Post's Functional Completeness Theorem

The paper provides a new proof, in a style accessible to modern logicians and teachers of elementary logic, of Post's Functional Completeness Theorem. Post's Theorem states the necessary and sufficient conditions for an arbitrary set of (2-valued) truth functional connectives to be expressively complete, that is, to be able to express every (2-valued) truth function or truth table. The theorem is stated in terms of five properties that an arbitrary connective may have, and claims that a set of connectives is expressively complete iff for each of the five properties there is a connective that lacks that property. Everyone knows the technique whereby, given an arbitrary (2-valued) truth table, one can construct a conjunctive (or disjunctive) normal form formula (using only connectives from {V,Λ,~ }) which has exactly that truth table. This proves that the set of connectives {V,Λ,~} is functionally complete: any (2valued) truth table can be constructed from them. Everyone also knows the definitions of Λ in terms of {v,~} and of v in terms of {Λ,~ j . This shows that {Λ,~ } and {v,~} are also functionally complete sets of connectives. Everyone also knows that the sheffer stroke functions, ΐ and I, are each functionally complete. Most everyone knows that {-sF} is functionally complete and that {->,γ} is functionally complete (F is the constant-false truth function, y is "exclusive or"). Some people, having worked through Church ([1], p. 131f.), even know that {[ ],T,F} is functionally complete ([ ] is the ternary connective of "conditional disjunction": [p,q9r] means "if 47, then/? else r"). However, what is not generally known is why these things are so. What is it about these particular sets of connectives that makes them functionally complete while (say) {<-•,-} is not func-