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June 30th, 2016, 06:31 PM   #1
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Can a simple (atomic) proposition be a tautology?

Definition: "A tautology is a propositional formula that is true under any truth assignment to each of the atomic propositions in the domain of propositional function."

Let $\displaystyle p$ be a simple (or atomic) proposition (e.g. "9 is a square root of 81").

I understand that a proposition may be either true or false (but not both true and false at the same time). That is, $\displaystyle p$ may be either true or false, exclusively. Under all possible truth assignments, $\displaystyle p$ is not always true. Therefore, from the definition of tautology, $\displaystyle p$ is not a tautology.

However, suppose I proved p to be true. I am tempted to write $\displaystyle p \iff \top$, but this means "$\displaystyle p$ is a tautology". However, the previous paragraph's conclusions was that "$\displaystyle p$ is not a tautology".

What's going on here?

Using the notation in symbolic logic, how does one write that $\displaystyle p$ is indeed true?

Thanks in advance for your help.
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