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(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?

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  • $\begingroup$ I mean values True and False, I've edited my answer to clarify! $\endgroup$
    – Ayyware
    Jun 13, 2023 at 0:58

2 Answers 2

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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.

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  • $\begingroup$ Thank you for your response! $\endgroup$
    – Ayyware
    Jun 13, 2023 at 4:23
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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 $T\land F$.

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