just curious if there is a formal name for the results that:
a) Two wffs in Sentence Logic are equivalent iff their truth tables are equal , as binary functions of {T,F}
b) Two wffs A,B in Predicate logic are equivalent when : I is an interpretation for A iff I is an interpretation for B.
Thanks.
We have that [see Enderton, page 88] :
$\varphi$ and $\psi$ are logically equivalent ($\varphi \equiv \psi$) iff $\varphi \vDash \psi$ and $\psi \vDash \varphi$,
where :
Let $\Gamma$ be a set of wffs, $\varphi$ a wff. Then $\Gamma$ logically implies $\varphi$, written $\Gamma \vDash \varphi$, iff for every structure $\mathfrak A$ for the language and every function $s : V \to |\mathfrak A|$ such that $\mathfrak A$ satisfies every member of $\Gamma$ with $s$, $\mathfrak A$ also satisfies $\varphi$ with $s$.
We have that $\varphi \equiv \psi$ iff $\vDash \varphi \leftrightarrow \psi$ (where : $\leftrightarrow$ is the bi-conditional connective).
Note
A structure $\mathfrak A$ for a first-order language, sometimes called an interpretation, is
a function whose domain is the set of symbols of the language and such that
$\mathfrak A$ assigns to the quantifier symbol ∀ a nonempty set $|\mathfrak A|$ called the universe (or domain) of $\mathfrak A$.
$\mathfrak A$ assigns to each $n$-place predicate symbol $P$ an $n$-ary relation $P^A ⊆ |\mathfrak A|^n$, i.e., $P^A$ is a set of $n$-tuples of members of the universe.
$\mathfrak A$ assigns to each constant symbol $c$ a member $c^A$ of the universe $|\mathfrak A|$.
$\mathfrak A$ assigns to each $n$-place function symbol $f$ an $n$-ary operation $f^A$ on $|\mathfrak A|$, i.e., $f^A : |\mathfrak A|^n \to |\mathfrak A|$.
