An algebraic approach to a Kripkean theory of probability and truth

(Logic & Philosophy of Mathematics, English)

n natural languages, we can formulate sentences as

(1) Sentence (1) is false.

(2) The probability of sentence (2) is less than __.

The first sentence is called the liar sentence and the paradox which arises from the question which truth value should be assigned to it led to the development of different formal theories of truth in the last century. Among those is a very influential fixed point construction by Saul Kripke. As a standard option, it is formulated by means of a three-valued set of truth values and the way the logical connectives and quantifiers operate on the truth values is summarised in the Strong Kleene scheme. But this is not the only scheme available. Other possible schemes include the Weak Kleene scheme and a four-valued extension of Strong Kleene. With an algebraic approach, it is possible to define the Kripkean construction for all the four mentioned schemes at once, as demonstrated in Chapter 2 of Gupta and Belnap 1993.

If one also wants to accommodate probability sentences like (2) in the formal system, the Kripkean construction needs to be extended. For the Strong Kleene scheme, Catrin Campbell-Moore introduced such an account in Campbell-Moore 2015 and showed that it satisfies several desirable properties.

In this talk, I want to extend the account of Chapter 2 of Gupta and Belnap 1993 to the probability case. This will result in an algebraic characterisation of the fixed points of a joint Kripkean theory of truth and probability. It complements the system of Campbell-Moore because also the Weak Kleene and the four-valued scheme are included. Apart from suggesting extensions to the definitions of Gupta and Belnap, I will begin to show that the resulting theory satisfies minimal desiderata like the fixed point property.

References:

Campbell-Moore, C., 2015, How to express self-referential probability. A Kripkean proposal,

The Review of Symbolic Logic, 8: 680-704.

Gupta, A., and Belnap, N., 1993, The Revision Theory of Truth, Cambridge, MA: MIT Press.

Chair:

Time: 14:40-15:10, 18 September 2019 (Wednesday)

Location: SR 1.007

Fabian Heimann

( University of Göttingen , Germany)

I am a student of math and philosophy in Göttingen. More precisely, I study math in the M.Sc. programme of Göttingen University and math and philosophy towards a B.A. Before all that, I obtained a B.Sc. in physics in 2017. Some of my interests are numerics of partial differential equations on the math side and know-how, (multi)modal logic, and theories of truth and other circular concepts in philosophy.

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