So, it's my understanding that Russell and Whitehead's project of logicism in the Principia Mathematica didn't work out. I understand that two reasons for this are (1) that some of their axioms don't seem to be derivable from pure logic and (2) Gödel's incompleteness theorems. However, particularly since symbolic logic and the philosophy of mathematics are not my area, it's hard for me to see how 1 & 2 work and defeat the project.

To follow on some of Richard's observations: I have never found it at all a compelling argument against logicism that it would have the existence of infinitely many natural numbers be a logical truth. That is not an argument against logicism so much as a restatement of the claim that it is incorrect. Richard's discussion of Boolos reminds me of Gödel's own caution with regard to what his Incompleteness Theorems establish with respect to Hilbert's Program (roughly, Hilbert's attempt to show that if a basic, or finitary, proposition of mathematics can be established using the powerful, or infinitary, methods of classical mathematics, then it could already have been established using very basic, or finitary, reasoning). [For more on Hilbert's Program, you might see here or here .] I don't myself think that the phenomenon of incompleteness puts paid to Hilbert's project (as divorced from certain other beliefs that Hilbert may have held, such as the belief that all true mathematical...

I have trouble understanding what people mean when they use a phrase with the word exception. To me it sounds like a contradiction. So my question has two parts: A) Is using the term exception ever legitimate? B) Does the term "except" usually contradict the general rule that comes before it? For example, All ice cream should be taxed, except vanilla. This seems that the quantifier "all" is false if a member is excluded. For example, All students passed the final exam except Roy. Seems to me this means only Roy failed the final exam and the quantifier "all" makes the sentence false. Please help me make sense of the term exception. Thanks for your help.

I see what you're thinking: that in sentences such as: (1) All teams lost except Spain we give in one hand what we take with the other. We are affirming that all teams lost and also that Spain did not lose. You're right that this would indeed be a contradiction. But I don't think the logical structure of such sentences is as you propose. The issue depends on what logicians call the relative scope of the terms "all" and "except". You understand (1) to mean: (2) (all teams lost) and (Spain did not lose) which is indeed a contradiction. Logicians would actually make a few changes to bring out more clearly the logical structure of (2): (2') (each team is such that it lost) and (it is not the case that Spain lost) Again, this is a contradiction. But a more accurate analysis of how the sentence (1) is usually meant is this: (3) all teams except Spain lost where a more perspicuous representation of the logical structure of this is really: (3') each team is such that...

I have recently stumbled upon a short book written by the Catholic theologian named Peter Kreeft. He deductively argued for Jesus’ divinity through an approach he summarized as “Aut deus aut homo malus.” (Either God or a Bad Man.) Basically, his argument works only on the assumption made by most historians. Jesus was a teacher, he claimed divinity, and was executed. So, assuming this is true he says Jesus must’ve been one of three things. One possibility is that he was a liar. He said he was divine even though he knew it was not true. Another possibility is that he was insane. He believed he was divine even though he wasn’t. The final possibility is that he was telling the truth and he was correct. He was divine. He goes through and points out that Jesus shows no symptoms of insanity. He had no motive for lying. In fact, he was executed because of his claims. That gives him a motive to deny his divinity, which he apparently was given a chance to do by according to the Jewish and Roman sources on the...

I don't find this argument persuasive - for what it's worth, versions of it have been given for centuries. History and common experience present us with many individuals who function well in many circumstances despite the fact that they have delusions of divine (or other) grandeur. Usually such individuals suffer greatly in all kinds of ways on account of their delusions. We do not take this suffering to speak to the correctness of their perceptions but rather to the psychically entrenched nature of their delusions. I do not see in any of the alleged facts about Jesus that you point to any reason for not counting him to be one such individual. In some everyday sense of "plausible," it seems much more plausible to think, on the basis of the evidence you have put forward, that he was delusional than to think that he was divine. So what I would take issue with is the claim that according to "the historical information ... [Jesus] showed no signs of insanity." He did: he claimed to be capable of...

Is it possible to prove that something cannot be derived (considering only well-formed-formulas) in a natural derivation system? I mean a premise P cannot yield the conclusion Q since there isn't any logical rule that justifies the inference but how can someone prove this?

One way to show this would be (1) to prove that the derivation system is sound (that is, if Q can be derived from P in the system, then the inference from P to Q is a valid one); and (2) to show that the inference from P to Q is not valid.

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.

There is a connection between (1) and (2) in that there is a connection between an argument's being a good one and some statement's being logically true. It can be stated somewhat generally like this: The argument whose premises are X 1 , X 2 , ..., X n , and whose conclusion is Y is logically correct (or valid) if and only if the statement "If X 1 and X 2 and ... and X n , then Y" is logically true (or a tautology). For instance, the argument whose two premises are "Either A or B" and "not-B" and whose conclusion is "A" is logically correct. And this amounts to saying that the statement "If it is both the case that A or B and the case that not-B, then A" is logically true.

I studied philosophy in university and I recall that one of my tutors for symbolic logic was trying to walk me through a problem by saying that if you have a large enough set of premises, two of them will inevitably contradict one another. I've always had trouble understanding (and consequently, accepting) this proposition because: if one conceives of reality as a set of claims (e.g., I am right-handed, electron X is in position Y, 2 + 2 = 4, etc.) there are an infinite number of "premises" to the "argument" that is reality and consequently reality is self-contradictory. Am I missing something here? Can you explain which of us is right about this and in which sense? I should mention that I don't necessarily have a problem with reality being self-contradictory, but that really throws symbolic logic out the window (and doesn't throw it out the window at the same time)! Thanks to all respondents for their time. -JAK

I'm not sure what your tutor was getting at either. If your tutor meant that one can always enlarge a set of premises to make it an inconsistent set, that's obviously true: simply add the negation of one of the premises already in the set. If he meant that any axiom system with infinitely many premises (say, one that employs an axiom schema) is inconsistent, then there's no reason to believe that.

Is a computer conceivable that would cut down on Philosophers' work by immediately identifying logic mistakes in arguments? For example: you enter "The Ontological argument for God" or "David Hume's argument against Inductive Reasoning" (or, for that matter, scan in the entire text of Plato's Republic) into the machine, and it immediately uses its programming (which tells it to watch out for contradiction, and all those other logic laws, etc.) and spits out the mistakes in reasoning. Is the problem with this that it would be too difficult to program, or that the laws of logic are under respectable attack?

Philosophy would be much easier if we could program such a machine -- and boring too. But it's not going to happen. For one thing, there's your interesting point that philosophical disputes can go very deep, so deep as to include disagreement about what the laws of logic, of correct inference, ought to be. Secondly, even for first-order classical logic, there simply is no computer that can decide whether any given inference is correct. (This is known as Church's Theorem and was proved by Alonzo Church in 1936.) Finally, there's the fact that evaluating the logical cogency of arguments is only a (small) part of the business of figuring out what to think about someone's argument in philosophy: at the very least, one must also understand and evaluate the assumptions to which the logical reasoning is applied.