Probability Distinguishes Different Types of Conditional Statements

TL;DR

This paper introduces a probabilistic framework for analyzing various types of conditional statements, translating them into polynomial systems and enabling algebraic methods for logical deduction and handling inconsistencies.

Contribution

It defines a unified probabilistic approach to different conditionals, translating them into polynomial equations and inequalities for analysis and deduction.

Findings

Distinct mathematical representations for each conditional type.

Polynomial systems reveal different behaviors and relationships among conditionals.

Paraconsistent deduction methods avoid explosion from inconsistent premises.

Abstract

The language of probability is used to define several different types of conditional statements. There are four principal types: subjunctive, material, existential, and feasibility. Two further types of conditionals are defined using the propositional calculus and Boole's mathematical logic: truth-functional and Boolean feasibility (which turn out to be special cases of probabilistic conditionals). Each probabilistic conditional is quantified by a fractional parameter between zero and one that says whether it is purely affirmative, purely negative, or intermediate in its sense. Conditionals can be specialized further by their content to express factuality and counterfactuality, and revised or reformulated to account for exceptions and confounding factors. The various conditionals have distinct mathematical representations: through intermediate probability expressions and logical formulas, each conditional is eventually translated into a set of polynomial equations and inequalities (with real coefficients). The polynomial systems from different types of conditionals exhibit different patterns of behavior, concerning for example opposing conditionals or false antecedents. Interesting results can be computed from the relevant polynomial systems using well-known methods from algebra and computer science. Among other benefits, the proposed framework of analysis offers paraconsistent procedures for logical deduction that produce such familiar results as modus ponens, transitivity, disjunction introduction, and disjunctive syllogism; all while avoiding any explosion of consequences from inconsistent premises. Several example problems from Goodman and Adams are analyzed. A new perspective called polylogicism is presented: mathematical logic that respects the diversity among conditionals in particular and logic problems in general.