This is joint work with Jean-Yves Beziau
The concept of possible world has increasingly gained in popularity during the latter part of the 20th century. This notion has been used in a technical way to develop the semantics of modal logics, and it can be viewed as a major impetus behind the growth of (formal) modal logic. It spread in philosophy and further contaminated a variety of fields, from literary theory to quantum physics. Witness of this glory: a Nobel symposium has been held on this topic.
After presenting some historical comments about the resurgence of this notion we will show that:
Weak quantification over choice functions is widely employed in semantic theory of natural language, for example in semantics of branching quantifiers, representing definite and indefinite noun phrases, anaphora etc. In theory this requires use of second order logic.
We present a first order formal system for operating with indexed choice functions sufficient for many linguistic applications and give a completeness proof with respect to a natural semantics.
In an unpublished paper, 'Zur Frage der Konstruktivität von Beweisen' (1930), Heinrich Behmann presented a flawed proof of a conjecture by Felix Kaufmann, in Das Unendliche in der Mathematik und seine Ausschaltung (1930, p. 66), according to which proofs of existence claims which do not depend on the axiom of choice implicitly rely on the exhibition of an instance satisfying the existence claim. Behmann provided a method for transforming indirect proofs into direct ones. In the last part of his paper, he presented, as an example, a constructivization of the well-known proof by Euler of the infinity of the prime numbers. In a footnote, Behmann attributes this constructive version to Ludwig Wittgenstein. This version differs from the standard one given by Leopold Kronecker in Vorlesungen über Zahlentheorie (1901, pp. 270f.) I shall give textual evidence that Behmann learned about Wittgenstein's proof through Friedrich Waismann and Kaufmann. I shall then present the proof and show how it sheds light on Wittgenstein's remarks on Euler's theorem that are published in Philosophische Bemerkungen. Wittgenstein refers to Euler's proof as a "proof by circumstantial evidence" and adds that such proofs should "absolutely never be permitted" in mathematics. In these and surrounding passages he criticizes existence proofs and claims that philosophical clarity will "prune mathematics". Wittgenstein's constructivization of the proof is evidence of the depth of his thinking on these issues and a clear indication of his constructivist stance in the early 1930s.
Within mainstream analytic philosophy, in particular philosophy of mathematics, it has been claimed frequently that metamathematical theorems possess philosophical significance or yield philosophical conclusions. Yet the question how to bridge the gap between formal sciences and philosophical discourse has hardly been addressed. I will briefly sketch two answers to this question, one put forth by P. Benacerraf and the other by W.V.O. Quine. A close analysis of Putnam's model-theoretic argument in the context of the constructible universe of sets then serves as a test for both accounts. It turns out that the rather strong tools Putnam needs in the construction of his model threaten to defeat his whole argument. Moreover, this will show that neither Benacerraf's nor Quine's account is wholly satisfactory. - There is no simple, straightforward way to "squeeze out philosophical juice from metamathematics".
Interpretability Logic is a modal logic designed for deriving principles valid for the notion of interpretability between finite extensions of a given (arithmetical) theory. Interpretability logic is an extension of provability logic.
In my talk, I will introduce the basic ideas of interpretability logic. I will discuss one of the frontiers of the field: the interpretability logic of all reasonable theories. I will show how to derive the principle W of this logic.