Independence-friendly logic without Henkin quantification

We analyze the expressive resources of $$\mathrm {IF}$$ IF logic that do not stem from Henkin (partially-ordered) quantification. When one restricts attention to regular $$\mathrm {IF}$$ IF sentences, this amounts to the study of the fragment of $$\mathrm {IF}$$ IF logic which is individuated by the game-theoretical property of action recall (AR). We prove that the fragment of prenex AR sentences can express all existential second-order properties. We then show that the same can be achieved in the non-prenex fragment of AR, by using “signalling by disjunction” instead of Henkin or signalling patterns. We also study irregular IF logic (in which requantification of variables is allowed) and analyze its correspondence to regular IF logic. By using new methods, we prove that the game-theoretical property of knowledge memory is a first-order syntactical constraint also for irregular sentences, and we identify another new first-order fragment. Finally we discover that irregular prefixes behave quite differently in finite and infinite models. In particular, we show that, over infinite structures, every irregular prefix is equivalent to a regular one; and we present an irregular prefix which is second order on finite models but collapses to a first-order prefix on infinite models.

Comments on Gabriel Sandu’s “Indefinites, Skolem Functions, and Arbitrary Objects”

Gabriel Sandu has made important contributions to the development of independence-friendly logic and I am grateful to him for his searching and sympathetic critique of my own work on arbitrary objects in relation both to independence friendly logic and to other treatments of quantificational and anaphoric dependence....

Hintikka, Jaakko (1929–2015)

Jaakko Hintikka was a Finnish philosopher who developed important new methods and systems in mathematical and philosophical logic. Over a distinguished career in universities in Finland and the USA, he was one of the most cited analytic philosophers and published prolifically in mathematical and philosophical logic, philosophy of language, formal epistemology, philosophy of science and history of philosophy. Hintikka was a pioneer of possible-worlds semantics, epistemic logic, inductive logic, game-theoretical semantics, the interrogative approach to inquiry and independence-friendly logic. He was an expert on Aristotle, Leibniz, Kant, Peirce and Wittgenstein. He also influenced philosophy as a successful teacher and the long-time editor of the journal Synthese.