Aristotle's Logic Computed by Parametric Probability and Linear Optimization

TL;DR

This paper introduces a novel computational approach that models Aristotelian logic as parametric probability networks, enabling the use of linear optimization to determine logical consequences numerically.

Contribution

It presents a new method translating Aristotelian logic problems into linear constraints on probabilities, allowing deductions via linear optimization techniques.

Findings

Successfully models Aristotelian logic as probability networks

Enables numerical computation of logical deductions

Demonstrates precise determination of necessary conclusions

Abstract

A new computational method is presented to implement the system of deductive logic described by Aristotle in Prior Analytics. Each Aristotelian problem is interpreted as a parametric probability network in which the premises give constraints on probabilities relating the problem's categorical terms (major, minor, and middle). Each probability expression from this network is evaluated to yield a linear function of the parameters in the probability model. By this approach the constraints specified as premises translate into linear equalities and inequalities involving a few real-valued variables. The problem's figure (schema) describes which specific probabilities are constrained, relative to those that are queried. Using linear optimization methods, the minimum and maximum feasible values of certain queried probabilities are computed, subject to the constraints given as premises. These computed solutions determine precisely which conclusions are necessary consequences of the premises. In this way, Aristotle's logical deductions can be accomplished by means of numerical computation.