The course is divided into two parts: (i) the truth-valuational approach to logic; (ii) the quantified argument calculus.
(i) The standard semantics for Predicate Logic and other formal systems is model-theoretic, using domains and interpretation functions to explain when a sentence is true, and with this theory of truth define validity and prove various properties of the calculus, primarily soundness and completeness (“adequacy”). We shall criticise this semantics, claiming that its concepts of object and of reference are empty. We shall then replace it with a truth-valuational substitutional approach, which does away with models. Like the truth-table approach in propositional logic, the truth-valuational approach is a theory of truth-value relations between sentences and not a theory of truth. We shall prove the adequacy of the first order predicate calculus on the truth-valuational approach. We shall next apply this approach to modal logic and prove the adequacy of modal propositional logic without recourse to possible worlds. Our conclusions will be, among other things, that models (in the sense of Model Theory) and possible worlds can be eliminated from logic
(ii) Frege’s logic allots the role of logical subject term or argument only to singular terms, and in this way departs from Natural Language, in which quantified arguments also occupy that role. By contrast to Natural Language, Frege introduced quantification into his calculus as a sentential operator. We shall follow Natural Language in having the quantifier join a one-place predicate to form a quantified argument. This departure has far-reaching consequences, which we shall pursue. We develop a formal system which is closer in many respects to Aristotle’s logic than to Frege’s, the Quantified Argument Calculus (Quarc). It will incorporate elements analogous to Natural Language’s negative predication, converse relation terms, anaphora, and more. We also develop a deductive system which we prove to be adequate. We then apply the system to modal logic, show how it incorporates a de dicto – de re distinction, how it invalidates the Barcan formulas, and more. We also consider extensions of the system to three-valued logic, plural logic, incorporation of the ‘there is’ structure, and more; much of this is work-in-progress. A conclusion shall be that this system should replace the Predicate Calculus as a tool for representing and studying the logic of Natural Language.