Material conditional

The material conditional (also known as material implication) is a binary operation commonly used in logic. When the conditional symbol \to is interpreted as material implication, a formula P \to Q is true unless P is true and Q is false. Material implication is used in all the basic systems of classical logic as well as some nonclassical logics. It is assumed as a model of correct conditional reasoning within mathematics and serves as the basis for commands in many programming languages. However, many logics replace material implication with other operators such as the strict conditional and the variably strict conditional. Due to the paradoxes of material implication and related problems, material implication is not generally considered a viable analysis of conditional sentences in natural language.

The material conditional (also known as material implication) is a binary operation commonly used in logic. When the conditional symbol \to is interpreted as material implication, a formula P \to Q is true unless P is true and Q is false. Material implication is used in all the basic systems of classical logic as well as some nonclassical logics. It is assumed as a model of correct conditional reasoning within mathematics and serves as the basis for commands in many programming languages. However, many logics replace material implication with other operators such as the strict conditional and the variably strict conditional. Due to the paradoxes of material implication and related problems, material implication is not generally considered a viable analysis of conditional sentences in natural language. ==Semantics== ===Truth table=== From a classical semantic perspective, material implication is the binary truth functional operator which returns "true" unless its first argument is true and its second argument is false. This semantics can be shown graphically in the following truth table: One can also consider the equivalence A \to B \equiv \neg (A \land \neg B) \equiv \neg A \lor B . The conditionals (A \to B) where the antecedent A is false, are called "vacuous truths". Examples are ... ... with B false: "If Marie Curie is a sister of Galileo Galilei, then Galileo Galilei is a brother of Marie Curie." ... with B true: "If Marie Curie is a sister of Galileo Galilei, then Marie Curie has a sibling." ===Analytic tableaux=== Formulas over the set of connectives \{\to, \bot\} are called f-implicational. In classical logic the other connectives, such as \neg (negation), \land (conjunction), \lor (disjunction) and \leftrightarrow (equivalence), can be defined in terms of \to and \bot (falsity): \begin{align} \neg A & \quad \overset{\text{def}}{=} \quad A \to \bot \\ A \land B & \quad \overset{\text{def}}{=} \quad (A \to (B \to \bot)) \to \bot \\ A \lor B & \quad \overset{\text{def}}{=} \quad (A \to \bot) \to B \\ A \leftrightarrow B & \quad \overset{\text{def}}{=} \quad \{(A \to B) \to [(B \to A) \to \bot]\} \to \bot \\ \end{align} The validity of f-implicational formulas can be semantically established by the method of analytic tableaux. The logical rules are F[p → ((p → ⊥) → ⊥)] | T[p] F[(p → ⊥) → ⊥] | T[p → ⊥] F[⊥] ┌────────┴────────┐ F[p] T[⊥] | | CONTRADICTION CONTRADICTION (T[p], F[p]) (⊥ is true) F[((p → ⊥) → ⊥) → p] | T[(p → ⊥) → ⊥] F[p] ┌────────┴────────┐ F[p → ⊥] T[⊥] | | T[p] CONTRADICTION (⊥ is true) F[⊥] | CONTRADICTION (T[p], F[p]) Hilbert-style proofs can be found here or here. 1. F[(p → q) → ((q → r) → (p → r))] | // from 1 2. T[p → q] 3. F[(q → r) → (p → r)] | // from 3 4. T[q → r] 5. F[p → r] | // from 5 6. T[p] 7. F[r] ┌────────┴────────┐ // from 2 8a. F[p] 8b. T[q] X ┌────────┴────────┐ // from 4 9a. F[q] 9b. T[r] X X A Hilbert-style proof can be found here. Intuitionistic logic: By adding Falsum Elimination ( \bot E) as a rule, one obtains (the implicational fragment of) intuitionistic logic. The statement P \to \neg \neg P is valid (already in minimal logic), unlike the reverse implication which would entail the law of excluded middle. Classical logic: If Double Negation Elimination ( \neg\neg E) is also permitted,{{refn|name="RAA"|Instead of \neg\neg E one can add reductio ad absurdum as a rule to obtain (full) classical logic: \frac{\begin{array}{c} [A \to \bot] \\ \vdots \\ \bot \end{array}}{A} (RAA)}} the system defines (full!) classical logic. ==A selection of theorems (classical logic)== In classical logic material implication validates the following: Import-export: P \to (Q \to R) \equiv (P \land Q) \to R Negated conditionals: \neg(P \to Q) \equiv P \land \neg Q Or-and-if: P \to Q \equiv \neg P \lor Q Commutativity of antecedents: \big(P \to (Q \to R)\big) \equiv \big(Q \to (P \to R)\big) Left distributivity: \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: P \to Q \models (P \land R) \to Q Transitivity: (P \to Q) \land (Q \to R) \models P \to R Simplification of disjunctive antecedents: (P \lor Q) \to R \models (P \to R) \land (Q \to R) Tautologies involving material implication include: Reflexivity: \models P \to P Totality: \models (P \to Q) \lor (Q \to P) Conditional excluded middle: \models (P \to Q) \lor (P \to \neg Q) ==The relationship between the material conditional and logical consequence== The material conditional is a sentential connective within a formal language. It should not be confused with the relation of logical consequence (also called logical implication or entailment), which is standardly treated as a relation between sentences expressed in a metalanguage. The relationship between the material conditional and the logical consequence relation is given by the deduction theorem. \Gamma \cup \{A\} \vdash B \; if and only if \; \Gamma \vdash A \to B This can be read as stating that the set of sentences \Gamma together with A logically implies B if and only if \Gamma logically implies the material conditional A \to B . In the special case where \Gamma is empty, this reduces to: A \vdash B \; if and only if \; \vdash A \to B This states A logically implies B if and only if the material conditional A \to B is a theorem of the logic. Many textbooks reserve the term logical consequence (or logical implication) for the semantic consequence relation with the symbol \models . In which case the relation becomes A \models B \; if and only if \; \models A \to B A logically implies B if and only if the material conditional A \to B is a tautology. ==See also== Boolean domain Boolean function Boolean logic Conditional quantifier Implicational propositional calculus Laws of Form Logical graph Logical equivalence Material implication (rule of inference) Peirce's law Propositional calculus Sole sufficient operator ===Conditionals=== Corresponding conditional Counterfactual conditional Indicative conditional Strict conditional ==External links==

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