Is there an explanation of how logic works?

Question

To get the answer we use the laws of logic. What tool can help to get rid of recursion and tautology when we discuss logic itself? I'll be greatful if you advise the books that have helped you to lift the veil:)

Answer 1

If I understand well you're asking "How can we be sure that the logical rules we use to prove things about Logic are valid ? How to escape that strange recursion phenomena ?"

They are many possible answers and no consensus. The foundations of Logic are not clear at all and diverges in a lot of directions and oppositions (realism vs anti-realism vs radical anti-realism, holism vs reductionism...). Moreover they're so many logical systems... it's almost absurd (classical, intuitionistic, linear, relevant, paraconsistant, quantum, modal, fuzzy, epistemic, ...).

I think it's fair to say that after few thousands of years we don't really understand Logic. That's why some people threw the foundations away to focus on other things.

Two major points of view :

• Logic as based on language. It's the path we took in the "foundational crisis of mathematics" led by mathematicians and philosophers such as Russell, Hilbert, Frege in the early 20th century. In that point of view, we don't give any explanations. We reduce Logic to a syntactic mathematical objects and assume there's a "meta world" above Logic giving an interpretation to symbols without any meaning. We live in the "meta world" and use basic mathematics to prove things about logic as an object. For instance the sentence (A /\ B) will be interpreted to [True] if (A) and (B) are interpreted to [True]. We escape what you call "recursion" by dividing the world into two parts : the world of semantic and the world of syntax (a bit like a duality between subject and object) so we don't have to deal with justifications. That view was strongly related to how the world was seen in the 20th century (see for instance "Sense and Reference" from Frege).

Afterword : maybe some philosophers developped these ideas by using concepts from the Philosophy of Language (see for instance the texts from Ludwig Wittgenstein). But is logic only about language ?

• Logic as computation. This is a recent turnover in Logic. It seems that a lot of philosophers and even logicians doesn't know that but around the 70s, with the emergence of Computer Science, we discovered a link between Logic and Computation : actually, proofs can be seen as computer programs, formulas can be seen as types for programs, reduction of proof (cut-elimination) can be seen as the execution of programs and vice-versa. That link is called the "Curry-Howard isomorphism") and throws away the old view we had about Logic. Logic can now rely on the natural phenomena called "computation". It seems that logic is something already present in the nature that we only approximated with our systems. Some philosophers in that point of view explain Logic by a system of rules interacting with each other in a dual way (no external reference, see Hegel's conception of negation). Most of the work in that field are written in french only (see LIGC group).

Afterword : Today, Linear Logic seems to be the closest system to what Logic truly is (see for instance the texts from Jean-Yves Girard).

Philosophers, Logicians, Mathematicians and Computer Scientists and even more should work together and share their ideas to understand what Logic truly is. It seems that it was already done before but only in France with french texts for some reasons I don't know... (see LIGC group)