I am trying to learn propositional logic. I have read that axiomatic system is defined since there are some problems which can not be solved using truth tables. I have found such a problem in predicate logic when we use quantifiers. There we should use axioms to derive a formula which contains quantifier, since we can't construct truth table of that formula. But I am interested why we need axiomatic system in propositional logic? We can check if a formula is tautology using its truth table.
[Math] Why is axiomatic system needed in propositional logic
axiomslogicpropositional-calculus
Best Answer
As you said we can check if a formula is a tautology just looking at it's truth table, anyway studying logical system (i.e. a system with has axioms and inference rules) also for propositional logic can be interesting for different reason:
first of all there's the didactic reason: it's a simple deductive system, indeed it can be seen as just a fragment of the system for first order logic, so it helps to familiarize with deductive systems;
a second reason it's complexity: sometimes proving the logic validity of a formula through such deductive system can be easier then proving it via truth tables, which involve calculate the truth values of a formula for all possible valuation of the variables which can become easy a troublesome for formula with huge number of variables, while in many cases writing a proof can be much easier involving just few application of inference rules.
Probably there other reasons too that I'm not remembering right now, but I reserve to myself the right to add them later. :)