When is an argument without premises valid?

Question

So the question is how do we know if an argument without premises is valid.

First of all, how would that go? I mean, what would an argument without premises look like (in terms of propositional logic)? Would there just be a conclusion? I also read that in case of a tautology, that would be a valid argument and I simply don't understand how there even can be a truth table created if there are no premises.

Also, why is $p$ or not $p$ an argument without premises? Isn't $p$ itself a premise?

Excuse my probably very simple questions, I'm very new to propositional logic or rather discrete math as a whole.

Answer 1

I simply don't understand how there even can be a truth table created if there are no premises.

With $n$ statements, there are $2^n$ ways to conjoin each statement or its negation with the others, and $2^{2^n}$ to disjoin these i.e. $2^{2^n}$ truth functions. With no premises, set $n=0$ so there are $2$ truth functions, true and false (or if you prefer, tautology and contradiction). The argument will just be a conclusion, and is valid iff the conclusion is tautological.

Also, why is $p$ or not $p$ an argument without premises? Isn't $p$ itself a premise?

To assume $p$ would in general be a premise, but that's not what you're doing. The simplest explanation is a comment by Ludwig Wittgenstein: "For example, I know nothing about the weather when I know that it is either raining or not raining."

Answer 2

An argument without premises is a single sentence : the conclusion.

A sentence is valid

if and only if it is true under every possible interpretation of the language. In propositional logic, they are tautologies.

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Regarding truth table, there is no issue with a truth table for a single formula.

Tautologies are exactly those formulas whose rightmost column in the truth table shows only the value TRUE.