Hintikka’s Logical Revolution

• Published in: Frontiers of philosophy in China Vol. 14; no. 4; pp. 630 - 648
• Main Author: XU Difei
• Format: Journal Article
• Language: English
• Published: Beijing Higher Education Press 2019; Higher Education Press Limited Company
• Subjects: game theoretical semantics; Identity; independence friendly logic (IFL); Logic; Mathematics; second-order logic; the definition of truth
• ISSN: 1673-3436; 1673-355X
• DOI: 10.3868/s030-008-019-0037-9

Hintikka thinks that second-order logic is not pure logic, and because of Gödel’s incompleteness theorems, he suggests that we should liberate ourselves from the mistaken idea that first-order logic is the foundational logic of mathematics. With this background he introduces his independence friendly logic (IFL). In this paper, I argue that approaches taking Hintikka’s IFL as a foundational logic of mathematics face serious challenges. First, the quantifiers in Hintikka’s IFL are not distinguishable from Linström’s general quantifiers, which means that the quantifiers in IFL involve higher order entities. Second, if we take Wright’s interpretation of quantifiers or if we take Hale’s criterion for the identity of concepts, Quine’s thesis that second-order logic is set theory will be rejected. Third, Hintikka’s definition of truth itself cannot be expressed in the extension of language of IFL. Since second-order logic can do what IFL does, the significance of IFL for the foundations of mathematics is weakened.