Completeness Theorems, Representation Theorems: What's the Difference?

Most areas of logic can be approached either semantically or syntactically. Typically, the approaches are linked through a completeness or representation theorem. The two kinds of theorem serve a similar purpose, yet there also seems to be some residual distinction between them. In what respects do they differ, and how important are the differences? Can we have one without the other? We discuss these questions, with examples from a variety of different logical systems. 1. Introduction: Syntax versus Semantics Usually, the first serious course that a student takes in logic will introduce classical propositional and predicate calculi. The class learns that there are two ways of approaching such systems: semantic (alias model theoretic) or syntactic (alias axiomatic, postulational). Typically, the two are made to work like chopsticks. The teacher takes one of the two presentations as a firm base. The other is then introduced and the two are linked by means of a completeness theorem that establishes their equivalence. This theorem is often the culminating point of the course. The decision which of the two presentations to treat as basic is very much a matter of personal preference, influenced by philosophical perspectives and pedagogical experience. In the case of classical propositional logic it is customary to begin semantically with the definition of a tautology, and then show how this coincides with an approach in terms of axioms (or axiom schemes) and derivation rules. On the other hand, in the case of intuitionistic propositional logic, it is more common to proceed in the reverse direction, first indicating how one might question certain of the axioms of the classical system, then forming a reduced axiom set, and finally showing how the resulting set of derivable formulae may be characterized semantically, say in terms of relational model structures or a suitable family of algebras. Of course, there are also maverick authors who do the reverse in each case. As time goes on, the student also learns that the distinction between 'semantic' and 'syntactic' is not set in stone. There are presentations such as that of semantic decomposition trees (alias semantic tableaux) which can be seen as somewhere between the two. Notwithstanding their semantic name, there is something syntactic about these trees, and indeed it is possible to map the account into Gentzen sequent systems on the one hand - a flagship of the syntactic fleet - and into truth-tables on the other. Moreover, when taught completeness proofs, one learns that even such a paradigmatically semantic object as a classical valuation can be identified with a syntactic item, namely a set of formulae that is well-behaved with respect to each of