Semantics of logical connectives (FOL)

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Definition

Let $\mathscr{L}$ be a given first order language and let $C$ denote the collection of all logical connective symbols, the semantics of a logical connective symbol is a function from the Cartesian product of one or more set of truth values to the truth values, written $\mathbf{B}_*$ where $*$ is a logical connective symbol, $*\in\{\neg,\vee,\wedge,\rightarrow,\leftrightarrow\}$, this mapping is commonly called the truth table of the connective.

Suppose $X$ and $Y$ are variables which may only take on truth values (eg, are the semantics of a formula) then we define $\mathbf{B}_*$ as follows:

Uniary connectives

Truth table for logical negation

$\mathbf{X}$	$\mathbf{B}_\neg(X)$
$T$	$F$
$F$	$T$

Binary connectives

Truth table for $\vee$, $\wedge$, $\rightarrow$, $\leftrightarrow$, that is logical or, logical and, logical implication and logical equivalence respectively:

$\mathbf{X}$	$\mathbf{Y}$	$\mathbf{B}_\vee(X,Y)$	$\mathbf{B}_\wedge(X,Y)$	$\mathbf{B}_\rightarrow(X,Y)$	$\mathbf{B}_\leftrightarrow(X,Y)$
$F$	$F$	$F$	$F$	$T$	$T$
$F$	$T$	$T$	$F$	$T$	$F$
$T$	$F$	$T$	$F$	$F$	$F$
$T$	$T$	$T$	$T$	$T$	$T$

See next

• Semantics of formulas