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Mathematical Information Content (Oliver Marshall)

The Logic and Metaphysics Workshop will meet on October 5th from 4:15-6:15 (NY time) via Zoom for a talk by Oliver Marshall (UNAM).

Title: Mathematical Information Content

Abstract: Alonzo Church formulated several logistic theories of propositions based on three alternative criteria of identity (1949, 1954, 1989, 1993). The most coarse grained of these criteria is Alternative (2), according to which two propositions are identical iff the sentences that express them are necessarily materially equivalent. Alternative (1) is more discerning. According to Alternative (1), two propositions are identical iff the sentences that express them can be obtained from one another by the substitution of synonyms for synonyms and λ-conversion. Church said that he intended this to limn a notion of proposition closely related to Frege’s notion of gedanke, but added that it will not be sufficiently discerning if propositions in the sense of Alternative (1) are taken as objects of assertion and belief (1993). Alternative (0), the most discerning criterion, says that two propositions are identical iff the sentences that express them can be obtained from one another by the substitution of synonyms for synonyms. I argue that Alternative (1) does indeed provide insight into one of the topics that concerned Frege (1884) – namely, abstraction. Then I discuss various counterexamples to Church’s criteria (including one due to Paul Bernays, 1961). I close by proposing a criterion of identity for mathematical information content based on the various examples under discussion.

Coin flips, Spinning Tops and the Continuum Hypothesis (Daniel Hoek)

The Logic and Metaphysics Workshop will meet on September 28th from 4:15-6:15 (NY time) via Zoom for a talk by Daniel Hoek (Virginia Tech).

Title: Coin flips, Spinning Tops and the Continuum Hypothesis

Abstract: By using a roulette wheel or by flipping a countable infinity of fair coins, we can randomly pick out a point on a continuum. In this talk I will show how to combine this simple observation with general facts about chance to investigate the cardinality of the continuum. In particular I will argue on this basis that the continuum hypothesis is false. More specifically, I argue that the probabilistic inductive methods standardly used in science presuppose that every proposition about the outcome of a chancy process has a certain chance between 0 and 1. I also argue in favour of the standard view that chances are countably additive. A classic theorem from Banach and Kuratowski (1929), tells us that it follows, given the axioms of ZFC, that there are cardinalities between countable infinity and the cardinality of the continuum. (Get the paper here: https://philpapers.org/archive/HOECAT-2.pdf).

Arithmetical Semantics for Non-Classical Logic (Yale Weiss)

The Logic and Metaphysics Workshop will meet on September 21st from 4:15-6:15 (NY time) via Zoom for a talk by Yale Weiss (CUNY).

Title: Arithmetical Semantics for Non-Classical Logic

Abstract: I consider logics which can be characterized exactly in the lattice of the positive integers ordered by division. I show that various (fragments of) relevant logics and intuitionistic logic are sound and complete with respect to this structure taken as a frame; different logics are characterized in it by imposing different conditions on valuations. This presentation will both cover and extend previous/forthcoming work of mine on the subject.

Cantor’s Theorem, Modalized (Chris Scambler)

The Logic and Metaphysics Workshop will meet on September 14th from 4:15-6:15 (NY time) via Zoom for a talk by Chris Scambler (NYU).

Title: Cantor’s Theorem, Modalized

Abstract: I will present a modal axiom system for set theory that (I claim) reconciles mathematics after Cantor with the idea there is only one size of infinity. I’ll begin with some philosophical background on Cantor’s proof and its relation to Russell’s paradox. I’ll then show how techniques developed to treat Russell’s paradox in modal set theory can be generalized to produce set theories consistent with the idea that there’s only one size of infinity.

(The slides are available here.)