Probability Distinguishes Different Types of Conditional Statements

Conditional statements, including subjunctive and counterfactual conditionals, are the source of many enduring challenges in formal reasoning. The language of probability can distinguish among several different kinds of conditionals, thereby strengthening our methods of analysis. Here we shall use probability to define four principal types of conditional statements: SUBJUNCTIVE, MATERIAL, EXISTENTIAL, and FEASIBILITY. 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. We shall consider also TRUTH-FUNCTIONAL conditionals constructed as statements of material implication from the propositional calculus; these constitute a fifth principal type. Finally there is a sixth type called BOOLEAN-FEASIBILITY conditionals, which use Boole’s mathematical logic to analyze the sets of possible truth values of formulas of the propositional calculus. Each TRUTH-FUNCTIONAL or BOOLEAN-FEASIBILITY conditional can be affirmative or negative, with this sense indicated by a binary true/false parameter. There are other kinds of conditional statements besides the six types addressed here. In particular, some conditionals ought to be regarded as recurrence relations that generate discrete dynamical systems [12]. Besides its principal type and the value of its (fractional or binary) sense-parameter, each conditional statement is further characterized by its content and by its role in analysis. Two important aspects of content are factuality and exception handling. We shall consider the factuality of a conditional relative to some proposition whose truth value is known. If the known proposition is included in the conditional statement (usually as part of the antecedent clause), then the conditional is declared ‘factual’ relative to that proposition; if the negation of the proposition is included then the conditional is ‘antifactual’ (strongly counterfactual) relative to it; and if the proposition is omitted then the conditional is ‘afactual’ (weakly counterfactual) relative to it. Like their indicative counterparts, subjunctive conditionals may be factual or counterfactual; and furthermore any given conditional statement may be correct or incorrect. These three distinct properties—mood, factuality, and correctness—may be correlated with one another. We shall consider two mechanisms to address potential exceptions that may confound conditional relationships: first, allowing the revision of old conditional statements as new information becomes available; and second, expressing conditionals in a cautious, defeasible manner in the first place. Regarding their use in analysis, there are two basic roles for conditional statements: a conditional may be asserted as a constraint itself, or provided as a query whose truth or falsity is to be evaluated subject to some other set of constraints. Recognizing the diverse types of conditional statements helps to clarify several subtle semantic distinctions. Using the tools of probability and algebra, each semantically distinct conditional statement is represented as a syntactically distinct mathematical expression. These expressions include symbolic probability expressions, polynomials with realnumber coefficients, sets of real numbers defined by polynomial constraints, and systems of equations and inequalities built from such formulas. Various algorithmic methods can be used to compute interesting results from conditional statements represented in mathematical form. These computational methods include linear and nonlinear optimization, arithmetic with symbolic polynomials, and manipulation of relational-database tables. Among other benefits, these methods of analysis offer paraconsistent procedures for logical deduction that produce such familiar results as modus ponens, transitivity, disjunction introduction, and disjunctive syllogism—while avoiding any explosion of consequences from inconsistent premises. The proposed method of analysis is applied to several example problems from Goodman and Adams [7, 2, 1].