SAT is best when there is one true logical answer. ANN are best when you want "best advice".

A good guide (in my experience) -- with SAT while the answer is hard to find it's easy to check (like a Sudoku, or building a timetable with no clashes), while ANN give answers where verifying the answer isn't easy to perform computstionally (if I give you a translation of a text into Latin, verifying it is about as hard as just making a new one from scratch)

There are also probabilistic SAT solvers for cases where there isn't just one true logical answer, e.g., https://github.com/MatthiasNickles/delSAT

I am trying to implement a natural language (English) logic checker. It would detect https://en.m.wikipedia.org/wiki/Formal_fallacy

How could a SAT/SMT help me in this task? I know it's already used for math theorem checking.

E.g All men are mortal. Socrates is a man. Therefore Socrate is mortal. I have currently two way to solve syllogisms

One hard-coded Disjonction of cases (but not flexible enough for non canonical forms of syllogisms)

And a flexible one that generate a graph of transitivity.

Both are 100% accurate.

Could SAT/SMT give me a third way of determining if the conclusion is logically valid? (not non sequitur)

This is exactly what SAT and SMT are for.

SMT is some extra stuff on top of SAT. There are a few frameworks which extend SAT. I'd look at minizinc, it is a fairly easy to use language which can produce SAT (and SMT, and others) as output. Often encoding things in SAT directly gets arkward.

Thank you but as you say, encoding is limiting. SMT are a progress versus SAT. But I believe I would need an SMT that take a graph and rules as input and verify if the graph or parts of the graph satisfy the rules/constraints. And output me the indexes of the satisfied parts of the graph. I believe such a thing does not yet exist.

This is what model checkers are for. They internally (might) use SAT solvers to reason over transition graphs. The general problem for model checkers is of this form:

System description: Transition system like automata or a guarded command language which induces a graph.

System Property: Stated in some logic, CTL/LTL

Output: Satisfied, or Unsatisfied with a counter example (usually a path in the graph which is false).

If your problem can fit into such a framework, have a look at model checkers: https://en.wikipedia.org/wiki/List_of_model_checking_tools

Deep neural networks don't help with problems where there is no solution; neural networks approximate functions where the explicit form of the function is unknown. For example nobody knows how to implement isCat(animal) without using machine learning, but we have enough data on cats to approximate said function without much trouble.

SAT solvers work out if it is possible to satisfy a logical model (hence the name).

In theory a neural network could approximate a function that solves SAT. In practice it is a much better idea to use a SAT solver. Neural networks aren't that powerful at solving logic puzzles.

There are many ways to see differences between the two.

One is heuristic vs algorithm. If SAT solver finds a solution, it's correct. If ANN finds a solution, it might be correct.

ANN is suitable for pattern recognition, statistical learning and representation learning. Learning continuous functions for example. SAT solvers work best for discrete problems.