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# Material conditional

Material conditional
IMPLY
Definition${\displaystyle x\rightarrow y}$
Truth table${\displaystyle (1011)}$
Logic gate
Normal forms
Disjunctive${\displaystyle {\overline {x}}+y}$
Conjunctive${\displaystyle {\overline {x}}+y}$
Zhegalkin polynomial${\displaystyle 1\oplus x\oplus xy}$
Post's lattices
0-preservingno
1-preservingyes
Monotoneno
Affineno

The material conditional (also known as material implication) is an operation commonly used in logic. When the conditional symbol ${\displaystyle \rightarrow }$ is interpreted as material implication, a formula ${\displaystyle P\rightarrow Q}$ is true unless ${\displaystyle P}$ is true and ${\displaystyle Q}$ is false. Material implication can also be characterized inferentially by modus ponens, modus tollens, conditional proof, and classical reductio ad absurdum.[citation needed]

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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• PHILOSOPHY - Language: Conditionals #1 [HD]
• Conditional Statements: if p then q
• Conditionals: the material conditional
• Is the indicative conditional the same as the material conditional?
• What are the Truth Definitions for the Material Conditional?

#### Transcription

(electronic music) - My name is Justin Khoo, and I'm an Assistant Professor of Philosophy at MIT. Today we are going to look at conditionals, which are a class of sentences that have puzzled philosophers for thousands of years. Here's an example of a conditional sentence from a speech by former Presidential candidate, Mitt Romney. "If the safety net needs repair, I will fix it." Conditional sentences, like one, consist of two parts, an antecedent, "the safety net needs repair," and a consequent, "I will fix it." Our question today is, what do conditional sentences, like one, mean? In other words, by uttering this sentence, what has Mitt Romney told us? Here's a way to think about questions of meaning like this. When I say, "The cat is on the mat," I tell you that the cat is on the mat rather than not on the mat. This is because the meaning of two is that the cat is on the mat. Okay, that's pretty easy. What about our conditional sentence one? What has Mitt Romney told us by uttering it? One way of figuring out what Romney has told us is to get clear on what he has not told us. He hasn't told us that the safety net needs repair, and he also hasn't told us that he will fix the safety net. Rather, what he said is that there is some connection between the safety net needing repair and his fixing it, but what connection? Here's the simple answer. By saying the sentence one, Romney has told us that it is not the case that the safety net needs repair and he won't fix it. Equivalently, he said that either the safety net doesn't need repair, or that he will fix it. Let's call this theory the material conditional theory. Philosophers, at least as far back as the Hellenistic philosopher, Philo of Megara, have been attracted to this theory about what conditionals mean. In order to state the material conditional theory more precisely, we will make use of a device, from logic, called a truth table. A truth table is a way of representing how the truth of a complex sentence, in this case, the conditional one, depends on the truth values of its parts, in this case, the antecedent and consequent of one. Let's start with a simple example of a conjunction. Take the sentence, "The cat is on the mat, and the cat is fat." Naturally, someone who says this tells you two things, that the cat is on the mat, and that the cat is fat, thus the sentence three is true if both the cat is on the mat and the cat is fat, and false if either the cat isn't on the mat or the cat isn't fat. We draw this dependence of the truth value of the whole sentence on its parts in our truth table as follows, noting T for true, when the sentence is true, and F for false, when the sentence is false. Notice that, since we want to represent how the truth value of three depends on the truth values of its parts, the first two columns contain every possible combination of assigning either T or F to the parts of three. Furthermore, notice three only has a T in the row where both of its parts have Ts. This captures the fact that conjunctions are true only if both of their conjuncts are true, and false otherwise. It also captures the fact that three tells us both that the cat is on the mat and that the cat is fat, since that is the only condition under which it is true. Okay, so now what about our conditional sentence one? According to the material conditional theory, one tells us that it is not the case that the safety net needs repair and Romney won't fix it, so, in our truth table, we assign F to the conditional only on row two, where it is true that the safety net needs repair, and false that Romney will fix it. We assign T to it on all other rows. This assignment of truth values, therefore, entirely captures the meaning of the conditional one according to the material conditional theory. Now, although the material conditional theory has been endorsed by many philosophers, it faces several difficult challenges. You might have noticed that, according to the theory, the sentence one is true in all rows besides the second, does this seem right to you? In the next video, we will explore some challenges facing the material conditional theory, and see how some other theories, about what conditionals mean, may fare better.

## 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 ${\displaystyle Cpq}$. 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 ${\displaystyle (p\to q)\to (r\to s)}$.

### History

In Arithmetices Principia: Nova Methodo Exposita (1889), Peano expressed the proposition "If ${\displaystyle A}$, then ${\displaystyle B}$" as ${\displaystyle A}$ Ɔ ${\displaystyle B}$ with the symbol Ɔ, which is the opposite of C.[3] He also expressed the proposition ${\displaystyle A\supset B}$ as ${\displaystyle A}$ Ɔ ${\displaystyle B}$.[a][4][5] Hilbert expressed the proposition "If A, then B" as ${\displaystyle A\to B}$ in 1918.[1] Russell followed Peano in his Principia Mathematica (1910–1913), in which he expressed the proposition "If A, then B" as ${\displaystyle A\supset B}$. Following Russell, Gentzen expressed the proposition "If A, then B" as ${\displaystyle A\supset B}$. Heyting expressed the proposition "If A, then B" as ${\displaystyle A\supset B}$ at first but later came to express it as ${\displaystyle A\to B}$ with a right-pointing arrow. Bourbaki expressed the proposition "If A, then B" as ${\displaystyle A\Rightarrow B}$ in 1954.[6]

## 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 ${\displaystyle A\to B\equiv \neg (A\land \neg B)\equiv \neg A\lor B}$.

### Truth table

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

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

The logical cases where the antecedent A is false and AB is true, are called "vacuous truths". Examples are ...

### Deductive definition

Material implication can also be characterized deductively in terms of the following rules of inference.[citation needed]

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, ${\displaystyle (A\to B)\Rightarrow \neg A\lor B}$ is not a propositional theorem, but the material conditional is used to define negation.[clarification needed]

## Formal properties

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: ${\displaystyle {\big (}P\to (Q\to R){\big )}\equiv {\big (}Q\to (P\to R){\big )}}$
• Left 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: ${\displaystyle \models (P\to Q)\lor (P\to \neg Q)}$

## 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.[7] 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.[8]

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.[7][9] Recent work in formal semantics and philosophy of language has 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 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.[7] Thus semantic analyses of conditionals typically propose alternative interpretations built on foundations such as modal logic, relevance logic, probability theory, and causal models.[9][7][10]

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.[11][12][13]

## Notes

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

## References

1. ^ a b 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.
4. ^ Michael Nahas (25 Apr 2022). "English Translation of 'Arithmetices Principia, Nova Methodo Exposita'" (PDF). GitHub. p. VI. Retrieved 2022-08-10.
5. ^ 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. Retrieved 2022-08-10.
6. ^ Bourbaki, N. (1954). Théorie des ensembles. Paris: Hermann & Cie, Éditeurs. p. 14.
7. ^ a b c d Edgington, Dorothy (2008). "Conditionals". In Edward N. Zalta (ed.). The Stanford Encyclopedia of Philosophy (Winter 2008 ed.).
8. ^ Starr, Will (2019). "Counterfactuals". In Zalta, Edward N. (ed.). The Stanford Encyclopedia of Philosophy.
9. ^ a b c 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.
10. ^ 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.
11. ^ 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.
12. ^ 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.
13. ^ 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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