In logic, a four-valued logic is any logic with four truth values. Several types of four-valued logic have been advanced.
Nuel Belnap considered the challenge of question answering by computer in 1975. Noting human fallibility, he was concerned with the case where two contradictory facts were loaded into memory, and then a query was made. "We all know about the fecundity of contradictions in two-valued logic: contradictions are never isolated, infecting as they do the whole system." Belnap proposed a four-valued logic as a means of containing contradiction.
He called the table of values A4: Its possible values are true, false, both (true and false), and neither (true nor false). Belnap's logic is designed to cope with multiple information sources such that if only true is found then true is assigned, if only false is found then false is assigned, if some sources say true and others say false then both is assigned, and if no information is given by any information source then neither is assigned. These four values correspond to the elements of the power set based on {T, F}.
T is the supremum and F the infimum in the logical lattice where None and Both are in the wings. Belnap has this interpretation: "The worst thing is to be told something is false simpliciter. You are better off (it is one of your hopes) in either being told nothing about it, or being told both that it is true and also that it is false; while of course best of all is to be told that it is true." Belnap notes that "paradoxes of implication" (A&~A)→B and A→(B∨~B) are avoided in his 4-valued system.
Belnap addressed the challenge of extending logical connectives to A4. Since it is the power set on {T, F}, the elements of A4 are ordered by inclusion making it a lattice with Both at the supremum and None at the infimum, and T and F on the wings. Referring to Dana Scott, he assumes the connectives are Scott-continuous or monotonic functions. First he expands negation by deducing that ¬Both = Both and ¬None = None. To expand And and Or the monotonicity goes only so far. Belnap uses equivalence (a&b = a iff avb = b) to fill out the tables for these connectives. He finds None & Both = F while None v Both = T.
The result is a second lattice L4 called the "logical lattice", where A4 is the "approximation lattice" determining Scott continuity.
Let one bit be assigned for each truth value: 01=T and 10=F with 00=N and 11=B.
Then the subset relation in the power set on {T, F} corresponds to order ab