# Is T V F a tautology?

(Consider $$T=true$$, $$F=false$$) I apologize for the simple question but I'm confused as to what is a tautology and what isn't and I haven't found a clear definition. I think that a tautology is a propositional well-defined formula which is true in all states. Does this allow $$T$$ v $$F$$ to be a tautology because it is true in all states and $$T$$ and $$F$$ are propositions themselves? Or does a tautology imply that there exists variables say $$E_{0}$$ to $$E_{i}$$ because it is a well-defined formula and not a closed formula?

• I mean values True and False, I've edited my answer to clarify! Jun 13, 2023 at 0:58

Probably the best way to think of $$T$$ and $$F$$ is that they are "constants".

If you think of arithmetic-type algebra, $$x$$ and $$y$$ are pronumerals or variables, but $$0$$ and $$1$$ are constants. $$T$$ and $$F$$ are not unlike $$1$$ and $$0$$ (respectively). They are propositions in the same way that $$x$$ and $$y$$ are numbers.

A "tautology" is a statement that is true under all possible valuations of its variables. That means that a closed formula is always a tautology, a contradiction (which is the opposite of a tautology), or formally undecidable in the system.

$$T \vee F$$ is closed formula; it has no free variables. Therefore, yes, it is a tautology. Any possible boolean valuation of its variables (all zero of them) is true.

• Thank you for your response! Jun 13, 2023 at 4:23

A tautology is something that is always true. $$T\lor F$$ is an almost-trivial example of one (the trivial example would be $$T$$ by itself).

As a bonus, a contradiction is something that is always false. The trivial example is $$F$$ and an almost-trivial example like the one you asked about would be $$T\land F$$.