A Logical Consequence Informed by Probability - presented by Dr. Neil Hallonquist

A Logical Consequence Informed by Probability

Dr. Neil Hallonquist

NH
Slide at 52:02
Neil Hallonquist Contrasting Conclusions (cont.) 'Correspondence 1': (sentences probabilistic events) Suppose we have a function space Fs, a sentence space SL, and a mapping T : SL Fn (that assigns valuations to sentences). The placement of probabilities on sentences amounts to the presupposition that, for sentences E SL, their binary valuation functions () E F n correspond to probabilistic events - subsets of N - and it is this correspondênce that defines the relationship between probability and logic. 'Correspondence 2': (sentences probability densities) Other possibility: the binary valuation functions T () E Fs correspond to probability densities, the valuation functions used in probability. That is, since both sentence valuations in logic and densities in probability define functions over a set N of possible worlds, except one binary and the other real valued, sentences could be taken to correspond, not to probabilistic events (the usual correspondence), but directly to probability measures.
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Summary (AI generated)

The motivation for this work stems from the ongoing research theme of integrating probability and logic, specifically in defining probability measures over propositions. This approach assumes a specific relationship between these systems, where the correspondence between sentences and events influences our understanding of their connection. Traditionally, this leads to the conclusion that probability lacks a consequence relation similar to that of classical logic.

In contrast, our work explores an alternative correspondence that may redefine this relationship. We propose that binary evaluation functions correspond to probability densities rather than to probabilistic events. Both sentence valuations in logic and densities in probability operate over a set of possible worlds, with one being binary and the other real-valued. This perspective suggests that sentences may correspond directly to probability measures, challenging the conventional understanding.

This notion, while unconventional, carries significant implications for interpreting the relationship between probability and logic. It invites us to consider the validity of this approach and its potential to provide a compelling abstraction of logic. This framework is not only simple and fitting but also possesses desirable qualities that merit further exploration. In summary, we encourage a serious examination of this perspective.