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first order logic
A logic is first order if it has exactly one type. Usually the term refers specifically to the logic with connectives $\neg$, $\vee$, $\wedge$, $\rightarrow$, and $\leftrightarrow$ and the quantifiers $\forall$ and $\exists$, all given the usual semantics:

$\neg\phi$ is true iff $\phi$ is not true

$\phi\vee\psi$ is true if either $\phi$ is true or $\psi$ is true

$\forall x\phi(x)$ is true iff $\phi^{t}_{x}$ is true for every object $t$ (where $\phi^{t}_{x}$ is the result of replacing every unbound occurrence of $x$ in $\phi$ with $t$)

$\phi\wedge\psi$ is the same as $\neg(\neg\phi\vee\neg\psi)$

$\phi\rightarrow\psi$ is the same as $(\neg\phi)\vee\psi$

$\phi\leftrightarrow\psi$ is the same as $(\phi\rightarrow\psi)\wedge(\psi\rightarrow\phi)$

$\exists x\phi(x)$ is the same as $\neg\forall x\neg\phi(x)$
However languages with slightly different quantifiers and connectives are sometimes still called first order as long as there is only one type.
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