tautology vs axiom vs premise

Question

I am trying to sharpen the line among the definitions of tautology, axiom and premise.

What I have understood so far is this:

Tautology: A statement that is proven to be true without relying on any axiom.

Axiom: A statement that is assumed to be true without a proof or by proof using at least one axiom.

Premise: A statement that is assumed to be true to get a conclusive statement. So it has a scope different from that of an axiom.

Are these correct, and if not what would be the way to describe them?

Answer 1

I am not convinced by your definition of tautology. The context is missing. For example in propositional logic it is a statement/formula that is true under any assignment of true/false values for the atomic statements out of which it is 'made'. In first order propositional logic, a statement is a tautology if it can be 're-formulated' in the language of propositional logic so that in that languate it becomes a tautology.

Regarding the axiom I agree with the above comment.

For the premise, it is not clear what you mean by 'get a conclusive statement'. Premise could be statement $p $ and then I'd use 'legitimate' steps to show that it implies $q$ say (it is what we usually do in mathematics). Rules of inference allow us to deduce that $q $ is true when $p $ is (Modus ponens).

Answer 2

I think your definition of Tautology in particular is lacking. Here is how I would define each of these terms (with examples):

1. Tautology: A statement which is necessarily true given its logical structure and definitions.
   • Example 1: I was born in the United States, or I was not born in the United States. (True by logical structure.)
   • Example 2: All dogs are animals. (True by definition of terms.)
2. Axiom: A statement that is assumed to be true without rigorous proof (I would add, potentially using un-defined terms).
   • Example: If two distinct lines are not parallel, they intersect at exactly one point.
3. Premises: A set of statements which, when assumed true, are (supposedly) used to (logically) lead to a set of true statements (called the conclusion or consequent).
   • Example 1: Consider the following argument. "If today is Monday, tomorrow is Tuesday. Today is Monday. Therefore, tomorrow is Tuesday." In this example, we have two premises (1. "If today is Monday, tomorrow is Tuesday" and 2. "Today is Monday") and one conclusion/consequent (i.e. "Tomorrow is Tuesday").
   • Example 2: Note that a statement being a premise does not require that statement to be true or false, nor do they absolutely require that the conclusion necessarily follows (though in any good argument, the conclusion does logically follow). Consider the following. "If today is Tuesday, tomorrow is Saturday. Today is Thursday. Therefore yesterday was Sunday."
     • In this case, we again have two premises (1. "If today is Tuesday, tomorrow is Saturday" and 2. "Today is Thursday") as well as one conclusion (i.e. "Yesterday was Sunday.")