The Truth of a Mathematical-Logical System, part 2

CD/mp3 106, part 2 of lecture 3 on mathematical logic 1957. Corresponds to CWL 18, pp. 79-86 and 337-43. The various types of mathematical-logical systems contain fragments of factual truth. What has been going on in mathematic logic is an empirical inquiry. The various systems that have been proposed are provisionally analytic. This seems confirmed by the succession of mathematical-logical systems. Some of these are equivalent, differing just in the axioms through which they proceed. But there are different types that are mutually incompatible and so not all the one and only logic. Each, as far as it goes, is true enough, but we cannot consider any one of them as exclusive. Lonergan treats four different types: the classical propositional or sentential calculus, modal logic that includes strict implication, three-valued logic that acknowledges the contingent future, and logics that weaken excluded middle or exclude it entirely. The questions that followed the lecture address a number of issues. Key points include the clarification that the factual references that give the truth of analytic principle in these matters are universal mental facts: judgments of fact regarding the occurrence of propositions, negations, strict implications, contingent future, and so on.

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