Material conditional

Material conditional
IMPLY
Definition
Truth table
Logic gate
Normal forms
Disjunctive
Conjunctive
Zhegalkin polynomial
Post's lattices
0-preserving	no
1-preserving	yes
Monotone	no
Affine	no
Self-dual	no
	v; t; e;

Thematerial conditional(also known asmaterial implication) is anoperationcommonly used inlogic.When the conditional symbol $\rightarrow$ isinterpretedas material implication, a formula $P\rightarrow Q$ is true unless $P$ is true and $Q$ is false. Material implication can also be characterized inferentially bymodus ponens,modus tollens,conditional proof,andclassicalreductio ad absurdum.^{[citation needed]}

Material implication is used in all the basic systems ofclassical logicas well as somenonclassical logics.It is assumed as a model of correct conditional reasoning within mathematics and serves as the basis for commands in manyprogramming languages.However, many logics replace material implication with other operators such as thestrict conditionaland thevariably strict conditional.Due to theparadoxes of material implicationand related problems, material implication is not generally considered a viable analysis ofconditional sentencesinnatural language.

Notation

In logic and related fields, the material conditional is customarily notated with an infix operator $\to$ .^[1]The material conditional is also notated using the infixes $\supset$ and $\Rightarrow$ .^[2]In the prefixedPolish notation,conditionals are notated as $Cpq$ .In a conditional formula $p\to q$ ,the subformula $p$ is referred to as theantecedentand $q$ is termed theconsequentof the conditional. Conditional statements may be nested such that the antecedent or the consequent may themselves be conditional statements, as in the formula $(p\to q)\to (r\to s)$ .

History

InArithmetices Principia: Nova Methodo Exposita(1889),Peanoexpressed the proposition "If $A$ ,then $B$ "as $A$ Ɔ $B$ with the symbol Ɔ, which is the opposite of C.^[3]He also expressed the proposition $A\supset B$ as $A$ Ɔ $B$ .^[a]^[4]^[5]Hilbertexpressed the proposition "IfA,thenB"as $A\to B$ in 1918.^[1]Russellfollowed Peano in hisPrincipia Mathematica(1910–1913), in which he expressed the proposition "IfA,thenB"as $A\supset B$ .Following Russell,Gentzenexpressed the proposition "IfA,thenB"as $A\supset B$ .Heytingexpressed the proposition "IfA,thenB"as $A\supset B$ at first but later came to express it as $A\to B$ with a right-pointing arrow.Bourbakiexpressed the proposition "IfA,thenB"as $A\Rightarrow B$ in 1954.^[6]

Definitions

Semantics

From aclassical semantic perspective,material implication is thebinary truth functionaloperator which returns "true" unless its first argument is true and its second argument is false. This semantics can be shown graphically in atruth tablesuch 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

Thetruth tableof $A\rightarrow B$ :

$A$	$B$	$A\rightarrow B$
F	F	T
F	T	T
T	F	F
T	T	T

The logical cases where the antecedent $A$ is false and $A \to B$ is true, are called "vacuous truths". Examples are...

... with $B$ false:"IfMarie Curieis a sister ofGalileo Galilei,then Galileo Galilei is a brother of Marie Curie ",
... with $B$ true:"IfMarie Curieis a sister ofGalileo Galilei,then Marie Curie has a sibling. ".

Deductive definition

Material implication can also be characterizeddeductivelyin terms of the followingrules of inference.^{[citation needed]}

Unlike the semantic definition, this approach to logical connectives permits the examination of structurally identical propositional forms in variouslogical systems,where somewhat different properties may be demonstrated. For example, inintuitionistic logic,which rejects proofs by contraposition as valid rules of inference, $(A\to B)\Rightarrow \neg A\lor B$ is not a propositional theorem, butthe material conditional is used to define negation.^{[clarification needed]}

Formal properties

Whendisjunction,conjunctionandnegationare 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 followingentailments:

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: $(P\lor Q)\to R\models (P\to R)\land (Q\to R)$

Tautologiesinvolving 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 ofconditional sentencesinnatural language.For example, even though material conditionals with false antecedents arevacuously 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 theparadoxes of material implication.^[7]In addition to the paradoxes, a variety of other arguments have been given against a material implication analysis. For instance,counterfactual conditionalswould all be vacuously true on such an account.^[8]

In the mid-20th century, a number of researchers includingH. P. GriceandFrank Jacksonproposed thatpragmaticprinciples could explain the discrepancies between natural language conditionals and the material conditional. On their accounts, conditionalsdenotematerial implication but end up conveying additional information when they interact with conversational norms such asGrice's maxims.^[7]^[9]Recent work informal semanticsandphilosophy of languagehas generally eschewed material implication as an analysis for natural-language conditionals.^[9]In particular, such work has often rejected the assumption that natural-language conditionals aretruth functionalin the sense that the truth value of "IfP,thenQ"is determined solely by the truth values ofPandQ.^[7]Thus semantic analyses of conditionals typically propose alternative interpretations built on foundations such asmodal logic,relevance logic,probability theory,andcausal models.^[9]^[7]^[10]

Similar discrepancies have been observed by psychologists studying conditional reasoning, for instance, by the notoriousWason selection taskstudy, 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.^[11]^[12]^[13]

Notes

^Note that the horseshoe symbol Ɔ has been flipped to become a subset symbol ⊂.

References

^^a ^bHilbert, D. (1918).Prinzipien der Mathematik (Lecture Notes edited by Bernays, P.).
^Mendelson, Elliott(2015).Introduction to Mathematical Logic(6th ed.). Boca Raton: CRC Press/Taylor & Francis Group (A Chapman & Hall Book). p. 2.ISBN 978-1-4822-3778-8.
^Jean van Heijenoort, ed. (1967).From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931.Harvard University Press. pp. 84–87.ISBN 0-674-32449-8.
^Michael Nahas (25 Apr 2022)."English Translation of 'Arithmetices Principia, Nova Methodo Exposita'"(PDF).GitHub. p. VI.Retrieved2022-08-10.
^Mauro ALLEGRANZA (2015-02-13)."elementary set theory – Is there any connection between the symbol ⊃ when it means implication and its meaning as superset?".Mathematics Stack Exchange.Stack Exchange Inc. Answer.Retrieved2022-08-10.
^Bourbaki, N. (1954).Théorie des ensembles.Paris: Hermann & Cie, Éditeurs. p. 14.
^^a ^b ^c ^dEdgington, Dorothy (2008)."Conditionals".In Edward N. Zalta (ed.).The Stanford Encyclopedia of Philosophy(Winter 2008 ed.).
^Starr, Will (2019)."Counterfactuals".In Zalta, Edward N. (ed.).The Stanford Encyclopedia of Philosophy.
^^a ^b ^cGillies, Thony (2017)."Conditionals"(PDF).In Hale, B.; Wright, C.; Miller, A. (eds.).A Companion to the Philosophy of Language.Wiley Blackwell. pp. 401–436.doi:10.1002/9781118972090.ch17.ISBN 9781118972090.
^von Fintel, Kai (2011)."Conditionals"(PDF).In von Heusinger, Klaus; Maienborn, Claudia; Portner, Paul (eds.).Semantics: An international handbook of meaning.de Gruyter Mouton. pp. 1515–1538.doi:10.1515/9783110255072.1515.hdl:1721.1/95781.ISBN 978-3-11-018523-2.
^Oaksford, M.; Chater, N. (1994). "A rational analysis of the selection task as optimal data selection".Psychological Review.101(4): 608–631.CiteSeerX10.1.1.174.4085.doi:10.1037/0033-295X.101.4.608.S2CID 2912209.
^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.CiteSeerX10.1.1.13.1854.doi:10.1016/j.cogsci.2004.02.002.
^von Sydow, M. (2006).Towards a Flexible Bayesian and Deontic Logic of Testing Descriptive and Prescriptive Rules(doctoralThesis). Göttingen: Göttingen University Press.doi:10.53846/goediss-161.S2CID 246924881.

External links

Media related toMaterial conditionalat Wikimedia Commons
Edgington, Dorothy."Conditionals".InZalta, Edward N.(ed.).Stanford Encyclopedia of Philosophy.

[4] Note that the horseshoe symbol Ɔ has been flipped to become a subset symbol ⊂.

[hilbert1918-1] Hilbert, D. (1918).Prinzipien der Mathematik (Lecture Notes edited by Bernays, P.).

[2] Mendelson, Elliott(2015).Introduction to Mathematical Logic(6th ed.). Boca Raton: CRC Press/Taylor & Francis Group (A Chapman & Hall Book). p. 2.ISBN 978-1-4822-3778-8.

[3] Jean van Heijenoort, ed. (1967).From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931.Harvard University Press. pp. 84–87.ISBN 0-674-32449-8.

[5] Michael Nahas (25 Apr 2022)."English Translation of 'Arithmetices Principia, Nova Methodo Exposita'"(PDF).GitHub. p. VI.Retrieved2022-08-10.

[6] Mauro ALLEGRANZA (2015-02-13)."elementary set theory – Is there any connection between the symbol ⊃ when it means implication and its meaning as superset?".Mathematics Stack Exchange.Stack Exchange Inc. Answer.Retrieved2022-08-10.

[bourbaki1954a-7] Bourbaki, N. (1954).Théorie des ensembles.Paris: Hermann & Cie, Éditeurs. p. 14.

[sep-conditionals-8] Edgington, Dorothy (2008)."Conditionals".In Edward N. Zalta (ed.).The Stanford Encyclopedia of Philosophy(Winter 2008 ed.).

[9] Starr, Will (2019)."Counterfactuals".In Zalta, Edward N. (ed.).The Stanford Encyclopedia of Philosophy.

[gillies-10] Gillies, Thony (2017)."Conditionals"(PDF).In Hale, B.; Wright, C.; Miller, A. (eds.).A Companion to the Philosophy of Language.Wiley Blackwell. pp. 401–436.doi:10.1002/9781118972090.ch17.ISBN 9781118972090.

[11] von Fintel, Kai (2011)."Conditionals"(PDF).In von Heusinger, Klaus; Maienborn, Claudia; Portner, Paul (eds.).Semantics: An international handbook of meaning.de Gruyter Mouton. pp. 1515–1538.doi:10.1515/9783110255072.1515.hdl:1721.1/95781.ISBN 978-3-11-018523-2.

[12] Oaksford, M.; Chater, N. (1994). "A rational analysis of the selection task as optimal data selection".Psychological Review.101(4): 608–631.CiteSeerX10.1.1.174.4085.doi:10.1037/0033-295X.101.4.608.S2CID 2912209.

[13] 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.CiteSeerX10.1.1.13.1854.doi:10.1016/j.cogsci.2004.02.002.

[vonSydow2006-14] von Sydow, M. (2006).Towards a Flexible Bayesian and Deontic Logic of Testing Descriptive and Prescriptive Rules(doctoralThesis). Göttingen: Göttingen University Press.doi:10.53846/goediss-161.S2CID 246924881.

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v t e Commonlogical connectives
Tautology/True $\top$
Alternative denial(NAND gate) $\uparrow$ Converse implication $\leftarrow$ Implication(IMPLY gate) $\rightarrow$ Disjunction(OR gate) $\lor$
Negation(NOT gate) $\neg$ Exclusive or(XOR gate) $\not \leftrightarrow$ Biconditional(XNOR gate) $\leftrightarrow$ Statement(Digital buffer)
Joint denial(NOR gate) $\downarrow$ Nonimplication(NIMPLY gate) $\nrightarrow$ Converse nonimplication $\nleftarrow$ Conjunction(AND gate) $\land$
Contradiction/False $\bot$
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