Abstract

1. Two kinds of logic. To a first approximation there are two main kinds of pursuit in logic. The first is the traditional one going back two millennia, concerned with characterizing the logically valid inferences. The second is the one that emerged most systematically only in the twentieth century, concerned with the semantics of logical operations. In the view of modern, model-theoretical eyes, the first requires the second, but not vice-versa. According to Tarski’s generally accepted account of logical consequence, inference from some statements as hypotheses to a statement as conclusion is logically valid if the truth of the hypotheses ensures the truth of the conclusion, in a way that depends only on the form of the statements involved, not on their content. Interpreted model-theoretically this means that every model of the hypotheses is a model of the conclusion. However, there is an ambiguity in Tarski’s explication, as he himself emphasized, since for the specification of form one needs to determine what are the logical notions. Once those are isolated and their semantical roles are settled, one can see how the truth of a statement is composed from the truth of its basic parts, in whatever way those are specified. The problem of what are the logical notions is an unsettled and controversial one. In the classical truthfunctional perspective, proposals range from those of first-order logic to generalized quantifiers to second and higher-order quantifiers to infinitary languages and beyond. Many of these stronger semantical notions have been treated in the volume Model Theoretic Logics. In a series of singular, thought-provoking publications in recent years, Jaakko Hintikka has vigorously promoted consideration of an extension of first-order logic called IF logic, along with claims that its adoption promises to have revolutionary consequences.