The following are notational conventions/abbreviations that we use when referring to sentences in first-order logic, for the sake of concision. First of all, we will use all of the conventions introduced to deal with sentential logic . In addition to these, we accept the following, as used by Chang and Kiesler:
$(\exists x)\varphi$ stands for $\neg(\forall x)\neg\varphi$ ("there exists $x$ such that $\varphi$")
$\varphi_1\land\varphi_2\land...\land\varphi_n$ stands for $\varphi_1\land (\varphi_2\land...\land\varphi_n)$ (the logical operation "and" is associative, so this will not get us into trouble)
$\varphi_1\lor\varphi_2\lor...\lor\varphi_n$ stands for $\varphi_1\lor (\varphi_2\lor...\lor\varphi_n)$ (again, the logical operation "or" is associative)
$(\forall x_1 x_2 ... x_n)\varphi$ stands for $(\forall x_1)(\forall x_2)...(\forall x_n)\varphi$
$(\exists x_1 x_2 ... x_n)\varphi$ stands for $(\exists x_1)(\exists x_2)...(\exists x_n)\varphi$
The following convention is very important: $t(v_0 \cdots v_n)$ is used to denote a term $t$ whose variables are included in ${v_0,...,v_n}$, and $\varphi(v_0\cdots v_n)$ to denote a formula whose free variables form a subset of ${v_0,...,v_n}$.