Logic has both a descriptive and a deductive use in mathematics. The descriptive use of logic consists in the use of logical notions for the purpose of capturing different structures studied in mathematical theories, with the aim of attaining a categorical axiomatization for a mathemathical theory S, i.e., any two models of S are isomorphic. The deductive use of logic consists in the use of logical inferences for systematizing and criticizing mathematicians reasoning about the structures they are interested in, with the aim of attaining a deductively complete system S, i.e. a system S such that for each sentence C in the language of S, either S ⊢ C or S ⊢ ~C. Both categoricity and deductive completeness are properties of the mathematical theory S formally axiomatized in a certain logical system. The underlying logical system L of S is semantically complete iff all valid formulas and all consequences can be derived in the system, i.e. if Г ⊨ C, then Г ⊢L C. Due to Gӧdel’s Incompleteness Theorem, however, if we have a categorical mathematical theory S (containing elementary arithmetic), then the underlying logic cannot be semantically complete. Thus, we can have categoricity only at the price of semantical completeness of the underlying logic. We can have either categoricity of S, or semantical completeness of L, but not both.
Logical inferentialism is the view based on the idea that formal rules of inference determine the meanings of the logical terms. In other words, if we know how to properly use the logical terms in logical inferences, then we know which their meanings are. Although the logical inferentialists agree that the meanings are fixed by the rules of inference, some of them consider that these meanings must be characterized only by using proof-theoretic concepts, such as deduction, proof, proof-theoretic conditions – these are the proof-theoretic inferentialists, while the others consider that these meanings should be characterized in model-theoretic terms, such as reference, truth-conditions, interpretations - these are the model-theoretic inferentialists. A challenge for the model-theoretic inferentialists is the existence of non-standard interpretations for the logical terms, i.e., interpretations for which the formal rules of inference preserve their soundness, but provide the logical terms with non-standard meanings. Consider for instance an interpretation of the standard classical propositional calculi that makes true all the theorems of the calculi and false all the non-theorems. In this interpretation all the rules of inference will preserve their soundness, but both ‘A’ and ‘~A’ will be false (being non-theorems) while ‘A v ~A’ will be true (being a theorem). In this interpretation a sentence and its negation are both false and a disjunction is true although both of its disjuncts are false. A model-theoretic logical inferentialist is interested in obtaining a categorical logical calculus, i.e. one that uniquely determines the intended meanings of the logical terms such that all non-standard interpretations are dismissed.
In his course, Constantin Brîncus shall first clarify the inferentialist idea that the rules of inference fix the meanings of the logical terms by showing that it depends both on the format of the logical system (axiomatic, natural deduction, and sequent calculi) and on the way validity is defined (by using deductive models, local models, and global models) and, secondly, he shall discuss some recent approaches for obtaining the categoricity of classical logic. Due to the tension between categoricity, deductive completeness and semantic completeness mentioned above, Constantin Brîncus will argue that the categoricity of a mathematical theory should not be the primary concern for the logical inferentialist and that he should accept a semantically complete logical system that uniquely determines the meanings of its logical terms.
Topics related to this course can be found here. Here are the slides.