Deduction, Induction, and beyond in Parametric Logic.

With parametric logic, we propose a unified approach to deduction and induction, both viewed as particular instances of a generalized notion of logical consequence. This generalized notion of logical consequence is Tarskian, in the sense that if a sentence phi is a generalized logical consequence of a set T of premises, then the truth of T implies the truth of phi. So in particular, if phi can be induced from T, then the truth of T implies the truth of phi. The difference between deductive and inductive consequences lies in the process of deriving the truth of phi from the truth of T. If phi is a deductive consequence of T, then phi can be conclusively inferred from T with absolute certainty that phi is true. If phi is an inductive consequence of T, then phi can be (correctly, though only hypothetically) inferred from T, but will also be (incorrectly, still hypothetically and provisionally only) inferred from other theories T' that might not have phi as an inductive consequence (but that have enough in common with T to 'provisionally force' the inference of phi). The hallmark of induction is that given such a theory T', phi can actually be refuted from T': -phi can be inferred from T' with absolute certainty that -phi is true. As a consequence, deduction and induction are both derivation processes that produce 'truths,' but the deductive process is characterized by the possibility of committing no mind change, whereas the inductive process is characterized by the possibility of committing one mind change at most. More generally, when a sentence phi is a generalized logical consequence of a set T of premises, it might be possible to infer the truth of phi from the truth of T with fewer than beta mind changes, for the least nonnull ordinal beta, or it might be possible to infer the truth of phi from the truth of T in the limit, though without any mind change bound, or it might be possible to discover the truth of phi from the truth of T thanks to still more complex notions of inference. 'Discovering with an ordinal mind change bound' and 'discovering in the limit' are concepts from formal learning theory, an illustration of the fact that parametric logic puts together fundamental notions from mathematical logic and formal learning theory. This paper presents the model-theoretic foundations of parametric logic and some of their connections to formal learning theory and topology.