A Logical Theory of Truth-Makers and Falsity-Makers

We explicate the different ways that a first-order sentence can be true (resp., false) in a model M, as formal objects, called (M-relative) truth-makers (resp., falsity-makers). M-relative truth-makers and falsity-makers are co-inductively definable, by appeal to the “atomic facts” in M, and to certain rules of verification and of falsification, collectively called rules of evaluation. Each logical operator has a rule of verification, much like an introduction rule; and a rule of falsification, much like an elimination rule. Applications of the rules (∀) and (∃) involve infinite furcation when the domain of M is infinite. But even in the infinite case, truth-makers and falsity-makers are tree-like objects whose branches are at most finitely long. A sentence φ is true (resp., false) in a model M (in the sense of Tarski) if and only if there existsπ such that π is an M-relative truth-maker (resp., falsity-maker) for φ. With “ways of being true” explicated as these logical truthmakers, one can re-conceive logical consequence between given premises and a conclusion. It obtains just in case there is a suitable method for transforming M-relative truthmakers for the premises into an M-relative truthmaker for the conclusion, whatever the model M may be.