Material conditional

In logic and related fields, the material conditional is customarily notated with an infix operator ${\displaystyle \to }$.[1] The material conditional is also notated using the infixes ${\displaystyle \supset }$ and ${\displaystyle \Rightarrow }$. In the prefixed Polish notation, conditionals are notated as ${\displaystyle Cpq}$. In a conditional formula ${\displaystyle p\to q}$, the subformula ${\displaystyle p}$ is referred to as the antecedent and ${\displaystyle q}$ is termed the consequent of the conditional. Conditional statements may be nested such that the antecedent or the consequent may themselves be conditional statements, as in the formula ${\displaystyle (p\to q)\to (r\to s)}$.

When disjunction, conjunction and negation are classical, material implication validates the following equivalences:

• Contraposition: ${\displaystyle P\to Q\equiv \neg Q\to \neg P}$
• Import-Export: ${\displaystyle P\to (Q\to R)\equiv (P\land Q)\to R}$
• Negated conditionals: ${\displaystyle \neg (P\to Q)\equiv P\land \neg Q}$
• Or-and-if: ${\displaystyle P\to Q\equiv \neg P\lor Q}$
• Commutativity of antecedents: ${\displaystyle {\big (}P\to (Q\to R){\big )}\equiv {\big (}Q\to (P\to R){\big )}}$
• Distributivity: ${\displaystyle {\big (}R\to (P\to Q){\big )}\equiv {\big (}(R\to P)\to (R\to Q){\big )}}$

Similarly, on classical interpretations of the other connectives, material implication validates the following entailments:

• Antecedent strengthening: ${\displaystyle P\to Q\models (P\land R)\to Q}$
• Vacuous conditional: ${\displaystyle \neg P\models P\to Q}$
• Transitivity: ${\displaystyle (P\to Q)\land (Q\to R)\models P\to R}$
• Simplification of disjunctive antecedents: ${\displaystyle (P\lor Q)\to R\models (P\to R)\land (Q\to R)}$

Tautologies involving material implication include:

• Reflexivity: ${\displaystyle \models P\to P}$
• Totality: ${\displaystyle \models (P\to Q)\lor (Q\to P)}$
• Conditional excluded middle: ${\displaystyle \models (P\to Q)\lor (P\to \neg Q)}$

Material implication does not closely match the usage of conditional sentences in natural language. For example, even though material conditionals with false antecedents are vacuously true, the natural language statement "If 8 is odd, then 3 is prime" is typically judged false. Similarly, any material conditional with a true consequent is itself true, but speakers typically reject sentences such as "If I have a penny in my pocket, then Paris is in France". These classic problems have been called the paradoxes of material implication.[6] In addition to the paradoxes, a variety of other arguments have been given against a material implication analysis. For instance, counterfactual conditionals would all be vacuously true on such an account.[7]

In the mid-20th century, a number of researchers including H. P. Grice and Frank Jackson proposed that pragmatic principles could explain the discrepancies between natural language conditionals and the material conditional. On their accounts, conditionals denote material implication but end up conveying additional information when they interact with conversational norms such as Grice's maxims.[6][8] Recent work in formal semantics and philosophy of language has generally eschewed material implication as an analysis for natural-language conditionals.[8] In particular, such work has often rejected the assumption that natural-language conditionals are truth functional in the sense that the truth value of "If P, then Q" is determined solely by the truth values of P and Q.[6] Thus semantic analyses of conditionals typically propose alternative interpretations built on foundations such as modal logic, relevance logic, probability theory, and causal models.[8][6][9]

Similar discrepancies have been observed by psychologists studying conditional reasoning. For instance, the notorious Wason selection task study, less than 10% of participants reasoned according to the material conditional. Some researchers have interpreted this result as a failure of the participants to confirm to normative laws of reasoning, while others interpret the participants as reasoning normatively according to nonclassical laws.[10][11][12]

• Brown, Frank Markham (2003), Boolean Reasoning: The Logic of Boolean Equations, 1st edition, Kluwer Academic Publishers, Norwell, MA. 2nd edition, Dover Publications, Mineola, NY, 2003.
• Edgington, Dorothy (2001), "Conditionals", in Lou Goble (ed.), The Blackwell Guide to Philosophical Logic, Blackwell.
• Quine, W.V. (1982), Methods of Logic, (1st ed. 1950), (2nd ed. 1959), (3rd ed. 1972), 4th edition, Harvard University Press, Cambridge, MA.
• Stalnaker, Robert, "Indicative Conditionals", Philosophia, 5 (1975): 269–286.

• Media related to Material conditional at Wikimedia Commons
• Edgington, Dorothy. "Conditionals". In Zalta, Edward N. (ed.). Stanford Encyclopedia of Philosophy.