# Discrete math: logical equivalent statement and statement forms

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 $$p=q$$?

• 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: $x:=5$). Commented Oct 25, 2021 at 9:34
• 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 $(\lnot p \land \lnot q)$. Commented Oct 25, 2021 at 9:45

$$p=q$$ 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.