Is it possible that there exist types or methods of argument/reasoning that have not been discovered or employed before? (I do not mean specific arguments for specific problems, but Forms of arguments, so to speak.)

How does Godel's incompleteness theorem affect the way that mathematicians understand and see mathematics as well as the world (if at all)? I'm not even close to a mathematician, but even a slight dose of the idea and theorem were enough to affect me so I suppose that I'm curious.

I want to start to study logic on my own to the level that I can understand a book like Enderton's and Michael Potter's Set Theory and Its Philosophy, also be able to study mathematical logic like recursive functions and model theory and Understand Logicist Reduction and why It failed because of Godel. Suppose that I've read nothing on logic and I'm not fair in math, but I want to be good at it. I'll be gleeful if you introduce me a series of books on a) Introductory books on logic followed by more professional ones, b) Mathematics of logic, c) books on Logicist Reduction and Godel or what is necessary to understand them. I have a plenty of time to study all these and will face all the troubles. Thank You Very Much Pouria From Turkish part of Iran (I think you guessed why I have to study on my own)

I am reading a book that explains Gödel's proofs of incompletenss, and I found something that disturbs me. There is a hidden premise that says something like "all statements of number theory can be expressed as Gödel numbers". How exactly do we know that? Can that be proved? The book did give few examples of translations of such kind (for example, how to turn statement "there is no largest prime number" into statement of formal system that resembles PM, and then how to turn that into Gödel number). So the question is: how do we know that every normal-language number theory statement has its equivalent in formal system such as PM? (it does seem intuitive, but what if there's a hole somewhere?)

There's a logical scenario which often comes up in discussions around the question of voting. We all know the conversation... Person 1: I don't vote because my vote has no impact on the outcome of the election. Person 2: Not on it's OWN it doesn't, but if everyone thought that, no one would vote, and THEN what would happen?! Person 1: But I don't decide whether all those other people vote, I only have control of my 1 vote! My question here relates not to whether or not one should or shouldn't vote, or to the voting example alone, but rather to the logic of this situation. For this example let us assume (for the sake of the point I am interested in) that it is universally agreed that all people (including Person 1 and 2) agree that nobody voting is an outcome that everyone wishes to avoid. And also assume (despite the conversation above!) that everyone decides privately whether to vote or not, such that their decision cannot influence others decisions) Finally assume that the election involved has...

Why do so many Anglo-American philosophy departments still prefer to teach ideas that depend on symbolic logic? Or in another light, why is so much contemporary philosophy in America still dedicated to analysis and ideals of "clarity" that depend on "higher order" languages?

It seems that many philosophers use the "socrates" argument to explain a simple deductive argument. This argument is P1: All men are mortal P2: Socrates is a man C: Therefore, Socrates is mortal. However, is this not begging the question because P1 assumes that Socrates is mortal?

Why Studying Model Theory is important to Philosophy? How much Math and Logic one needs to know to get down to study Model Theory? and does it have something to say about Hierarchical essence of scientific language?

I presently working through Grayling's Introduction to Philosophical Logic (Blackwell), after studying philosophy at university in the late 1960s. Can anyone recommend a follow-on text (for when I feel I have assimilated this book)? (I have seen the interesting replies to the August post about further reading on symbolic logic.) Peter

I have two question abouts logic. In occidental thought, logic generally is presented as bi-polar (no pun intended); however, I am not quite sure of the 'correct' formulation. Is it: either "A is true," or "not A is true" or is it: Either "A is true," or "A is not true." and if you could expand on the distinction a little perhaps? More interesting, to me, is that I have heard the assertion that there also exists a "quadrant" logical system that also works. Either: "A is true" or " Not A is true" or "Neither A nor Not A are true" or "Both A nor Not A are true" This latter form of logic seems to work for things like polarization of light, for example (if you have a vertical polarizer in front of a horizontal polarizer, no light gets through; but if you insert a polarizer at a 45 degree angle in between the two, some light gets through). I can think of some other examples as well; I prefer to hear your responses.