It is indeed contingent. The intuitive problem is that "a -> ~b" is not logically equivalent to "~(a -> b)". The "Counterexample" button shows a world in which the modified statement is false: there is a single man (Socrates) and he is mortal.

Proof Editor for Natural Deduction in First-order Logic The Evaluation of an Educational Aiding Tool for Students Learning Logic Bachelor’s thesis in Computer Science ELIN BJÖRNSSON, FREDRIK JOHANSSON, JAN LIU, HENRY LY, JESPER OLSSON, ANDREAS WIDBOM Department of Computer Science and Engineering C UNIVERSITY OF TECHNOLOGY NIVERSITY OF ...

The "natural deduction" proof systems allows you to (temporarily) eliminate the annoying implication without assuming the law of excluded middle. The problem with using "natural deduction" in a beginners course is that this system has desirable technical qualities beyond the scope of a beginners course.

Enter a formula of standard propositional, predicate, or modal logic. The page will try to find either a countermodel or a tree proof (a.k.a. semantic tableau).

Students will apply standard techniques of natural deduction for quantificational logic, involving important inference rules (e.g., universal instantiation, existential elimination) to determine important quantificational logical properties and relations: e.g., quantificational validity, quantificational consistency, quantificational equivalence.

Logic is the study of the principles and methods used to distinguish "good" reasoning from "bad" reasoning. As it is through good reasoning that we plan, explain, persuade, convince, solve, and prove things successfully through language, good reasoning matters. So too do arguments, for they are the main medium through which we reason.

Natural Deduction is a proof system that is sound and complete for e.g. classical propositional calculus. Sound means that if a formula is provable with ND, it is valid .