Logic and Modern Philosophy of Mathematics

The consequences of Gödel's theorems were profound. For example, logicism was replaced, especially by the American W. V. Quine, by elaborate systems of set theory and logic such as those developed in Quine's Mathematical Logic (1940); however, Russellian reductions of the former to the latter were not claimed.

Quine adhered to classical logic, with its two truth-values. But nonclassical logics gained adherents from the 1930s. The American C. I. Lewis had been a pioneer already in the 1910s, when he advocated "modal logics," where necessity and possibility were adjoined in various ways to truth and falsehood, as a means of clarifying the undoubtedly messy treatment of implication in Principia Mathematica. But from the 1930s these logics were viewed as viable alternatives to classicism in their own right. Another notable adherent was Gödel's friend Carnap, especially with his Meaning and Necessity (1947).

Since then logics classical and nonclassical have proliferated enormously, with their metalogics; many connections have been forged with computer science and aspects of semantics and linguistics. Notable among many cases are deontic logic, with its focus on sentences using "ought" and applications to law; temporal logics, seeking means to formulate deductions such as John eats; thus John will have eaten; many-valued logics, allowing for values other than true and false; non-monotonic logic or defeasible reasoning, defined by the property (in the simplest case) of propositions A, B, and C that given A C, then A and B C does not hold; paraconsistent logics, where various heresies are entertained, such as paradoxical propositions being both true and false; and fuzzy logic and set theory, where the vagueness of John is very tall is itself studied mathematically, initially by engineers rather than professional logicians.

This last topic bears upon control theory, which became associated with the discipline of cybernetics during the 1950s; logics of some kind were used in areas such as machine learning and brain modeling. Such concerns then developed further within the field of artificial intelligence, and were involved also in topics such as the representation of knowledge, automated theorem proving, and the relationship between complexity and recursion. These and very many more topics are now also linked to theoretical computer science, where many (meta)logics are used, with further connections to programming, formal and natural languages, and linguistics.

Partly due to these connections a renaissance of interest has occurred in higher-order logic, where quantification is executed over predicates as well as individuals. This revises a practice followed freely by Frege and by Russell and Whitehead. Theories of types have also been revived, appearing in several of the above contexts.