Why on earth not? Elementary mathematics is full of proofs of negative claims. There are familiar school-room proofs that no negative number has a real square root, that no fraction is the square root of two, that there is no largest prime number.
Is every statement true?
Consider the following argument:
If a statement is true, then it is a member of the set of true statements.
If a statement is false, then it implies a contradiction. Since anything follows from a contradiction, it follows that the statement is true. Thus the statement is a member of the set of true statements.
Since a statement is true or false, all statements therefore belong in the set of true statements. All statements are true, with the set of false statements being a subset of the set of true statements. A statement thus is either true and true only, or both true and false.
Does this mean that all statements are true?
It is false that Cambridge is a bigger city than Oxford. But that doesn't meant that the statement that Cambridge is a bigger city than Oxford entails a contradiction. It plainly doesn't. We can all imagine a world where history went just a bit differently and Cambridge ended up the bigger city; there's no internal incoherence at all in that counterfactual story, no contradiction is entailed. You might say, sloppily, that the claim that Cambridge is a bigger city than Oxford contradicts how things are (or some such). But contradicting how things are in this sense, i.e. being plain false, doesn't mean being necessarily false, and doesn't mean entailing a constradiction in that sense of "contradiction" in which, arguably, anything follows from a contradiction.
Modern logicians teach us that some of the inferences embodied by the Aristotelian square of opposition (i.e., the A-E-I-O scheme) are not valid. Take the inference from the Universal Affirmative "Every man is mortal" to the Particular Affirmative "Some men are mortal": the logical form of the first proposition is a conditional ("Every x is such that if x is a man, then x is mortal") and we know that a conditional is true whenever its antecedent is false. In other words, the proposition "Every x is such that if x is a man, then x is mortal" is true even if there were no man, so the aforementioned inference is invalid. But if the universal quantifier has not ontological import, why such a logical truth as "Everything is self-idential" implies that there is something self-identical? And, above all, why the classical first order logic needs to posit a non-empty domain?
What does it mean to say that the logical form of "Every man is mortal" is "Every x is such that, if x is a man, then x is mortal"? The content of this claim is, in fact, quite obscure! However what is true is that if we are going to translate the English "Every man is mortal" into a a standard single-sorted first-order language of the kind beloved by logicians, the best we can do is along the lines of (For all x)(Fx -> Gx). And, as you say, in so doing, we don't respect the existential commitment which arguably accrues to the "Every" proposition. OK, but that's one of the prices we pay for trying to shoehorn our everyday claims involving many sorted quantifiers (as in "Every man", "no woman", "some horse", "any natural number") into an artificial language where (in any application) all the quantifiers run over a single common domain. That's a price typically worth paying in order to get other benefits (ease of logical manipulation, etc.). But if, in some context, we don't want to pay the...
What is the relation between logic and good reasoning? I once thought that logic was the science or study of good reasoning, but I've read a few things (mostly online, I confess) saying that logic is only a matter of "formalizing" reasoning (making it clear and unambiguous, and perhaps making possible that computers reproduce it). But whether reasoning is good should not be a concern for logic. Is that so? And if it is so, what is the current name for the study of good reasoning?
The business of logic is the evaluation of reasoning -- "do these premisses really support that conclusion"? But we want a systematic theory, not just piecemeal case studies. It is difficult to be systematic about reasoning presented in a ordinary language (think e.g. of the different ways we have of expressing generalizations in English using all/any/every/each, and the subtly different ways these behave). So ever since Aristotle, logicians have been attracted by a "divide and rule" approach. Separate the task of rendering arguments into a clear, unambiguous, tidy formalized framework, from the task of evaluating the resulting formally regimented arguments. The prime point of the formalization, though, is to aid the evaluation of arguments and develop proof-techniques for warranting complex inferences. Logicians still care whether reasoning is good!
Is it impossible that there be two recursive sets T and T* of axioms (in the same language) such that their closures under the same recursive set of recursive rules is identical and yet there is no recursive proof of this fact? It seems impossible but a simple proof of this fact would help elucidate matters!
To decide whether T and T* have the same deductive closure involves deciding whether the axioms of T* are deducible in T. That, in general, will require have a decision procedure for determining whether a given sentence is a deductive consequence of T. But theoremhood in T could be recursively undecidable.
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?)
Gödel's first incompleteness theorem says that any particular formalized theory T which contains enough arithmetic and satisfies some other modest conditions is incomplete. The standard Gödelian proof depends on coding up T -formulae (including those T -formulae which are statements belonging to number theory) using the Gödel-numbering trick. And you can always do that if T is indeed a formalized theory. This just falls out of the conditions for counting as being properly -- in the jargon, 'effectively' -- formalized. In an effectively formalized theory, by definition, we can nicely and determinately regiment the strings of symbols that count as T -formulae and number them off. But note: it is only required for the standard proof of T 's incompleteness that we can code up T -formulae -- and hence code up any statements of number theory expressible in T . It is not claimed that we can code up "all statements of number theory", whatever that might mean. And...
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?
I'm not sure what is meant by "prefer to teach ideas that depend on symbolic logic". Most departments teach e.g. aesthetics, political philosophy, the history of early modern philosophy, the philosophy of mind, and so on and so forth -- and symbolic logic features little if at all in those courses. (When did you last see a quantifier when discussing how it is that we can apply emotion terms to music, or discussing whether we can justify more than a minimal state, etc., etc.?) And a concern for clarity has little to do with symbolic logic (and nothing at all to do with 'higher order' languages). Clarity matters because we want to seek the truth co-operating with other enquirers. And we can't co-operate with other enquirers by together subjecting our conjectures to stern test and criticism and proposing revisions if we can't manage to make ourselves very plainly understood to each other. Of course there are always intellectual pseuds who get off on talking to themselves with willful obscurity ...
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?
In response to the original question. We might have general grounds for thinking that all men are mortal -- e.g. general beliefs about the structure of human beings and about the relevant biological laws -- which we accept on inductive grounds (in a broad sense of inductive) and where our supporting evidence, as it happens, doesn't depend on inspecting Socrates in particular. So there need be nothing "question-begging" in any sense in then going on to deduce a claim about Socrates. In response to Sean Greenberg, (a) it should be noted that the Socrates argument is not a simple modus ponens of the form (1), (2), (3) (the main logical operator in the "all" premiss is a quantifier, not a conditional). Also (b) he uses "sound" and "valid" the wrong way round in the first part of his answer, though that slip seems to be corrected in the last sentence. For the record, in by far the dominant modern usage, an argument is "valid" if the conclusion follows logically from the premisses, and "sound" if it...
I have been reading discussions on this site about the Principia and about Godel's incompleteness theorem. I would really like to understand what you guys are talking about; it seems endlessly fascinating. I was an English/history major, though, and avoided math whenever I could. Consequently I have never even taken a semester of calculus. The good news (from my perspective) is that I have nothing to do for the rest of my life except for working toward the fulfillment of this one goal I have: to plow through the literature of the Frankfurt School and make sense of it all. Understanding the methods and arguments of logicians would seem to provide a strong context for the worldview that inspired Horkheimer, Fromm, et al.
So yeah, where should I start? Do I need to get a book on the fundamentals of arithmetic? Algebra? Geometry? Or do books on elementary logic do a good job explaining the mathematics necessary for understanding the material?
As I said, I'm not looking for a quick solution. I...
1. I don't think there is any reason to suppose that learning about mathematical logic from Principia to Gödel will be any help at all in understanding what is going on with the Frankfurt School. (The only tenuous connection I can think of is that the logical positivists were influenced by developments in logic, and the Frankfurt School were concerned inter alia to give a critique of positivism. But since neither the authors of Principia nor Gödel were positivists, it would be better to read some of the positivists themselves if you want to know what the Frankfurt School were reacting against). 2. Of course, I think finding out a bit about mathematical logic is fun for its own sake: but it is mathematics and to really understand I'm afraid there is not much for it other than working through some increasingly tough books called the likes of "An Introduction to Logic" followed by "Intermediate Logic" and then "Mathematical Logic". Still, you can get a distant impression of what's going on...
I aced a basic logic class in college that covered both sentential and predicate logic. I am interested in furthering my skills in symbolic logic, but I don't know how. My school doesn't offer any upper-level logic courses. I'm thinking I would like to buy a simple textbook for a more in-depth study of the more advanced concepts (I've heard the term "modal logic" thrown around, but I don't know what that is). Can you suggest a good text or author I should investigate?
Shame on your school! :-)After a basic logic you can either go deeper (more of the same, but pursued to greater depth), or go wider (look at logics that deal with more than do sentential and predicate logic -- modal logic, for example, which has primitive operators for "necessarily" and "possibly" -- and also look at rivals to classical logic. Going a bit deeper: try David Bostock Intermediate Logic , OUP ; Ian Chiswell & Wilfrid Hodges, Mathematical Logic, OUP (not as advanced as its title might suggest). Going a bit wider: try Rod Girle, Modal Logics and Philosophy , Acumen; Graham Priest, An Introduction to Non-Classical Logic (2nd edn: CUP ). Some of each: John Bell, David DeVidi, Graham Solomon, Logical Options (Broadview Press).