# Axiomatic system and proof.

• B
Hi, I have a question about axiomatic system and proof.

Let's say we have a finite sequence of propositions ai, which is an axiomatic system.
To prove a proposition P that is a finite sequence of propositions qi with axiomatic system {ai}, we can take 3 methodologies.

(A) qi itself is equivalent to one of the propositions of axiomatic system.
(T) Tautology
(M) Modus Ponens.

But what makes me uncomfortable is (T) tautology. It acts as deus ex machina "within" the proposition P that is examined with axiomatic system.

I believe an axiomatic system is justified by (T) because it justifies propositions by itself, but I wonder why we can use tautology within the propositions P which is under examination of axiomatic system. Because if we can use tautology, inside the proposition P, any proposition can be essentially justified within P regardless of the given axiomatic system (we can justify any qi with tautology).

Could anyone please enlighten me why we are allowed to use tautology (T) within a sequence of propositions P? which questions me why we have a separate axiomatic system although we can justify the sequence by itself.

Stephen Tashi