Speaker: Dr. Umberto Riveccio

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Logics of Indicative conditionals and ordinary discourse

Indicative conditionals are the simplest sentences of the if-then type that occur in natural language, concerning what could be true — in opposition to counterfactuals, which concern eventualities that are no longer possible. In Boolean propositional logic, an indicative conditional is formalized as a material implication or as an equivalent disjunction. This approach has several limitations: in particular, a number of authors have argued that conditionals having a false antecedent — true in Boolean logic, independently of the consequent — should instead be regarded as lacking a (classical) truth value.

A very simple way to formalize the above intuition consists in expanding the classical truth values (0 and 1) with a third "gap" value (1/2) assigned to conditional sentences with a false antecedent, and then extending the truth tables of propositional connectives in accordance with the above interpretation. As for the designated elements to be preserved in derivations, it is natural to include (besides 1) also 1/2, at least if one wants to retain basic classical tautologies such as the identity law.

These restrictions determine a family of three-valued propositional logics of indicative conditionals that are not, in general, subclassical (i.e., weaker) but rather incomparable with Boolean logic. In particular, they can be connective, since they validate the (classically contingent) formulas known as Aristotle's and Boethius' theses, paraconsistent, and also contradictory in the sense of Wansing (in that they admit valid formulas whose negation is also valid). I will discuss prospects and problems of some of the main alternatives introduced in the literature to formalize indicative conditionals, in particular, the systems proposed by Cantwell, De Finetti, Farrell, and Cooper. I will devote special space to Cooper's Logic of Ordinary Discourse, analyzing a sample from the many natural language arguments proposed by the author.

Speaker: Dr. Johanna N.Y. Franklin

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Algorithmic randomness and mind changes

The most frequently encountered definitions of algorithmic randomness are based on tests, martingales, or variants of Kolmogorov complexity that are inherently computably enumerable and are thus defined using only positive information. I will discuss difference randomness, which is defined using tests in which elements may enter a test component and then exit it, and which thus allow the use of both positive information and negative information. I will further present some ways in which difference randomness is intrinsically connected to various ways of viewing computational strength.