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Material conditional
Logical connective

The material conditional (also known as material implication) is a binary operation commonly used in logic. When the conditional symbol → {\displaystyle \rightarrow } is interpreted as material implication, a formula P → Q {\displaystyle P\rightarrow Q} is true unless P {\displaystyle P} is true and Q {\displaystyle 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.

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Notation

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 } .2 In the prefixed Polish notation, conditionals are notated as C p q {\displaystyle Cpq} . In a conditional formula p → q {\displaystyle p\to q} , the subformula p {\displaystyle p} is referred to as the antecedent and q {\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 ( p → q ) → ( r → s ) {\displaystyle (p\to q)\to (r\to s)} .

History

In Arithmetices Principia: Nova Methodo Exposita (1889), Peano expressed the proposition "If A {\displaystyle A} , then B {\displaystyle B} " as A {\displaystyle A} Ɔ B {\displaystyle B} with the symbol Ɔ, which is the opposite of C.3 He also expressed the proposition A ⊃ B {\displaystyle A\supset B} as A {\displaystyle A} Ɔ B {\displaystyle B} .456 Hilbert expressed the proposition "If A, then B" as A → B {\displaystyle A\to B} in 1918.7 Russell followed Peano in his Principia Mathematica (1910–1913), in which he expressed the proposition "If A, then B" as A ⊃ B {\displaystyle A\supset B} . Following Russell, Gentzen expressed the proposition "If A, then B" as A ⊃ B {\displaystyle A\supset B} . Heyting expressed the proposition "If A, then B" as A ⊃ B {\displaystyle A\supset B} at first but later came to express it as A → B {\displaystyle A\to B} with a right-pointing arrow. Bourbaki expressed the proposition "If A, then B" as A ⇒ B {\displaystyle A\Rightarrow B} in 1954.8

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 → B ≡ ¬ ( A ∧ ¬ B ) ≡ ¬ A ∨ B {\displaystyle A\to B\equiv \neg (A\land \neg B)\equiv \neg A\lor B} .

Truth table

The truth table of A → B {\displaystyle A\rightarrow B} :

A {\displaystyle A} B {\displaystyle B} A → B {\displaystyle A\rightarrow B}
FFT
FTT
TFF
TTT

The conditionals ( A → B ) {\displaystyle (A\to B)} where the antecedent A {\displaystyle A} is false, are called "vacuous truths". Examples are ...

  • ... with B {\displaystyle B} false: "If Marie Curie is a sister of Galileo Galilei, then Galileo Galilei is a brother of Marie Curie."
  • ... with B {\displaystyle B} true: "If Marie Curie is a sister of Galileo Galilei, then Marie Curie has a sibling."

Syntactical properties

The semantic definition by truth tables does not permit the examination of structurally identical propositional forms in various logical systems, where different properties may be demonstrated. The system considered here is implicational propositional calculus to which a nullary connective ⊥ {\displaystyle \bot } for falsity is added.

  • Minimal logic: By limiting the natural deduction rules of this logic to Implication Introduction ( → {\displaystyle \to } I) and Implication Elimination ( → {\displaystyle \to } E), one obtains minimal logic (as discussed by Johansson).9 See below.
  • Intuitionistic logic: By adding the rule Falsum Elimination ( ⊥ {\displaystyle \bot } E), one obtains intuitionistic logic. See below.
  • Classical logic: If Reductio ad Absurdum (RAA) is also permitted, the result is classical logic. See below.

The well-formed formulas are:

  1. Each sentence-letter is a formula.
  2. " ⊥ {\displaystyle \bot } " is a formula.
  3. If A {\displaystyle A} and B {\displaystyle B} are formulas, so is ( A → B ) {\displaystyle (A\to B)} .
  4. Nothing else is a formula.

Formulas over the set of connectives { → , ⊥ } {\displaystyle \{\to ,\bot \}} are called f-implicational.10 All other connectives, such as ¬ {\displaystyle \neg } (negation), ∧ {\displaystyle \land } (conjunction) and ∨ {\displaystyle \lor } (disjunction), are defined in terms of " → {\displaystyle \to } " and " ⊥ {\displaystyle \bot } ". ¬ A = def A → ⊥ A ∧ B = def ( A → ( B → ⊥ ) ) → ⊥ A ∨ B = def ( A → ⊥ ) → B {\displaystyle {\begin{aligned}\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\end{aligned}}}

Consider the following (candidate) natural deduction rules.

Implication Introduction ( → {\displaystyle \to } I)

If assuming A {\displaystyle A} one can derive B {\displaystyle B} , then one can conclude A → B {\displaystyle A\to B} .

[ A ] ⋮ B A → B {\displaystyle {\frac {\begin{array}{c}[A]\\\vdots \\B\end{array}}{A\to B}}} ( → {\displaystyle \to } I)

[ A ] {\displaystyle [A]} is an assumption that is discharged when applying the rule.

Implication Elimination ( → {\displaystyle \to } E)

This rule corresponds to modus ponens.

A → B A B {\displaystyle {\frac {A\to B\quad A}{B}}} ( → {\displaystyle \to } E)

A A → B B {\displaystyle {\frac {A\quad A\to B}{B}}} ( → {\displaystyle \to } E)

Reductio ad absurdum (RAA)

[ A → ⊥ ] ⋮ ⊥ A {\displaystyle {\frac {\begin{array}{c}[A\to \bot ]\\\vdots \\\bot \end{array}}{A}}} (RAA)

[ A → ⊥ ] {\displaystyle [A\to \bot ]} is an assumption that is discharged when applying the rule.

Falsum Elimination ( ⊥ {\displaystyle \bot } E)

From falsum ( ⊥ {\displaystyle \bot } ) one can derive any formula.(ex falso quodlibet)

⊥ A {\displaystyle {\frac {\bot }{A}}} ( ⊥ {\displaystyle \bot } E)

  • When only → {\displaystyle \to } I and → {\displaystyle \to } E are admitted as deduction rules, the system corresponds to minimal logic as defined by Johansson.11
  • When ⊥ {\displaystyle \bot } E is added to the rules, the system defines intuitionistic logic.
⊥ {\displaystyle \bot } E allows to prove A → ¬ ¬ A {\displaystyle A\to \neg \neg A} , but not the reverse implication which would entail the law of excluded middle.
Proof of p → ¬ ¬ p {\displaystyle p\to \neg \neg p\quad } , using ⊥ {\displaystyle \bot } E
1. [ p ] // Assume
2. [ p → ⊥ ] // Assume
3. ⊥ // → {\displaystyle \to } E (1, 2)
4. (p → ⊥) → ⊥) // → {\displaystyle \to } I (2, 3), discharging 2
5. p → ((p → ⊥) → ⊥) // → {\displaystyle \to } I (1, 4), discharging 1
  • When all four natural deduction rules are admitted, the system defines classical logic.
The extra RAA allows to prove ¬ ¬ A → A {\displaystyle \neg \neg A\to A} .
Proof of ¬ ¬ p → p {\displaystyle \neg \neg p\to p\quad } , using RAA
1. [ (p → ⊥) → ⊥ ] // Assume
2. [ p → ⊥ ] // Assume
3. ⊥ // → {\displaystyle \to } E (1, 2)
4. p // RAA (2, 3), discharging 2
5. ((p → ⊥) → ⊥) → p // → {\displaystyle \to } I (1, 4), discharging 1

A selection of theorems (classical logic)

In classical logic material implication validates the following equivalences:

  • Contraposition: P → Q ≡ ¬ Q → ¬ P {\displaystyle P\to Q\equiv \neg Q\to \neg P}
  • Import-export: P → ( Q → R ) ≡ ( P ∧ Q ) → R {\displaystyle P\to (Q\to R)\equiv (P\land Q)\to R}
  • Negated conditionals: ¬ ( P → Q ) ≡ P ∧ ¬ Q {\displaystyle \neg (P\to Q)\equiv P\land \neg Q}
  • Or-and-if: P → Q ≡ ¬ P ∨ Q {\displaystyle P\to Q\equiv \neg P\lor Q}
  • Commutativity of antecedents: ( P → ( Q → R ) ) ≡ ( Q → ( P → R ) ) {\displaystyle {\big (}P\to (Q\to R){\big )}\equiv {\big (}Q\to (P\to R){\big )}}
  • Left distributivity: ( R → ( P → Q ) ) ≡ ( ( R → P ) → ( R → Q ) ) {\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: P → Q ⊨ ( P ∧ R ) → Q {\displaystyle P\to Q\models (P\land R)\to Q}
  • Vacuous conditional: ¬ P ⊨ P → Q {\displaystyle \neg P\models P\to Q}
  • Transitivity: ( P → Q ) ∧ ( Q → R ) ⊨ P → R {\displaystyle (P\to Q)\land (Q\to R)\models P\to R}
  • Simplification of disjunctive antecedents: ( P ∨ Q ) → R ⊨ ( P → R ) ∧ ( Q → R ) {\displaystyle (P\lor Q)\to R\models (P\to R)\land (Q\to R)}

Tautologies involving material implication include:

  • Reflexivity: ⊨ P → P {\displaystyle \models P\to P}
  • Totality: ⊨ ( P → Q ) ∨ ( Q → P ) {\displaystyle \models (P\to Q)\lor (Q\to P)}
  • Conditional excluded middle: ⊨ ( P → Q ) ∨ ( P → ¬ Q ) {\displaystyle \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.12 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, when in fact some are false.13

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.1415</ref> Recent work in formal semantics and philosophy of language has generally eschewed material implication as an analysis for natural-language conditionals.16 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.17 Thus semantic analyses of conditionals typically propose alternative interpretations built on foundations such as modal logic, relevance logic, probability theory, and causal models.181920

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.212223

See also

Conditionals

Notes

Bibliography

  • Bourbaki, N. (1954). Théorie des ensembles. Paris: Hermann & Cie, Éditeurs. p. 14.
  • Edgington, Dorothy (2008). "Conditionals". In Edward N. Zalta (ed.). The Stanford Encyclopedia of Philosophy (Winter 2008 ed.).
  • Van Heijenoort, Jean, 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.
  • Hilbert, 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.
  • Starr, Will (2019). "Counterfactuals". In Zalta, Edward N. (ed.). The Stanford Encyclopedia of Philosophy.

Further reading

References

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

  2. Mendelson 2015. - 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. Van Heijenoort 1967. - Van Heijenoort, Jean, 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. ↩

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

  5. Nahas 2022, p. VI. - Nahas, Michael (25 Apr 2022). "English Translation of 'Arithmetices Principia, Nova Methodo Exposita'" (PDF). GitHub. Retrieved 2022-08-10. https://github.com/mdnahas/Peano_Book/blob/46e27bdb5aed51c078ad99e5a78d134fd2a0c3ca/Peano.pdf ↩

  6. Allegranza 2015. - Allegranza, Mauro (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. Retrieved 2022-08-10. https://math.stackexchange.com/a/1146502/186330 ↩

  7. Hilbert 1918. - Hilbert, D. (1918). Prinzipien der Mathematik (Lecture Notes edited by Bernays, P.). ↩

  8. Bourbaki 1954, p. 14. - Bourbaki, N. (1954). Théorie des ensembles. Paris: Hermann & Cie, Éditeurs. p. 14. ↩

  9. Johansson 1937. - Johansson, Ingebrigt (1937). "Der Minimalkalkül, ein reduzierter intuitionistischer Formalismus". Compositio Mathematica (in German). 4: 119–136. http://www.numdam.org/item/CM_1937__4__119_0 ↩

  10. Franco et al. 1999. - Franco, John; Goldsmith, Judy; Schlipf, John; Speckenmeyer, Ewald; Swaminathan, R.P. (1999). "An algorithm for the class of pure implicational formulas". Discrete Applied Mathematics. 96–97: 89–106. doi:10.1016/S0166-218X(99)00038-4. https://doi.org/10.1016%2FS0166-218X%2899%2900038-4 ↩

  11. Johansson 1937. - Johansson, Ingebrigt (1937). "Der Minimalkalkül, ein reduzierter intuitionistischer Formalismus". Compositio Mathematica (in German). 4: 119–136. http://www.numdam.org/item/CM_1937__4__119_0 ↩

  12. Edgington 2008. - Edgington, Dorothy (2008). "Conditionals". In Edward N. Zalta (ed.). The Stanford Encyclopedia of Philosophy (Winter 2008 ed.). http://plato.stanford.edu/archives/win2008/entries/conditionals/ ↩

  13. For example, "If Janis Joplin were alive today, she would drive a Mercedes-Benz", see Starr (2019) /wiki/Janis_Joplin ↩

  14. Edgington 2008. - Edgington, Dorothy (2008). "Conditionals". In Edward N. Zalta (ed.). The Stanford Encyclopedia of Philosophy (Winter 2008 ed.). http://plato.stanford.edu/archives/win2008/entries/conditionals/ ↩

  15. Gillies 2017. - 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. http://www.thonygillies.org/wp-content/uploads/2015/11/gillies-conditionals-handbook.pdf ↩

  16. Gillies 2017. - 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. http://www.thonygillies.org/wp-content/uploads/2015/11/gillies-conditionals-handbook.pdf ↩

  17. Edgington 2008. - Edgington, Dorothy (2008). "Conditionals". In Edward N. Zalta (ed.). The Stanford Encyclopedia of Philosophy (Winter 2008 ed.). http://plato.stanford.edu/archives/win2008/entries/conditionals/ ↩

  18. Gillies 2017. - 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. http://www.thonygillies.org/wp-content/uploads/2015/11/gillies-conditionals-handbook.pdf ↩

  19. Edgington 2008. - Edgington, Dorothy (2008). "Conditionals". In Edward N. Zalta (ed.). The Stanford Encyclopedia of Philosophy (Winter 2008 ed.). http://plato.stanford.edu/archives/win2008/entries/conditionals/ ↩

  20. Von Fintel 2011. - 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. http://mit.edu/fintel/fintel-2011-hsk-conditionals.pdf ↩

  21. Oaksford & Chater 1994. - Oaksford, M.; Chater, N. (1994). "A rational analysis of the selection task as optimal data selection". Psychological Review. 101 (4): 608–631. CiteSeerX 10.1.1.174.4085. doi:10.1037/0033-295X.101.4.608. S2CID 2912209. https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.174.4085 ↩

  22. Stenning & van Lambalgen 2004. - 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. CiteSeerX 10.1.1.13.1854. doi:10.1016/j.cogsci.2004.02.002. https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.13.1854 ↩

  23. Von Sydow 2006. - 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. https://ediss.uni-goettingen.de/handle/11858/00-1735-0000-0006-AC29-9 ↩