but merely reading it in books is not enough!!! Thanks!

In deduction, the move from premises to conclusions is such that if the premises are true, then the conclusion must also be true. For example, take the following argument: 1. Elvis Presley lives in a secret location in Idaho. 2. All people who live in secret locations in Idaho are miserable. 3. Therefore Elvis Presley is miserable. If the two premises are true, then it must be true that Elvis is miserable. Note, however, that if Elvis doesn’t exist any longer, then the conclusion need not be true either. But IF 1 and 2 are true, then so is 3. Unlike deductive inferences, induction involves an inference where the conclusion follows from the premises not with certainty but only with probability. Often, induction involves reasoning from a limited number of observations to wider, probable generalisations. Reasoning this way is commonly called "inductive generalization." It’s a kind of inference that usually involves reasoning from past regularities to future regularities. One...

How are we sure that anything is true because our whole world is based off of the circular logic that we assume our logic is right?

It's a great question, and the way you pose it raises a number of fascinating issues. About the concern whether "anything" is true, I assume you mean any truth claim. Things can be said to be "true" in certain contexts ("That's a true Rembrandt"; "That way is true north"; "Mine is true love"), but typically in philosophical contexts it's statements that are either true or false. Here I'm inclined to say that it's not unwise to withold some measure of judgment on virtually any truth claim. If by "sure" you mean, impossible to doubt, then I think we might be better off not trying to achieve it. Skeptical philosophers like Sextus Empiricus, Hume, and Montaigne have in various ways advanced ideas like this. On the other hand, Ludwig Wittgenstein and other philosophers have held that in many contexts it's simply senseless to raise doubts our to try to raise doubts. I must say, I've not been persuaded yet by this strategy. But, as Hume reminds us, we must be as diffident of our doubts as of our...

According to Descartes' demon hypothesis, would it be possible for the demon to deceive us about the rules of logical inference e.g. could my belief in the law of non-contradiction be caused by the demon?

May I weigh in a bit? I think that panelists are right to suggest that while the dream argument addresses the veracity of perception about the world, the demon argument goes farther and addresses mathematical and logical inferences. I'd like, however, to return to Peter Lipton's question about why the cogito survives the demon argument. There's a bit more to say on that score that explains Descartes's position. There's a difference between mathematical and logical inferences and the cogito, and that is that what Descartes finds persuasive about the cogito is not a matter of inference but rather direct intuition. The demon argument works but pointing out that in any discursive movement of thought, from one idea to another, the demon might interfere. Discursive thought, therefore, is dubious. N.B. that's why the example Alexander George quotes is about discursive movements like "adding" and "counting." In this way, Descartes anticipates, in a way, the point Quine makes in his article, "Two...