Logical connective

Remarks

Conjunction $\land$ is a binary connective: $f_\land(a, b) = \mathsf{T}$ exactly when $a = b = \mathsf{T}$. So $p \land q$ is true only when propositions $p$ and $q$ are both true. The modal operator "it is necessary that" is not a connective of classical propositional logic. It has no truth function: the truth of $p$ does not determine whether $p$ is necessary. Binary connectives are written in infix form, such as $\varphi \to \psi$. The set $\set{\neg, \land}$ is functionally complete: every truth function is expressed by a formula built with these two. The single connective NAND, with $f(a, b) = \mathsf{F}$ exactly when $a = b = \mathsf{T}$, is also functionally complete.