How do you solve logical deduction?

Answer 1

I don't know what you mean by solving logical deduction. Do you mean how do you tell, given an allegedly logical deduction, whether it really is logical? Or do you mean, given a theorem, how do you logically prove it, that is, prove that it logically follows from the axioms?

The last question is very complicated. Some theorems have taken centuries to prove (like Fermat's last theorem and the independence of Euclid's Parallel Postulate), and some have not yet been proven, like the Goldbach conjecture and Riemann's hypothesis.

The first question is much simpler, but to describe exactly how to verify the validity of a deduction, we would need to know what kind of deduction it is. For example, a deduction involving only logical connectives like and, or, if-then, not can be verified with a truth table. Those involving quantification or non-logical symbols like set membership require looking at the proof and seeing that each step can be justified on the basis of the axioms of the system, whether it is the system of Euclidean Geometry, of the field of real numbers, or of Zermelo-Frankel Set Theory, etc.