On Predicate Logic as Modal Logic

Abstract

The clause for existential formulas in the Tarskian definition of satisfaction for predicate logic can be written as follows:

$${\text{A}},{\kern 1pt} \alpha \left| { = {\kern 1pt} \exists x\varphi {\kern 1pt} \Leftrightarrow {\kern 1pt} {\text{there}}{\kern 1pt} is{\kern 1pt} \beta {\kern 1pt} {\text{such that }}\alpha {\kern 1pt} {{\text{ = }}_x}\beta {\kern 1pt} {\text{and A, }}\beta } \right| = \varphi ,$$

(1)

where A is a structure, α and β assignments in the universe A, i.e., elements of A ^Var, and α = _x β means that for all variables z except possibly x, α(z) = β(z). A modal perspective on predicate logic starts from the observation that (1) is similar to the clause for diamonds in Kripke semantics:

$$M,s\left| { = {\diamondsuit _x}\varphi {\kern 1pt} \Leftrightarrow {\kern 1pt} {\text{there}}{\kern 1pt} {\text{is}}{\kern 1pt} t{\kern 1pt} {\text{such}}{\kern 1pt} {\text{that}}{\kern 1pt} s{R_x}t{\kern 1pt} {\text{and}}{\kern 1pt} M,t} \right| = \varphi .$$

(2)