Abstract

We investigate how to assign probabilities to sentences that contain a type-free truth predicate. These probability values track how often a sentence is satisfied in transfinite revision sequences, following Gupta and Belnap’s revision theory of truth. This answers an open problem by Leitgeb which asks how one might describe transfinite stages of the revision sequence using such probability functions. We offer a general construction, and explore additional constraints that lead to desirable properties of the resulting probability function. One such property is Leitgeb’s Probabilistic Convention T, which says that the probability of φ equals the probability that φ is true.

Highlights

  • The revision theory of truth is an influential way to account for a type-free truth predicate

  • The sentence T 0 = 0, for example, settles down on being true, whereas the liar sentence will have its truth value continuing to switch throughout the revision sequence

  • In this paper we construct such probability functions that measure how often a sentence is true in a revision sequence of hypotheses

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Summary

Introduction

The revision theory of truth is an influential way to account for a type-free truth predicate. In this paper we construct such probability functions that measure how often a sentence is true in a revision sequence of hypotheses This idea was pioneered in Leitgeb [9], where the finite stages of the revision theory of truth are used to define probabilities for sentences that may include a type-free truth predicate. We explore ways that Leitgeb’s construction can be extended to determine probability values dependent on the transfinite stages of the revision sequence, answering a question in Leitgeb [10]. Our proposal extends his by assigning probabilities at a transfinite ordinal stage, measuring how often the sentence is true in the revision sequence up to that ordinal.

Some Preliminaries
Semantic Probability
The Revision Theory of Truth
Leitgeb
Two Proposals
The Idea Behind the Proposals
Real-Valued Probabilities
Hyperrational Probabilities
Constraints on Ultrafilters
Non-Principal
Stability
Banachness
Grand Loop
How These Properties of Ultrafilters Relate
Further Research
Full Text
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