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{{short description|Logical connective}}{{Redirect|Logical conditional|other related meanings|Conditional statement (disambiguation){{!}}Conditional statement}}{{distinguish|Material inference|Material implication (rule of inference)}}

factoids

{{Logical connectives sidebar}}The material conditional (also known as material implication) is an operation commonly used in logic. When the conditional symbol rightarrow is interpreted as material implication, a formula P rightarrow Q is true unless P is true and Q is false. Material implication can also be characterized inferentially by modus ponens, modus tollens, conditional proof, and classical reductio ad absurdum.{{citation needed|reason=This is copied from below, but is it really the minimal + sufficient characterization?|date=February 2021}}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.

Notation

In logic and related fields, the material conditional is customarily notated with an infix operator to.BOOK, Hilbert, D., Prinzipien der Mathematik (Lecture Notes edited by Bernays, P.), 1918, The material conditional is also notated using the infixes supset and Rightarrow.BOOK, Mendelson, Elliott, Elliott Mendelson, Introduction to Mathematical Logic, 2015, 6th, Boca Raton, CRC Press/Taylor & Francis Group (A Chapman & Hall Book), 978-1-4822-3778-8, 2, In the prefixed Polish notation, conditionals are notated as Cpq. In a conditional formula pto q, the subformula p is referred to as the antecedent and 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 (pto q)to(rto s).

History

In (Arithmetices principia, nova methodo exposita|Arithmetices Principia: Nova Methodo Exposita) (1889), Peano expressed the proposition “If A, then B” as A Ɔ B with the symbol Ɔ, which is the opposite of C.BOOK, Jean van Heijenoort, From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931, 1967, Harvard University Press, 0-674-32449-8, 84–87, He also expressed the proposition Asupset B as A Ɔ B.{{efn|Note that the horseshoe symbol Ɔ has been flipped to become a subset symbol ⊂.}}WEB,github.com/mdnahas/Peano_Book/blob/46e27bdb5aed51c078ad99e5a78d134fd2a0c3ca/Peano.pdf, English Translation of ‘Arithmetices Principia, Nova Methodo Exposita’, 2022-08-10, Michael Nahas, 25 Apr 2022, GitHub, VI, WEB,math.stackexchange.com/a/1146502/186330, elementary set theory – Is there any connection between the symbol ⊃ when it means implication and its meaning as superset?, 2022-08-10, Mauro ALLEGRANZA, 2015-02-13, Mathematics Stack Exchange, Stack Exchange Inc, en, Answer, Hilbert expressed the proposition “If A, then B” as Ato B in 1918. Russell followed Peano in his Principia Mathematica (1910–1913), in which he expressed the proposition “If A, then B” as Asupset B. Following Russell, Gentzen expressed the proposition “If A, then B” as Asupset B. Heyting expressed the proposition “If A, then B” as Asupset B at first but later came to express it as Ato B with a right-pointing arrow. Bourbaki expressed the proposition “If A, then B” as ARightarrow B in 1954.BOOK, Bourbaki, N., Théorie des ensembles, 1954, Hermann & Cie, Éditeurs, Paris, 14,

Definitions

Semantics

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 a truth table such as the one below. One can also consider the equivalence A to B equiv neg (A land neg B) equiv neg A lor B.

Truth table

The truth table of A rightarrow B:{{2-ary truth table|1|1|0|1|A rightarrow B}}The logical cases where the antecedent {{mvar|A}} is false and {{math|A → B}} is true, are called “vacuous truths”.Examples are ...

• ... with {{math|B}} false: “If Marie Curie is a sister of Galileo Galilei, then Galileo Galilei is a brother of Marie Curie”,
• ... with {{math|B}} true: “If Marie Curie is a sister of Galileo Galilei, then Marie Curie has a sibling.”.

Deductive definition

Material implication can also be characterized deductively in terms of the following rules of inference.{{citation needed|reason=As in the lead, is this really the minimal + sufficient characterization?|date=February 2021}}

• Modus ponens
• Conditional proof
• Classical contraposition
• Classical reductio ad absurdum

Unlike the semantic definition, this approach to logical connectives permits the examination of structurally identical propositional forms in various logical systems, where somewhat different properties may be demonstrated. For example, in intuitionistic logic, which rejects proofs by contraposition as valid rules of inference, (A to B) Rightarrow neg A lor B is not a propositional theorem, but the material conditional is used to define negation.{{clarify|date=February 2021 |reason= Why is proof theory necessary for this?}}

Formal properties

{{Expand section|date=February 2021}}When disjunction, conjunction and negation are classical, material implication validates the following equivalences:

• Contraposition: P to Q equiv neg Q to neg P
• 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
• Vacuous conditional: neg P models P 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)

Discrepancies with natural language

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.ENCYCLOPEDIA, Dorothy, Edgington, Edward N. Zalta, 2008, Conditionals, The Stanford Encyclopedia of Philosophy, Winter 2008,plato.stanford.edu/archives/win2008/entries/conditionals/, 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.ENCYCLOPEDIA, Starr, Will, Zalta, Edward N., The Stanford Encyclopedia of Philosophy, Counterfactuals, 2019,plato.stanford.edu/archives/fall2019/entries/counterfactuals, 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.ENCYCLOPEDIA, Gillies, Thony, Hale, B., Wright, C., Miller, A., A Companion to the Philosophy of Language, Conditionals,www.thonygillies.org/wp-content/uploads/2015/11/gillies-conditionals-handbook.pdf, 2017, 401–436, Wiley Blackwell, 10.1002/9781118972090.ch17, 9781118972090, Recent work in formal semantics and philosophy of language has generally eschewed material implication as an analysis for natural-language conditionals. 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. Thus semantic analyses of conditionals typically propose alternative interpretations built on foundations such as modal logic, relevance logic, probability theory, and causal models.ENCYCLOPEDIA, von Fintel, Kai, von Heusinger, Klaus, Maienborn, Claudia, Paul, Portner, Semantics: An international handbook of meaning, Conditionals,mit.edu/fintel/fintel-2011-hsk-conditionals.pdf, 2011, 1515–1538, de Gruyter Mouton, 10.1515/9783110255072.1515, 1721.1/95781, 978-3-11-018523-2, free, Similar discrepancies have been observed by psychologists studying conditional reasoning, for instance, by the notorious Wason selection task study, where less than 10% of participants reasoned according to the material conditional. Some researchers have interpreted this result as a failure of the participants to conform to normative laws of reasoning, while others interpret the participants as reasoning normatively according to nonclassical laws.JOURNAL, Oaksford, M., Chater, N., 1994, A rational analysis of the selection task as optimal data selection, Psychological Review, 101, 4, 608–631, 10.1037/0033-295X.101.4.608, 10.1.1.174.4085, 2912209, JOURNAL, Stenning, K., van Lambalgen, M., 2004, A little logic goes a long way: basing experiment on semantic theory in the cognitive science of conditional reasoning, Cognitive Science, 28, 4, 481–530, 10.1016/j.cogsci.2004.02.002, 10.1.1.13.1854, THESIS, von Sydow, M., Towards a Flexible Bayesian and Deontic Logic of Testing Descriptive and Prescriptive Rules, 2006, Göttingen, Göttingen University Press, 10.53846/goediss-161, 246924881,ediss.uni-goettingen.de/handle/11858/00-1735-0000-0006-AC29-9, doctoralThesis, free,

See also

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• 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

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Conditionals

• Corresponding conditional
• Counterfactual conditional
• Indicative conditional
• Strict conditional

Notes

{{Notelist}}

References

{{Reflist}}

Further reading

• 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.

External links

• {{Commons category-inline}}
• SEP, conditionals, Conditionals, Edgington, Dorothy,

{{Logical connectives}}{{Common logical symbols}}{{Mathematical logic}}