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I'm just starting a study of mathematical logic, primarily through 'First Order Mathematical Logic' by Angelo Margaris. In the text, axioms for the statement calculus and the predicate calculus are given and properties of the resulting system are argued. However, it seems a bit circular, in that we are necessarily using logic to argue about logic. This leads to the question, are there axioms for the metalogic used to argue about the logic, and then axioms for the metametalogic, ad infinitum?

user695931
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  • In mathematics they dont really use terms like meta logic, because as you stated you would get meta meta logic and so on. One example would be the incompleteness theorem which is a statement about a formal system in a larger system. The statement exists in the larger one and the inner system unaware. – marshal craft May 18 '17 at 04:42
  • The text uses the term metalanguage and metatheorem. I guess I'm not certain I see the distinction. Doesn't the same question apply to the logic used to argue the incompleteness theorem? – user695931 May 18 '17 at 05:05
  • Well the incompleteness theorem posses strong limitations on the kinds of sets which can be discussed formally but at the same time itself uses set and some notion of set theory. – marshal craft May 18 '17 at 05:19
  • Also as asked this isn't a mathematics question. That is meaningless mathematically. It is entirely philosophical. So what kind of mathematical answer do you expect? Some meaningless symbols delivered by a person of authority? To me this i'd like if you and I just in plain words sat and argued if there is infinite quantity in the universe. We can't readily check? But we sure can assume there is or isn't and then agree on possibilities. Regardless of the prior questions solution – marshal craft May 18 '17 at 05:29
  • We still can consider possibilities inside a formal system with out the answer to your question. I'm not saying it isn't a perplexing question. – marshal craft May 18 '17 at 05:31
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    Yes; metalogic can be formalized in a quite "weak" theory: arithmetic or the theory of hereditary finite set (a subsystem of set-theory). – Mauro ALLEGRANZA May 18 '17 at 06:10
  • @MauroALLEGRANZA This sounds interesting. Would you mind expanding on this, or posting a link that does? – user695931 May 20 '17 at 03:12

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Note that you are using the term "logic" here in two separate but related meanings. One meaning is the way we argue, and the other meaning is a formalization of the way we argue.

The first one indeed is more or less a given; it basically consists of all arguments that "are supposed to convince you". This type of logic is largely explored in philosophy. In mathematics, you indeed have to assume that you already have a way to argue.

The second one is what mathematical logic is about: You define a formal system that captures the essential parts of your logical thinking/argumentation process. That is, you formulate a formal language in which you can describe all statements you want to talk about, and manipulation rules where you can convince yourself that when you encode a statement in that language, and apply the manipulation rules, then the resulting statement necessarily will also be the encoding of a valid statement in the formal language.

Why would you do something like that? Well, on one hand, by formalizing,you can make sure that you don't accidentally make a step that is not justified: Your formal system only contains those rule where you are absolutely convinced that they hold; therefore if you follow those rules, you know for sure that if you translate each step back into an argument, that argument will be absolutely convincing. Also, by being conservative what you include, you minimize the risk of later finding that some of your used rules being found problematic. And finally, if you do steps formally, you always know exactly what assumptions you used, so if you later indeed start to doubt some of the used assumptions (for example, you might start to doubt the law of the excluded middle), you can go through your proofs and for each one see whether you used it, and be sure that any proof where you didn't use it remains valid.

Note that you are also creating a formal language, but to describe that language, you still need a language to describe it. Ultimately, you resort to your native language (e.g. English) to do so. There may be several layers between your original language and the formal language (e.g. the first step will likely be a more formalized version of the natural language where you give a more precise definition to common terms), but ultimately you are drawing on your native language, just as you are drawing on your ability to think/argue when defining logic.

celtschk
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  • I can kind of see what you are saying, in the sense that in the end it is just a means of creating more convincing arguments. In the extreme though, we can ignore all interpretations and reduce mathematics to nothing more than symbols and rules for their manipulation right? If we were to take that approach, would we need to resort to our native language? Could each successive metalanguage be just a set of symbols and rules for their manipulation? – user695931 May 20 '17 at 04:31
  • Let me ask a different but related question. One of the things that we argue is that a formula is a theorem of the statement calculus if and only if it is a tautology. How do we know that the same thing holds for the metalogic used in the argument? – user695931 May 20 '17 at 04:43
  • @user695931: We have to resort to some pre-existing structure. Using our natural language is the easiest because it is already developed; we of course at one time also learned that, by resorting to a more primitive language of pointing to things, face expressions, gestures, etc., ultimately using our native shared interpretation of sensual information. But the point is, we need a starting point from which we build a common base, otherwise we cannot give meaning to the formal language, nor describe the formal rules. Quick, is $aX\circ^\otimes \xi_\wedge$ well-formed, and if so, is it true? – celtschk May 20 '17 at 08:12
  • @user695931: On all theorems being tautologies: In a formal language, that is by construction: We only allow manipulations that turn tautologies to tautologies, and the axioms are tautologies by definition. If the meta-language is a formal one, then this is also true by construction. Of course, ultimately you'll reach a non-formal language. In that, being a tautology is basically part of the definition of the term "theorem". So the claim is still true, but it is true by definition, instead of by construction. – celtschk May 20 '17 at 08:53
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I haven't thought about this long and I don't know if this is correct.

However it seems to me the point of mathematics is to set about some axioms and an agreed decidable system of logic so that we can explore the consequences of such.

For example discussing the broad existence of infinity in reality would be philosophy whilst the discussion of the possibilities of infinite classes in a decidable manner, mathematics. Which I believe a specific subset of philosophy.

Perhaps this can at least explain then how to decide a mathematical approach to solving this problem which satisfies the intuition.

marshal craft
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