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I'm reading Susanna Epps book on discrete mathematics and I have a question about the notation of logical equivalence.

In the books definition: "Two statement forms are called logically equivalent if, and only if, they have identical truth values for each possible substitution of statements for their statement variables... denoted $P\equiv Q$.

Two statements are called logically equivalent if, and only if , they have logically equivalent forms when identical component statement variables are used to replace identical component statements."

Later in the exercise section she writes: $p="x>5"$. Do you not use $\equiv$ for statement definitions and if so, how do you symbolise equivalence between two statements, $p\equiv q$ or $p=q$.

Since p and q by them selves could technically be seen as statement forms, is there a difference between $p\equiv q$ and $p=q$?

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    $\begingroup$ When the author writes $p="x>5"$ she means that the "value" $x>5$ is assigned to statement variable $p$ (compare with the assignment in programming language: $x:=5$). $\endgroup$ Commented Oct 25, 2021 at 9:34
  • $\begingroup$ Please, consider that the definition is based on "statement forms". In propositional logic, two atoms, i.e. propositional variables ($p,q,\ldots$) cannot be logical equivalent. An example in propositional logic (classical logic) will be $\lnot (p \lor q)$ and $(\lnot p \land \lnot q)$. $\endgroup$ Commented Oct 25, 2021 at 9:45

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$p=q$ means the two statements are identical, i.e. that they are one single statement.

$p\equiv q$ means that the two statements are equivalent, but not (necessarily) identical.

For example, if $p$ is the statement "$x>5$", while $q$ is the statement "$\neg (x\leq 5)$", then the two statements are equivalent, but they are not identical.

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