Material Conditional - Philosophical Problems With Material Conditional

Philosophical Problems With Material Conditional

Outside of mathematics, it is a matter of some controversy as to whether the truth function for material implication provides an adequate treatment of ‘conditional statements in English’ (a sentence in the indicative mood with a conditional clause attached, i.e., an indicative conditional, or false-to-fact sentences in the subjunctive mood, i.e., a counterfactual conditional). That is to say, critics argue that in some non-mathematical cases, the truth value of a compound statement, "if p then q", is not adequately determined by the truth values of p and q. Examples of non-truth-functional statements include: "p because q", "p before q" and "it is possible that p". “ the sixteen possible truth-functions of A and B, is the only serious candidate. First, it is uncontroversial that when A is true and B is false, "If A, B" is false. A basic rule of inference is modus ponens: from "If A, B" and A, we can infer B. If it were possible to have A true, B false and "If A, B" true, this inference would be invalid. Second, it is uncontroversial that "If A, B" is sometimes true when A and B are respectively (true, true), or (false, true), or (false, false)… Non-truth-functional accounts agree that "If A, B" is false when A is true and B is false; and they agree that the conditional is sometimes true for the other three combinations of truth-values for the components; but they deny that the conditional is always true in each of these three cases. Some agree with the truth-functionalist that when A and B are both true, "If A, B" must be true. Some do not, demanding a further relation between the facts that A and that B.”

The truth-functional theory of the conditional was integral to Frege's new logic (1879). It was taken up enthusiastically by Russell (who called it "material implication"), Wittgenstein in the Tractatus, and the logical positivists, and it is now found in every logic text. It is the first theory of conditionals which students encounter. Typically, it does not strike students as obviously correct. It is logic's first surprise. Yet, as the textbooks testify, it does a creditable job in many circumstances. And it has many defenders. It is a strikingly simple theory: "If A, B" is false when A is true and B is false. In all other cases, "If A, B" is true. It is thus equivalent to "~(A&~B)" and to "~A or B". "A ⊃ B" has, by stipulation, these truth conditions.

— Dorothy Edgington, The Stanford Encyclopedia of Philosophy, “Conditionals”

The meaning of the material conditional can sometimes be used in the natural language English "if condition then consequence" construction (a kind of conditional sentence), where condition and consequence are to be filled with English sentences. However, this construction also implies a "reasonable" connection between the condition (protasis) and consequence (apodosis) (see Connexive logic).

There are various kinds of conditionals in English; e.g., there is the indicative conditional and the subjunctive or counterfactual conditional. The latter do not have the same truth conditions as the material conditional. For an overview of some the various analyses, formal and informal, of conditionals, see the "References" section below.