can we reason about logic?

Question

People who study mathematical logic make arguments about logic itself. So it seems that people take for granted an "intuitive logic" (otherwise, how would they form arguments?). So the observation is this: there is an accepted way of reasoning all throughout math, which I'm calling "intuitive logic", and we apply that same reasoning towards logic itself.

My concern is this:

Can "intuitive logic" be studied? Any form of reasoning about "intuitive logic" would have to use "intuitive logic". This seems paradoxical because reasoning/argumentation is used as a method of justification and it doesn't really make sense to provide justification for something that has been taken for granted.

I think my question is basically equivalent to

How to justify the use of logic?

Answer 1

Mathematical logic isn't logic, but a formalisation thereof; one view is to think of it as mathematics as inspired by logic, in the same way that mathematics has been inspired by physics but is not synonymous with it - indeed they often have difficult understanding each other despite the laymans view that they are virtually synonymous.

There is such a thing as naive set theory, and mathematicians that aren't set theorists use this all the time when they reason about sets. Likewise, one can say that there is a naive or intuitive logic that mathematicians use when they reason about mathematics.

Logic remain a field of study in philosophy, and it isn't neccessarily mathematical; linguistics and semantics for example come into it as well as others.

Answer 2

We have to assume some finitary arithmetic, otherwise there's no way to even begin making sense of symbols, formulas, etc.

According to Hilbert, there is a privileged part of mathematics, contentual elementary number theory, which relies only on a “purely intuitive basis of concrete signs.” Whereas the operating with abstract concepts was considered “inadequate and uncertain,” there is a realm of

extra-logical discrete objects, which exist intuitively as immediate experience before all thought. If logical inference is to be certain, then these objects must be capable of being completely surveyed in all their parts, and their presentation, their difference, their succession (like the objects themselves) must exist for us immediately, intuitively, as something which cannot be reduced to something else.

These objects were, for Hilbert, signs. The domain of contentual number theory consists in the finitary numerals, i.e., sequences of strokes. These have no meaning, i.e., they do not stand for abstract objects, but they can be operated on (e.g., concatenated) and compared. Knowledge of their properties and relations is intuitive and unmediated by logical inference. Contentual number theory developed this way is secure, according to Hilbert: no contradictions can arise simply because there is no logical structure in the propositions of contentual number theory.

The intuitive-contentual operations with signs forms the basis of Hilbert's metamathematics. Just as contentual number theory operates with sequences of strokes, so metamathematics operates with sequences of symbols (formulas, proofs). Formulas and proofs can be syntactically manipulated, and the properties and relationships of formulas and proofs are similarly based in a logic-free intuitive capacity which guarantees certainty of knowledge about formulas and proofs arrived at by such syntactic operations. [...]

The question of what Hilbert thought the epistemological status of the objects of finitism was is equally difficult. In order to carry out the task of providing a secure foundation for infinitistic mathematics, access to finitary objects must be immediate and certain. Hilbert's philosophical background was broadly Kantian, as was Bernays's, who was closely affiliated with the neo-Kantian school of philosophy around Leonard Nelson in Göttingen. Hilbert's characterization of finitism often refers to Kantian intuition, and the objects of finitism as objects given intuitively. [...] Both Bernays and Hilbert justify finitary knowledge in broadly Kantian terms (without however going so far as to provide a transcendental deduction), characterizing finitary reasoning as the kind of reasoning that underlies all mathematical, and indeed, scientific, thinking, and without which such thought would be impossible.

https://plato.stanford.edu/entries/hilbert-program/

Answer 3

I think you have this backwards. 'Reason' is inherently linguistic: reasoning is what we do to systematize and structure out language use. Logic and mathematics are particular forms of reasoning done within particular linguistic domains with particular linguistic rules. There is obviously some feedback — it's useful sometimes to take insights from logic and mathematics and try to extend them to the larger domain of reasoning proper — but let's not put the cart in front of the horse.

Wondering whether 'intuitive logic' can be studied the way we use reasoning in maths is like driving down the road in one's car wondering whether it's possible to make a functional wheel.

Answer 4

You are right to say there is an intuitive use of logic. Even little children can understand basic deductions and use logical operators in their speech. Consider a child who says, "If I go to bed, then tomorrow I can have breakfast AND desert?". While not the only example, clear a mastery of the conditional and the conjunction. It is a trope that children quickly acquire the skills of a lawyer around 3 or 4. The basic operations of quantification, qualification, classification, and logic are a function of neural development.

Keeping in mind that logic is grown from neurons, it helps keep perspective on what logic really is and isn't. Logic isn't some program transferred to the mind like software is uploaded to a computer. Logic is more like the skill of playing a violin: some natural talents are present, but then the skill is slowly developed through practice when the brain is ready. As a former teacher, particularly enlightening are the ideas of Jean Piaget's theory of cognitive development. While your question is about logic, which is a field of study generally placed in philosophy departments, in some ways your question is a question of human development.

Ultimately, there is no monolith called "logic", but rather a series of "logics" that vary in terms of complexity and are adopted one after another. For instance, Aristotelian syllogisms are rather simple and intuitive, but modal logics require some studying, particularly as formalisms are used. Mathematical logic is just another formalism in a sense.

Can the "intuitive logics" be studied? Yes, they are, but not by philosophers. Rather the basis of how the brain infers is more a question pursued by cognitive scientists like those with backgrounds in cognitive semantics, artificial intelligence researchers, and psychologists. Generally the intuition-provided logics are part of a philosophers intuition or metaphysics and often aren't scrutinized.