Problems with using logic to study logic

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?

Solution 1:

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!

Solution 2:

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.