What are some practical applications of mathematical/formal logic to science and humanities?

Question

I am studying a bit of this and so far it seems that, apart from math and computer science, the discipline of Logic is very self facing, with logicians proving things for other logicians. It left me wondering about interdiscipliary work. Specifically, can classical(propositional and first order predicate) and/or non-classical (i.e., fuzzy, intuitionist, relevant etc) logics provide unique insights or analysis in the following domains:

1. History,
2. Law,
3. Psychology,
4. Engineering?

I know this is a bit broad, just looking for smattering of concrete examples indexed to these domains.

Thanks

Answer 1

I think that Mathematical Logic is not the "foundation" of Mathematics; refer to Y.Manin, A Course in Mathematical Logic for Mathematicians (2nd ed, Springer - 2010; pag.xi) :

Foundational problems are for the most part passed over in silence. Most likely, logic is capable of justifying mathematics to no greater extent than biology is capable of justifying life.

Math Logic is Mathematics : Proof Theory, Model Theory, Computability Theory. But ML is a "strange" branch of Math because it has as his object of study Math itself.

The impressive success of Math is with application (through physics, engineering, economy) to the understanding of the external world.

ML has its application in the study of a particular "human activity" : the mathematician's one and its "products" : mathematical theories.