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Does using logic as a tool to study logic itself lead to problems/paradoxes? Similarly to how self-referencing sentences sometimes make no sense, e.g. This statement is false.

When we try to study logic we use logic to arrive at any conclusions about itself. Isn't it somewhat circular?

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  • $\begingroup$ Good point but isn't it the case that when studying logic, when we arrive at some conclusions they may change the way we apply logic itself to study other objects? 'Other object' can be logic itself what makes the whole reasoning circular. Am I missing something? $\endgroup$
    – twowo
    Jul 10 '13 at 21:18
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Typically, mathematicians and logicians take the following attitude: When we set out the details of formal logic, we implicitly adopt in the background a meta-logic. The metalogic is the logic we use in English: it's way we reason about things as mathematicians, as humans, etc., and the hope, then, is that our formal theory of logic (set out in a formal language, not English) will match up with the metalogic we are using (in English). So yes, while it's true that we are using logic to set out the formal details of a theory of logic, still the hope is that the theory of logic accurately represents the way we are using logic in the background.

This, of course, requires admitting at the beginning what kind of logic you're employing as your metalogic. A clear manifestation of this comes when defining what the various formal connectives $\wedge, \vee, \neg$, etc. mean. Often, we'll give an inductive definition as follows:

$\mathcal{M} \vDash \varphi \vee \psi$ if and only if either $\mathcal{M} \vDash \varphi$ or $\mathcal{M} \vDash \psi$

This may look innocent enough, until we ask what the "either...or..." clause of the definition is. Is it a classical disjunction, or an iniutionisitic one, or neither? Surely, answers to such questions will affect our understanding of the given definition. Similarly, when we write:

$\mathcal{M} \vDash \neg\varphi$ if and only if it's not the case that $\mathcal{M} \vDash \varphi$

how are we to understand the English-negation "it's not the case that..."? If our metalogic is classical logic, then the negation is classical; but different choices of metalogic will reflect a difference in our understanding of these definitions.

There's a lot of philosophical literature revolving around this point. For instance, it really takes centerfield in certain nonclassical logics, e.g. relevant logic, where people working in this field typically wish for their theories of logic to be "taken seriously" as genuine logic (i.e. the logic we do and/or should use in English). There are theorems about the formal theory of relevant logic (for instance, the admissibility of distribution) for which there is a classical proof but not a relevant proof (or rather, there is a relevant proof of the classical admissibility, but no proof of the relevant admissibility). So which metalogic one adopts in describing a formal theory of relevant logic becomes crucial. In general, which metalogic one adopts, as well as which formal logic one wishes to study, can affect the results that are obtainable in one's pursuits (intuitionistic logicians have been particularly concerned with this matter as well).

Most of the time, for the mathematician, it suffices to just adopt a classical metalogic, since even then there's a lot that can be said about the formal theory of logic!

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    $\begingroup$ So does this mean that if we change the metalogic we use to study the "theory of logic" we might get completely different results and a different theory? If that's the case, then isn't it just "guessing" with no way of knowing whether the metalogic used is the right one? $\endgroup$
    – Ormi
    Jul 10 '13 at 22:03
  • $\begingroup$ Yes, we might get different results, but that doesn't mean it's "just guessing". Rather, what that means is that the importance (or even the validity) of some of these results in formal/mathematical logic is ultimately a philosophical question that depends on an answer to the question "Which, if any, is the correct metalogic to adopt?". Many seem to think it's classical logic; some think it's intuitionistic logic; others think there is no one right logic. That's not a matter of guessing; it's a matter of holding a different philosophical viewpoint (which is different, I hope!). $\endgroup$ Jul 10 '13 at 22:25
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We use ordinary mathematical thinking to study certain mathematical objects, such as the ones called proofs in certain formal systems. Or else we study, in Model Theory, the relationship between algebraic structures and certain types of strings called formulas.

This is no different than using mathematical thinking to study rings. In particular there is no one step regress, let alone infinite regress.

Added: As an example, when we study (first-order) Peano arithmetic, we are only indirectly studying the natural numbers. In principle, we are discovering properties of an axiom system that has many models not isomorphic to the natural numbers. * In showing the incompleteness of Peano arithmetic, we are studying what sorts of strings can be obtained as endformulas in certain syntactic objects called formal proofs.

Many if not most people who work in Mathematical Logic believe, not unreasonably, that the technical results shed some light on mathematical thinking. However, that is a matter of belief. In any case, the technical results stand.

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  • $\begingroup$ Thank you for your answer but I still do not get it. Are your examples fully analogous to my problem? When you use mathematics to study certain mathematical objects it is different than using logic to study logic. Analogous example would involve using mathematical objects to study the same mathematical objects. $\endgroup$
    – twowo
    Jul 10 '13 at 21:24
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    $\begingroup$ They are decidedly not the same mathematical objects. The natural numbers are the natural numbers. When we study (first-order) Peano arithmetic, we are looking at an axiom system that has many models not isomorphic to the natural numbers. In showing the incompleteness of Peano arithmetic, we are studying what sorts of strings can be obtained as endformulas in certain syntactic objects called formal proofs. $\endgroup$ Jul 10 '13 at 21:31
  • $\begingroup$ @AndréNicolas: I suspect you could expand that comment into an answer. $\endgroup$
    – dfeuer
    Jul 10 '13 at 21:33
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When we study logic, is it not true that we impose certain axioms and rules that we follow, and from those conclude other things? We don't just say "Once there was nothing, and then.... LOGIC!"

However, in regards to the question you're asking... I understand this from a historical point of view moreso than I know about the mathematical details, but Hilbert tried to prove the consistency of mathematics using nothing but the axioms of mathematics in the early 30s (as part of the whole constructivist vs intuitionist math war that was all the rage). It was a big thing at the time, but broke down after Godel's incompleteness theorems showed that it is impossible to prove the consistency of any formal system from the interior of that system. And I think this is where your example "This sentence is false" falls under. You can read more at http://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems

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