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A propositional type is a collection of propositional formulas to be jointly realized by a Boolean valuation. A valuation realizes it when every member is true and omits it when some member is false. For propositional omission arguments, the collection need not itself be consistent.

A satisfying assignment is a complete Boolean valuation $v$ for which a Boolean formula $F$ has truth value one, written $v\models F$. The notation describes a relation between the Boolean valuation and the Boolean formula. For the example in Boolean satisfiability problem, $(0,1,0)$ is a model of a Boolean formula $F$, whereas $(0,0,0)$ is not. SAT asks for existence of at least one satisfying assignment, rather than for every Boolean valuation to work; the latter condition makes $F$ a tautology.

A tautology is a propositional formula true under every Boolean valuation in classical propositional logic, such as the law of excluded middle $p\vee \neg p$. A primitive proposition $p$ is not a tautology, even if it represents a true statement. See forall x: Calgary, section 12.1.

A truth table lists a propositional formula's value for every Boolean valuation of its distinct primitive propositions. With $k$ primitive propositions, there are $2^k$ rows: each independently receives T or F. Evaluate compound columns from their subexpressions using the classical rules for logical connectives. This is exhaustive checking, so a formula true on every row is valid in classical propositional logic.

For a computed example over the natural numbers, let $A(n)$ mean that $n$ is an even number and $B(n)$ mean $n>2$. The choices $n=4,2,3,1$ realize respectively $(A,B)=(\mathrm{T},\mathrm{T}),(\mathrm{T},\mathrm{F}),(\mathrm{F},\mathrm{T}),(\mathrm{F},\mathrm{F})$: all four possible inputs.

The logical implication $A\to B$ and its contrapositive agree on every row; the converse implication $B\to A$ differs on two rows. A truth table therefore checks equivalence or supplies a counterexample. Classical truth conditions are given in forall x: Calgary, section 31.2. These two-valued truth tables do not characterize validity in Intuitionistic propositional logic.

In classical propositional logic, a truth value is false or true, encoded by a bit as zero or one. A Boolean valuation assigns these values to primitive propositions and extends to compound propositional formulas through logical connectives.

An unsatisfiable Boolean formula has no satisfying assignment. Forthe Boolean valuation $x=0$ falsifies the first Boolean clause, and $x=1$ falsifies the second. Both possible truth values fail, proving UNSAT. Each Boolean clause alone has a satisfying assignment, but their logical conjunction does not: compatibility is the central issue. By contrast, finding just one failed Boolean valuation for a general Boolean formula only rejects that candidate. Finding one successful Boolean valuation proves SAT; proving UNSAT requires excluding every candidate, possibly with a compact argument rather than a full truth table.

For a CNF input with $n$ Boolean variables, $m$ Boolean clauses, and $L$ Boolean literal occurrences, first check that the supplied Boolean valuation gives each input Boolean variable exactly one truth value. Evaluate each Boolean literal by lookup or logical negation, each Boolean clause by logical disjunction, and the Boolean formula by logical conjunction. Indexed Boolean variables and constant-cost lookups give an $O(n+m+L)$ operation count; with a finite binary string encoding the check still takes polynomial time. Thus a satisfying assignment is a certificate in computational complexity for SAT, proving that SAT belongs to NP. Finding it by exhaustive checking can require $2^n$ candidate Boolean valuations; this enumeration cost is not a lower bound on every possible SAT algorithm.^[ref]