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-preserving	no
1-preserving	yes
Monotone	no
Affine	no
Self-dual	no
v t e

Logical connectives
NOT ${\displaystyle \neg A}$, ${\displaystyle -A}$, ${\displaystyle {\overline {A}}}$, ${\displaystyle \sim A}$ AND ${\displaystyle A\land B}$, ${\displaystyle A\cdot B}$, ${\displaystyle AB}$, ${\displaystyle A\&B}$, ${\displaystyle A\&\&B}$ NAND ${\displaystyle A{\overline {\land }}B}$, ${\displaystyle A\uparrow B}$, ${\displaystyle A\mid B}$, ${\displaystyle {\overline {A\cdot B}}}$ OR ${\displaystyle A\lor B}$, ${\displaystyle A+B}$, ${\displaystyle A\mid B}$, ${\displaystyle A\parallel B}$ NOR ${\displaystyle A{\overline {\lor }}B}$, ${\displaystyle A\downarrow B}$, ${\displaystyle {\overline {A+B}}}$ XNOR ${\displaystyle A\ {\text{XNOR}}\ B}$ └ equivalent ${\displaystyle A\equiv B}$, ${\displaystyle A\Leftrightarrow B}$, ${\displaystyle A\leftrightharpoons B}$ XOR ${\displaystyle A{\underline {\lor }}B}$, ${\displaystyle A\oplus B}$ └nonequivalent ${\displaystyle A\not \equiv B}$, ${\displaystyle A\not \Leftrightarrow B}$, ${\displaystyle A\nleftrightarrow B}$ implies ${\displaystyle A\Rightarrow B}$, ${\displaystyle A\supset B}$, ${\displaystyle A\rightarrow B}$ converse ${\displaystyle A\Leftarrow B}$, ${\displaystyle A\subset B}$, ${\displaystyle A\leftarrow B}$
Related concepts
Propositional calculus Predicate logic Boolean algebra Truth table Truth function Boolean function Functional completeness Scope (logic)
Applications
Digital logic Programming languages Mathematical logic Philosophy of logic
Category
v t e

The material conditional (also known as material implication) is an

. When the conditional symbol ${\displaystyle \rightarrow }$ is

${\displaystyle P\rightarrow Q}$

${\displaystyle P}$

${\displaystyle Q}$

Material implication is used in all the basic systems of

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

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]

From a

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 ${\displaystyle A\rightarrow B}$:

${\displaystyle A}$	${\displaystyle B}$	${\displaystyle A\rightarrow B}$
F	F	T
F	T	T
T	F	F
T	T	T

The logical cases where the antecedent A is false and A → B is true, 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.".

Material implication can also be characterized

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

This section needs expansion. You can help by adding to it. (February 2021)

When

• 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: ${\displaystyle \models (P\to Q)\lor (P\to \neg Q)}$

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

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]

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