Material conditional: Difference between revisions

Revision as of 05:42, 24 August 2013

Latest revision as of 19:26, 28 July 2025

(303 intermediate revisions by more than 100 users not shown)

Line 1:

{{short description|Logical connective}}

{{Redirect|Logical conditional|other related meanings|Conditional statement (disambiguation){{!}}Conditional statement}}

{{distinguish|Material inference|Material implication (rule of inference)}}

{{Distinguish2|[[material inference]]}}

{{Infobox logical connective

{{Confusing|date=May 2013}}

| title = Material conditional

| other titles = IMPLY

| Venn diagram = Venn1011.svg

| wikifunction = Z10329

| definition = <math>x \to y</math>

| truth table = <math>(1011)</math>

| logic gate = IMPLY_ANSI.svg

| DNF = <math>\overline{x} + y</math>

| CNF = <math>\overline{x} + y</math>

| Zhegalkin = <math>1 \oplus x \oplus xy</math>

| 0-preserving = no

| 1-preserving = yes

| monotone = no

| affine = no

| self-dual = no

}}

{{Logical connectives sidebar}}

The '''material conditional''' (also known as '''material implication''') is a [[binary operation]] commonly used in [[mathematical logic|logic]]. When the conditional symbol <math>\to</math> is [[Interpretation (logic)|interpreted]] as material implication, a formula <math> P \to Q</math> is true unless <math>P</math> is true and <math>Q</math> is false.

Material implication is used in all the basic systems of [[classical logic]] as well as some [[nonclassical logic]]s. It is assumed as a model of correct conditional reasoning within mathematics and serves as the basis for commands in many [[programming language]]s. 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 sentence]]s in [[natural language]].

The '''material conditional''' (also known as "'''material implication'''", "'''material consequence'''", or simply "'''implication'''", "'''implies'''" or "'''conditional'''") is a [[logical connective]] (or a [[binary operator]]) that is often symbolized by a forward arrow "→". The material conditional is used to form [[statement (logic)|statements]] of the form "''p''→''q''" (termed a [[conditional statement]]) which is read as "if p then q" and conventionally compared to the English construction "If...then...". But unlike as the English construction may, the conditional statement "''p''→''q''" does not specify a causal relationship between ''p'' and ''q'' and is to be understood to mean "if ''p'' is true, then ''q'' is also true" such that the statement "''p''→''q''" is false only when ''p'' is true and ''q'' is false.<ref>{{cite web|title=forallx: An Introduction to Formal Logic|url=http://www.fecundity.com/codex/forallx.pdf|author=Magnus, P.D|date=January 6, 2012|publisher=Creative Commons|page=25|accessdate=28 May 2013}}</ref> The material conditional is also to be distinguished from [[logical consequence]].

== Notation ==

The material conditional is also symbolized using:

In logic and related fields, the material conditional is customarily notated with an infix operator <math>\to</math>.{{sfn|Hilbert|1918}} The material conditional is also notated using the infixes <math>\supset</math> and <math>\Rightarrow</math>.{{sfn|Mendelson|2015}} In the prefixed [[Polish notation]], conditionals are notated as <math>Cpq</math>. In a conditional formula <math>p\to q</math>, the subformula <math>p</math> is referred to as the ''[[antecedent (logic)|antecedent]]'' and <math>q</math> 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 <math>(p\to q)\to(r\to s)</math>.

# <math>p \supset q</math> (Although this symbol is confused with the superset symbol used by [[algebra of sets]].);

# <math>p \Rightarrow q</math> (Although this symbol is often used for [[logical consequence]] (i.e. logical implication) rather than for material conditional.)

== History ==

With respect to the material conditionals above, ''p'' is termed the ''[[antecedent (logic)|antecedent]]'', and ''q'' the ''[[consequent]]'' of the conditional. Conditional statements may be nested such that either or both of the antecedent or the consequent may themselves be conditional statements. In the example "(''p''→''q'')&nbsp;→&nbsp;(''r''→''s'')" both the antecedent and the consequent are conditional statements.

In ''[[Arithmetices principia, nova methodo exposita|Arithmetices Principia: Nova Methodo Exposita]]'' (1889), [[Giuseppe Peano|Peano]] expressed the proposition "If <math>A</math>, then <math>B</math>" as <math>A</math> Ɔ <math>B</math> with the symbol Ɔ, which is the opposite of C.{{sfn|Van Heijenoort|1967}} He also expressed the proposition <math>A\supset B</math> as <math>A</math> Ɔ <math>B</math>.<ref>Note that the horseshoe symbol Ɔ has been flipped to become a subset symbol ⊂.</ref>{{sfn|Nahas|2022|page=VI}}{{Citation needed|reason=Originally cited a Stack Exchange post, which is original research.|date=July 2025}} [[David Hilbert|Hilbert]] expressed the proposition "If ''A'', then ''B''" as <math>A\to B</math> in 1918.{{sfn|Hilbert|1918}} [[Bertrand Russell|Russell]] followed Peano in his ''[[Principia Mathematica]]'' (1910–1913), in which he expressed the proposition "If ''A'', then ''B''" as <math>A\supset B</math>. Following Russell, [[Gerhard Gentzen|Gentzen]] expressed the proposition "If ''A'', then ''B''" as <math>A\supset B</math>. [[Arend Heyting|Heyting]] expressed the proposition "If ''A'', then ''B''" as <math>A\supset B</math> at first but later came to express it as <math>A\to B</math> with a right-pointing arrow.<!-- check https://jeff560.tripod.com/set.html later --> [[Nicolas Bourbaki|Bourbaki]] expressed the proposition "If ''A'', then ''B''" as <math>A \Rightarrow B</math> in 1954.{{sfn|Bourbaki|1954|page=14}}<ref>{{cite web |last=Miller |first=Jeff |date=2020 |title=Earliest Uses of Symbols for Set Theory and Logic |url=https://mathshistory.st-andrews.ac.uk/Miller/mathsym/set/ |website=Maths History (University of St Andrews) |publisher=University of St Andrews |access-date=10 June 2025}}</ref>

==Semantics==

In [[classical logic]] <math>p \rightarrow q</math> is [[Logical equivalence|logically equivalent]] to <math>\neg(p \and \neg q)</math> and by [[De Morgan's Law]] to <math>\neg p \or q</math>.<ref>{{cite web|title=A Modern Formal Logic Primer: Sentence Logic Volume 1|url=http://tellerprimer.ucdavis.edu/pdf/1ch4.pdf|author=Teller, Paul|date=January 10, 1989|publisher=Prentice Hall|page=54|accessdate=28 May 2013}}</ref>

===Truth table===

From a [[classical logic|classical]] [[semantics of logic|semantic perspective]], material implication is the [[binary operator|binary]] [[truth function]]al operator which returns "true" unless its first argument is true and its second argument is false. This semantics can be shown graphically in the following [[truth table]]:

{{2-ary truth table|1|1|0|1|<math>A \to B</math>}}

One can also consider the equivalence <math>A \to B \equiv \neg (A \land \neg B) \equiv \neg A \lor B</math>.

The conditionals <math>(A \to B)</math> where the antecedent <math>A</math> is false, are called "[[vacuous truth]]s".

==Definitions of the material conditional==

Examples are ...

Logicians have many different views on the nature of material implication and approaches to explain its sense.<ref>{{cite web | url=http://www.cs.cornell.edu/Info/People/gries/symposium/clarke.htm | title=A Comparison of Techniques for Introducing Material Implication | publisher=Cornell University | date=March 1996 | accessdate=March 4, 2012 | author=Clarke, Matthew C.}}</ref>

* ... with <math>B</math> false: ''"If [[Marie Curie]] is a sister of [[Galileo Galilei]], then Galileo Galilei is a brother of Marie Curie."''

* ... with <math>B</math> true: ''"If Marie Curie is a sister of Galileo Galilei, then Marie Curie has a sibling."''

===AsAnalytic a truth functiontableaux===

{{further|Method of analytic tableaux}}

In [[classical logic]], the compound ''p''→''q'' is logically equivalent to the negative compound: not both ''p'' and not ''q''. Thus the compound ''p''→''q'' is ''false'' [[if and only if]] both ''p'' is true and ''q'' is false. By the same stroke, ''p''→''q'' is ''true'' if and only if either ''p'' is false or ''q'' is true (or both). Thus → is a function from pairs of [[truth value]]s of the components ''p'', ''q'' to truth values of the compound ''p''→''q'', whose truth value is entirely a function of the truth values of the components. Hence, the compound ''p''→''q'' is called ''[[Truth function|truth-functional]]''. The compound ''p''→''q'' is logically equivalent also to ¬''p''∨''q'' (either not ''p'', or ''q'' (or both)), and to ¬''q''&nbsp;→&nbsp;¬''p'' (if not ''q'' then not ''p''). But it is not equivalent to ¬''p''&nbsp;→&nbsp;¬''q'', which is equivalent to ''q''→''p''.

Formulas over the set of connectives <math>\{\to, \bot\}</math><ref>The [[well-formed formula]]s are:

# Each [[propositional variable]] is a formula.

# "<math>\bot</math>" is a formula.

# If <math>A</math> and <math>B</math> are formulas, so is <math>(A \to B)</math>.

# Nothing else is a formula.</ref> are called '''f-implicational'''.{{sfn|Franco|Goldsmith|Schlipf|Speckenmeyer|1999}} In [[classical logic]] the other connectives, such as <math>\neg</math> ([[negation]]), <math>\land</math> ([[logical conjunction|conjunction]]), <math>\lor</math> ([[disjunction]]) and <math>\leftrightarrow</math> ([[If and only if|equivalence]]), can be defined in terms of <math>\to</math> and <math>\bot</math> ([[False (logic)#False, negation and contradiction|falsity]]):<ref name="connective_needed">f-implicational formulas cannot express all valid formulas in [[Minimal logic|minimal]] (MPC) or [[intuitionistic logic|intuitionistic]] (IPC) propositional logic — in particular, <math>\lor</math> (disjunction) cannot be defined within it. In contrast, <math>\{\to, \lor, \bot \}</math> is a complete basis for MPC / IPC: from these, all other connectives (e.g., <math>\land, \neg, \leftrightarrow, \bot</math>) can be defined.</ref>

<math display="block">

\begin{align}

\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 \\

A \leftrightarrow B & \quad \overset{\text{def}}{=} \quad \{(A \to B) \to [(B \to A) \to \bot]\} \to \bot \\

\end{align}

</math>

The validity of f-implicational formulas can be semantically established by the [[method of analytic tableaux]]. The logical rules are

====Truth table====

:{| style="border: none; border-spacing: 1px; border-collapse: separate;"

The truth table associated with the material conditional '''not p or q''' (symbolized as '''p&nbsp;→&nbsp;q''') and the logical implication '''p implies q''' (symbolized as '''p&nbsp;→&nbsp;q''', or '''Cpq''') is as follows:

{|class="centered"

|

{|class="wikitable center hintergrundfarbe2"

! <math>p</math> || <math>q</math> || <math>p \rightarrow q</math>

|-

| style="vertical-align: top;" | <math>\frac{\boldsymbol{\mathsf{T}}(A \to B)}{\boldsymbol{\mathsf{F}}(A)

!T||T

\quad \mid \quad \boldsymbol{\mathsf{T}}(B)}</math> || valign="top" | <math>\frac{\boldsymbol{\mathsf{F}}(A \to B)}{\begin{array}{c} \boldsymbol{\mathsf{T}}(A) \\ \boldsymbol{\mathsf{F}}(B)\end{array}}</math>

|T

|-

|colspan="2" | <math>\boldsymbol{\mathsf{T}}(\bot)</math> : Close the branch (contradiction)<br/><math>\boldsymbol{\mathsf{F}}(\bot)</math> : Do nothing (since it just asserts no contradiction)

!T||F

|F

|-

!F||T

|T

|-

!F||F

|T

|}

|}

<div style="margin-left: 20px;">

===As a formal connective===

{{collapse top

The material conditional can be considered as a symbol of a [[theory (mathematical logic)|formal theory]], taken as a set of sentences, satisfying all the classical inferences involving →, in particular the following characteristic rules:

| title=<span style="display:block; text-align:left; margin-left: -50px; padding-left: 0;">Example: proof of <math>p \to \neg \neg p\quad</math>, by [[method of analytic tableaux]]</span>

| bg=#ffffff | fg=#000000

}}

<pre>

F[p → ((p → ⊥) → ⊥)]

|

T[p]

F[(p → ⊥) → ⊥]

|

T[p → ⊥]

F[⊥]

┌────────┴────────┐

F[p] T[⊥]

| |

CONTRADICTION CONTRADICTION

(T[p], F[p]) (⊥ is true)

</pre>

{{collapse bottom}}

</div>

<div style="margin-left: 20px;">

# [[Modus ponens]];

{{collapse top

# [[Conditional proof]];

| title=<span style="display:block; text-align:left; margin-left: -50px; padding-left: 0;">Example: proof of <math>\neg \neg p \to p\quad</math>, by method of analytic tableaux</span>

# [[contraposition|Classical contraposition]];

| bg=#ffffff | fg=#000000

# [[reductio ad absurdum|Classical reductio]].

}}

<pre>

F[((p → ⊥) → ⊥) → p]

|

T[(p → ⊥) → ⊥]

F[p]

┌────────┴────────┐

F[p → ⊥] T[⊥]

| |

T[p] CONTRADICTION (⊥ is true)

F[⊥]

|

CONTRADICTION (T[p], F[p])

</pre>

[[Hilbert system|Hilbert-style proofs]] can be found [[Implicational propositional calculus#An alternative axiomatization|here]] or [[Peirce's law|here]].

{{collapse bottom}}

</div>

<div style="margin-left: 20px;">

{{collapse top

| title=<span style="display:block; text-align:left; margin-left: -50px; padding-left: 0;">Example: proof of <math>(p \to q) \to ((q \to r) \to (p \to r))</math>, by method of analytic tableaux</span>

| bg=#ffffff | fg=#000000

}}

<pre>

1. F[(p → q) → ((q → r) → (p → r))]

| // from 1

2. T[p → q]

3. F[(q → r) → (p → r)]

| // from 3

4. T[q → r]

5. F[p → r]

| // from 5

6. T[p]

7. F[r]

┌────────┴────────┐ // from 2

8a. F[p] 8b. T[q]

X ┌────────┴────────┐ // from 4

9a. F[q] 9b. T[r]

X X

</pre>

A [[Hilbert system|Hilbert-style proof]] can be found [[Implicational propositional calculus#The Bernays–Tarski axiom system|here]].

{{collapse bottom}}

</div>

== Syntactical properties ==

Unlike the truth-functional one, this approach to logical connectives permits the examination of structurally identical propositional forms in various [[Formal system|logical system]]s, where somewhat different properties may be demonstrated. For example, in [[intuitionistic logic]] which rejects proofs by contraposition as valid rules of inference, {{math|(''p'' → ''q'') ⇒ ¬''p'' ∨ ''q''}} is not a propositional theorem, but [[False (logic)#False, negation and contradiction|the material conditional is used to define negation]].

{{further|Natural deduction}}

The semantic definition by truth tables does not permit the examination of structurally identical propositional forms in various [[Formal system|logical system]]s, where different properties may be demonstrated. The language considered here is restricted to '''f-implicational formulas'''.

Consider the following (candidate) [[natural deduction]] rules.

==Formal properties==

{| class="wikitable"

When studying logic formally, the material conditional is distinguished from the [[Logical consequence#Semantic consequence|semantic consequence]] relation <math>\models</math>. We say <math>A \models B</math> if every interpretation that makes A true also makes B true. However, there is a close relationship between the two in most logics, including [[classical logic]]. For example, the following principles hold:

|valign="top"| '''Implication Introduction''' (<math>\to</math>I)

If assuming <math>A</math> one can derive <math>B</math>, then one can conclude <math>A \to B</math>.

* If <math>\Gamma\models\psi</math> then <math>\varnothing\models(\varphi_1\land\dots\land\varphi_n\rightarrow\psi)</math> for some <math>\varphi_1,\dots,\varphi_n\in\Gamma</math>. (This is a particular form of the [[deduction theorem]]. In words, it says that if Γ models ψ this means that ψ can be deduced just from some subset of the theorems in Γ.)

<math>

* The converse of the above

\frac{\begin{array}{c}

[A] \\

\vdots \\

B

\end{array}}{A \to B}</math> (<math>\to</math>I)

<math>[A]</math> is an assumption that is discharged when applying the rule.

* Both <math>\rightarrow</math> and <math>\models</math> are [[Monotonic function|monotonic]]; i.e., if <math>\Gamma\models\psi</math> then <math>\Delta\cup\Gamma\models\psi</math>, and if <math>\varphi\rightarrow\psi</math> then <math>(\varphi\land\alpha)\rightarrow\psi</math> for any α, Δ. (In terms of structural rules, this is often referred to as [[weakening]] or ''thinning''.)

|valign="top"| '''Implication Elimination''' (<math>\to</math>E)

This rule corresponds to [[modus ponens]].

These principles do not hold in all logics, however. Obviously they do not hold in [[non-monotonic logic]]s, nor do they hold in [[relevance logic]]s.

<math>\frac{A \to B \quad A}{B}</math> (<math>\to</math>E)

Other properties of implication (the following expressions are always true, for any logical values of variables):

* [[distributivity]]: <math>(s \rightarrow (p \rightarrow q)) \rightarrow ((s \rightarrow p) \rightarrow (s \rightarrow q))</math>

<math>\frac{A \quad A \to B}{B}</math> (<math>\to</math>E)

* [[transitive relation|transitivity]]: <math>(a \rightarrow b) \rightarrow ((b \rightarrow c) \rightarrow (a \rightarrow c))</math>

|-

|valign="top"| '''[[Double negation|Double Negation Elimination]]''' (<math>\neg\neg</math>E)

* [[reflexive relation|reflexivity]]: <math>a \rightarrow a</math>

<math>

* [[total relation|totality]]: <math>(a \rightarrow b) \vee (b \rightarrow a)</math>

\frac{\begin{array}{c}

(A \to \bot) \to \bot \\

\end{array}}{A}</math> (<math>\neg\neg</math>E)

|valign="top"| '''Falsum Elimination''' (<math>\bot</math>E)

From falsum (<math>\bot</math>) one can derive any formula.<br/>(ex falso quodlibet)

* truth preserving: The interpretation under which all variables are assigned a truth value of 'true' produces a truth value of 'true' as a result of material implication.

<math>\frac{\bot}{A}</math> (<math>\bot</math>E)

* commutativity of antecedents: <math>(a \rightarrow (b \rightarrow c)) \equiv (b \rightarrow (a \rightarrow c))</math>

|}

* '''[[Minimal logic]]''': By limiting the [[natural deduction]] rules to ''Implication Introduction'' (<math>\to</math>I) and ''Implication Elimination'' (<math>\to</math>E), one obtains (the implicational fragment of)<ref name="connective_needed"/> minimal logic (as defined by [[Ingebrigt Johansson|Johansson]]).{{sfn|Johansson|1937}}

Note that <math>a \rightarrow (b \rightarrow c)</math> is [[Logical equivalence|logically equivalent]] to <math>(a \and b) \rightarrow c</math>; this property is sometimes called [[currying]]. Because of these properties, it is convenient to adopt a [[right-associative]] notation for → where <math>a \rightarrow b \rightarrow c</math> denotes <math>a \rightarrow (b \rightarrow c)</math>.

<div style="margin-left: 20px;">

{{collapse top

| title=<span style="display:block; text-align:left; margin-left: -50px; padding-left: 0;">Proof of <math>P \to \neg \neg P\quad</math>, within minimal logic</span>

| bg=#ffffff | fg=#000000

}}

{|

|1.{{spaces|1}}

|[ P ]

|{{spaces|1}}// Assume

|-

|2.{{spaces|1}}

|[ P → ⊥ ]

|{{spaces|1}}// Assume

|-

|3.{{spaces|1}}

|⊥

|{{spaces|1}}// <math>\to</math>E (1, 2)

|-

|4.{{spaces|1}}

|(P → ⊥) → ⊥)

|{{spaces|1}}// <math>\to</math>I (2, 3), discharging 2

|-

|5.{{spaces|1}}

|P → ((P → ⊥) → ⊥)

|{{spaces|1}}// <math>\to</math>I (1, 4), discharging 1

|}

{{collapse bottom}}

</div>

* '''[[Intuitionistic logic]]''': By adding ''Falsum Elimination'' (<math>\bot</math>E) as a rule, one obtains (the implicational fragment of)<ref name="connective_needed"/> intuitionistic logic.

:The statement <math>P \to \neg \neg P</math> is valid (already in minimal logic), unlike the reverse implication which would entail the [[law of excluded middle]].

* '''[[Classical logic]]''': If ''[[Double negation|Double Negation Elimination]]'' (<math>\neg\neg</math>E) is also permitted,{{refn|name="RAA"|Instead of <math>\neg\neg</math>E one can add '''[[reductio ad absurdum]]''' as a rule to obtain (full) classical logic:{{sfn|Prawitz|1965|p=21}}{{sfn|Ayala-Rincón|de Moura|2017|pp=17-24}}

Note also that comparison of the truth table shows that <math>a \rightarrow b</math> is equivalent to <math>\neg a \or b</math>, and it is sometimes convenient to replace one by the other in proofs. Such a replacement can be viewed as a [[rule of inference]].

:<math>

\frac{\begin{array}{c}

[A \to \bot] \\

\vdots \\

\bot

\end{array}}{A}</math> (RAA)}} the system defines (full!) classical logic.{{sfn|Prawitz|1965|p=21}}{{sfn|Ayala-Rincón|de Moura|2017|pp=17-24}}{{sfn|Tennant|1990|p=48}}

==A selection of theorems (classical logic)==

== Philosophical problems with material conditional ==

In [[classical logic]] material implication validates the following:

<div style="margin-left: 20px;">

{{collapse top

| title=<span style="display:block; text-align:left; margin-left: -50px; padding-left: 0;">Contraposition: <math>(\neg Q \to \neg P) \to (P \to Q)</math></span>

| bg=#ffffff | fg=#000000

}}

{|

|1.{{spaces|1}}

|[ (Q → ⊥) → (P → ⊥) ]

|{{spaces|1}}// Assume (to discharge at 9)

|-

|2.{{spaces|1}}

|[ P ]

|{{spaces|1}}// Assume (to discharge at 8)

|-

|3.{{spaces|1}}

|[ Q → ⊥ ]

|{{spaces|1}}// Assume (to discharge at 6))

|-

|4.{{spaces|1}}

|P → ⊥

|{{spaces|1}}// <math>\to</math>E (1, 3)

|-

|5.{{spaces|1}}

|⊥

|{{spaces|1}}// <math>\to</math>E (2, 4)

|-

|6.{{spaces|1}}

|(Q → ⊥) → ⊥

|{{spaces|1}}// <math>\to</math>I (3, 5) (discharging 3)

|-

|7.{{spaces|1}}

|Q

|{{spaces|1}}// <math>\neg\neg</math>E (6)

|-

|8.{{spaces|1}}

|P → Q

|{{spaces|1}}// <math>\to</math>I (2, 7) (discharging 2)

|-

|9.{{spaces|1}}

|((Q → ⊥) → (P → ⊥)) → (P → Q)

|{{spaces|1}}// <math>\to</math>I (1, 8) (discharging 1)

|}

{{collapse bottom}}

</div>

<div style="margin-left: 20px;">

{{collapse top

| title=<span style="display:block; text-align:left; margin-left: -50px; padding-left: 0;">[[Peirce's law]]: <math>((P \to Q) \to P) \to P</math></span>

| bg=#ffffff | fg=#000000

}}

{|

|1.{{spaces|1}}

|[ (P → Q) → P ]

|{{spaces|1}}// Assume (to discharge at 11)

|-

|2.{{spaces|1}}

|[ P → ⊥ ]

|{{spaces|1}}// Assume (to discharge at 9)

|-

|3.{{spaces|1}}

|[ P ]

|{{spaces|1}}// Assume (to discharge at 6)

|-

|4.{{spaces|1}}

|⊥

|{{spaces|1}}// <math>\to</math>E (2, 3)

|-

|5.{{spaces|1}}

|Q

|{{spaces|1}}// <math>\bot</math>E (4)

|-

|6.{{spaces|1}}

|P → Q

|{{spaces|1}}// <math>\to</math>I (3, 5) (discharging 3)

|-

|7.{{spaces|1}}

|P

|{{spaces|1}}// <math>\to</math>E (1, 6)

|-

|8.{{spaces|1}}

|⊥

|{{spaces|1}}// <math>\to</math>E (2, 7)

|-

|9.{{spaces|1}}

|(P → ⊥) → ⊥

|{{spaces|1}}// <math>\to</math>I (2, 8) (discharging 2)

|-

|10.{{spaces|1}}

|P

|{{spaces|1}}// <math>\neg \neg</math>E (9)

|-

|11.{{spaces|1}}

|((P → Q) → P) → P

|{{spaces|1}}// <math>\to</math>I (1, 10) (discharging 1)

|}

{{collapse bottom}}

</div>

<div style="margin-left: 20px;">

{{collapse top

| title=<span style="display:block; text-align:left; margin-left: -50px; padding-left: 0;">[[Vacuous truth|Vacuous conditional]] (IPC): <math>\neg P \to (P \to Q)</math></span>

| bg=#ffffff | fg=#000000

}}

{|

|1.{{spaces|1}}

|<math>[ P \to \bot ]</math>

|{{spaces|1}}// Assume

|-

|2.{{spaces|1}}

|<math>[ P ]</math>

|{{spaces|1}}// Assume

|-

|3.{{spaces|1}}

| <math>\bot</math>

|{{spaces|1}}// <math>\to</math>E (1, 2)

|-

|4.{{spaces|1}}

|<math>Q</math>

|{{spaces|1}}// <math>\bot</math>E (3)

|-

|5.{{spaces|1}}

|<math>P \to Q</math>

|{{spaces|1}}// <math>\to </math>I (2, 4) (discharging 2)

|-

|6.{{spaces|1}}

|<math>( P \to \bot ) \to ( P \to Q )</math>

|{{spaces|1}}// <math>\to </math>I (1, 5) (discharging 1)

|}

{{collapse bottom}}

</div>

* [[Import-Export (logic)|Import-export]]: <math>P \to (Q \to R) \equiv (P \land Q) \to R</math>

* Negated conditionals: <math>\neg(P \to Q) \equiv P \land \neg Q</math>

* Or-and-if: <math>P \to Q \equiv \neg P \lor Q</math>

* Commutativity of antecedents: <math>\big(P \to (Q \to R)\big) \equiv \big(Q \to (P \to R)\big)</math>

* [[Left distributivity]]: <math>\big(R \to (P \to Q)\big) \equiv \big((R \to P) \to (R \to Q)\big)</math>

Similarly, on classical interpretations of the other connectives, material implication validates the following [[Logical consequence#Semantic consequence|entailment]]s:

* Antecedent strengthening: <math>P \to Q \models (P \land R) \to Q</math>

* [[transitive relation|Transitivity]]: <math>(P \to Q) \land (Q \to R) \models P \to R</math>

* [[Simplification of disjunctive antecedents]]: <math>(P \lor Q) \to R \models (P \to R) \land (Q \to R)</math>

[[Tautology (logic)|Tautologies]] involving material implication include:

* [[reflexive relation|Reflexivity]]: <math>\models P \to P</math>

* [[connex relation|Totality]]: <math>\models (P \to Q) \lor (Q \to P)</math>

* [[Law of excluded middle|Conditional excluded middle]]: <math>\models (P \to Q) \lor (P \to \neg Q)</math>

== Discrepancies with natural language ==

Outside of mathematics, it is a matter of some controversy as to whether the [[truth function]] for [[material implication (rule of inference)|material implication]] provides an adequate treatment of conditional statements in English (a [[sentence (mathematical logic)|sentence]] in the [[indicative mood]] with a [[conditional clause]] attached, i.e., an [[indicative conditional]], or false-to-fact sentences in the [[subjunctive mood]], i.e., a [[counterfactual conditional]]).<ref name="sep-conditionals"/> That is to say, critics argue that in some non-mathematical cases, the truth value of a compound statement, "if ''p'' then ''q''", is not adequately determined by the truth values of ''p'' and ''q''.<ref name="sep-conditionals"/> Examples of non-truth-functional statements include: "''p'' because ''q''", "''p'' before ''q''" and "it is possible that ''p''".<ref name="sep-conditionals"/> “[Of] the sixteen possible truth-functions of ''A'' and ''B'', material implication is the only serious candidate. First, it is uncontroversial that when ''A'' is true and ''B'' is false, "If ''A'', ''B''" is false. A basic rule of inference is [[modus ponens]]: from "If ''A'', ''B''" and ''A'', we can infer ''B''. If it were possible to have ''A'' true, ''B'' false and "If ''A'', ''B''" true, this inference would be invalid. Second, it is uncontroversial that "If ''A'', ''B''" is sometimes true when ''A'' and ''B'' are respectively (true, true), or (false, true), or (false, false)… Non-truth-functional accounts agree that "If ''A'', ''B''" is false when ''A'' is true and ''B'' is false; and they agree that the conditional is sometimes true for the other three combinations of truth-values for the components; but they deny that the conditional is always true in each of these three cases. Some agree with the truth-functionalist that when ''A'' and ''B'' are both true, "If ''A'', ''B''" must be true. Some do not, demanding a further relation between the facts that ''A'' and that ''B''.”<ref name="sep-conditionals">{{cite web |first=Dorothy |last=Edgington |editor=Edward N. Zalta |year=2008 |title=Conditionals |work=The Stanford Encyclopedia of Philosophy |edition=Winter 2008 |url=http://plato.stanford.edu/archives/win2008/entries/conditionals/}}</ref>

Material implication does not closely match the usage of [[conditional sentence]]s in [[natural language]]. For example, even though material conditionals with false antecedents are [[vacuous truth|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]].{{sfn|Edgington|2008}} In addition to the paradoxes, a variety of other arguments have been given against a material implication analysis. For instance, [[counterfactual conditional]]s would all be vacuously true on such an account, when in fact some are false.{{refn|For example, "If [[Janis Joplin]] were alive today, she would drive a [[Mercedes-Benz]]", see {{harvtxt|Starr|2019}}}}

{{quotation|The truth-functional theory of the conditional was integral to [[Gottlob Frege|Frege]]'s new logic (1879). It was taken up enthusiastically by [[Bertrand Russell|Russell]] (who called it "material implication"), [[Ludwig Wittgenstein|Wittgenstein]] in the ''[[Tractatus Logico-Philosophicus|Tractatus]]'', and the [[logical positivist]]s, and it is now found in every logic text. It is the first theory of conditionals which students encounter. Typically, it does not strike students as ''obviously'' correct. It is logic's first surprise. Yet, as the textbooks testify, it does a creditable job in many circumstances. And it has many defenders. It is a strikingly simple theory: "If ''A'', ''B''" is false when ''A'' is true and ''B'' is false. In all other cases, "If ''A'', ''B''" is true. It is thus equivalent to "~(''A''&~''B'')" and to "~''A'' or ''B''". "''A'' ⊃ ''B''" has, by stipulation, these truth conditions.|[[Dorothy Edgington]]|The Stanford Encyclopedia of Philosophy|"Conditionals"<ref name="sep-conditionals"/>}}

In the mid-20th century, a number of researchers including [[Paul Grice|H. P. Grice]] and [[Frank Cameron Jackson|Frank Jackson]] proposed that [[pragmatics|pragmatic]] principles could explain the discrepancies between natural language conditionals and the material conditional. On their accounts, conditionals [[denotation|denote]] material implication but end up conveying additional information when they interact with conversational norms such as [[Cooperative principle#Grice's maxims|Grice's maxims]].{{sfn|Edgington|2008}}{{sfn|Gillies|2017}} Recent work in [[formal semantics (natural language)|formal semantics]] and [[philosophy of language]] has generally eschewed material implication as an analysis for natural-language conditionals.{{sfn|Gillies|2017}} In particular, such work has often rejected the assumption that natural-language conditionals are [[truth function]]al in the sense that the truth value of "If ''P'', then ''Q''" is determined solely by the truth values of ''P'' and ''Q''.{{sfn|Edgington|2008}} Thus semantic analyses of conditionals typically propose alternative interpretations built on foundations such as [[modal logic]], [[relevance logic]], [[probability theory]], and [[causal graph|causal models]].{{sfn|Gillies|2017}}{{sfn|Edgington|2008}}{{sfn|Von Fintel|2011}}

The meaning of the material conditional can sometimes be used in the [[natural language]] English "if ''condition'' then ''consequence''" construction (a kind of [[conditional sentence]]), where ''condition'' and ''consequence'' are to be filled with English sentences. However, this construction also implies a "reasonable" connection between the condition ([[Protasis (linguistics)|protasis]]) and consequence ([[Consequent|apodosis]]) (see [[Connexive logic]]).{{citation needed|date=February 2012}}

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.{{sfn|Oaksford |Chater|1994}}{{sfn|Stenning|van Lambalgen|2004}}{{sfn|Von Sydow|2006}}

The material conditional can yield some unexpected truths when expressed in natural language. For example, any material conditional statement with a false antecedent is true (see [[vacuous truth]]). So the statement "if 2 is odd then 2 is even" is true. Similarly, any material conditional with a true consequent is true. So the statement "if I have a penny in my pocket then Paris is in France" is always true, regardless of whether or not there is a penny in my pocket. These problems are known as the [[paradoxes of material implication]], though they are not really paradoxes in the strict sense; that is, they do not elicit logical contradictions. These unexpected truths arise because speakers of English (and other natural languages) are tempted to [[equivocation|equivocate]] between the material conditional and the [[indicative conditional]], or other conditional statements, like the [[counterfactual conditional]] and the [[logical biconditional |material biconditional]]. It is not surprising that a rigorously defined truth-functional operator does not correspond exactly to all notions of implication or otherwise expressed by 'if...then...' sentences in English (or their equivalents in other natural languages). For an overview of some the various analyses, formal and informal, of conditionals, see the "References" section below.

==See also==

{{Div col-begin|colwidth=20em}}

{{col-break}}

* [[Boolean algebra (logic)|Boolean algebra]]

* [[Boolean ___domain]]

* [[Boolean function]]

* [[Boolean logic]]

{{col-break}}

* [[Conditional quantifier]]

* [[Implicational propositional calculus]]

* ''[[Laws of Form]]''

* [[Logic gate]]

* [[Logical graph]]

* [[Logical equivalence]]

{{col-break}}

* [[Material implication (rule of inference)]]

* [[Paradoxes of material implication]]

* [[Peirce's law]]

* [[Propositional logiccalculus]]

* [[Sole sufficient operator]]

{{Div col- end}}

===Conditionals===

* [[Corresponding conditional]]

* [[Counterfactual conditional]]

* [[Indicative conditional]]

* [[Corresponding conditional]]

* [[Strict conditional]]

==References Notes ==

{{Reflist}}

== Bibliography ==

* {{Cite book |last1=Ayala-Rincón |first1=Mauricio |last2=de Moura |first2=Flávio L. C. |title=Applied Logic for Computer Scientists |date=2017 |publisher=Springer |series=Undergraduate Topics in Computer Science |isbn=978-3-319-51651-6 |doi=10.1007/978-3-319-51653-0 |url=https://link.springer.com/book/10.1007/978-3-319-51653-0 }}

*{{cite book |last=Bourbaki |first=N. |title=Théorie des ensembles |date=1954 |publisher=Hermann & Cie, Éditeurs |___location=Paris |page=14}}

*{{cite encyclopedia |first=Dorothy |last=Edgington |editor=Edward N. Zalta |year=2008 |title=Conditionals |encyclopedia=The Stanford Encyclopedia of Philosophy |edition=Winter 2008 |url=http://plato.stanford.edu/archives/win2008/entries/conditionals/}}

*{{cite encyclopedia|last=Von Fintel|first=Kai |editor-last1=von Heusinger |editor-first1= Klaus | editor-last2= Maienborn |editor-first2= Claudia | editor-first3=Paul |editor-last3=Portner |encyclopedia=Semantics: An international handbook of meaning |title=Conditionals |url=http://mit.edu/fintel/fintel-2011-hsk-conditionals.pdf |year=2011 |pages=1515–1538 |publisher= de Gruyter Mouton |doi=10.1515/9783110255072.1515|hdl=1721.1/95781 |isbn=978-3-11-018523-2 |hdl-access=free }}

*{{cite journal | doi=10.1016/S0166-218X(99)00038-4 | volume=96-97 | title=An algorithm for the class of pure implicational formulas | journal=Discrete Applied Mathematics | pages=89–106 | year=1999 | last1=Franco | first1=John | last2=Goldsmith | first2=Judy | last3=Schlipf | first3=John | last4=Speckenmeyer | first4=Ewald | last5=Swaminathan | first5=R.P. | doi-access=free}}

*{{cite encyclopedia |last=Gillies|first=Thony |editor-last1=Hale |editor-first1=B. | editor-last2=Wright |editor-first2=C. | editor-last3=Miller |editor-first3=A. |encyclopedia=A Companion to the Philosophy of Language |title=Conditionals |url=http://www.thonygillies.org/wp-content/uploads/2015/11/gillies-conditionals-handbook.pdf |year=2017 |pages=401–436 |publisher=Wiley Blackwell |doi=10.1002/9781118972090.ch17|isbn=9781118972090 }}

*{{Cite book |editor-first=Jean |editor-last=Van Heijenoort |title=From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931 |year=1967 |publisher=Harvard University Press |isbn=0-674-32449-8 |pages=84–87}}

*{{cite book |last=Hilbert |first=D. |title=Prinzipien der Mathematik (Lecture Notes edited by Bernays, P.) |date=1918}}

*{{cite journal|last= Johansson|first=Ingebrigt|author-link=Ingebrigt Johansson|year=1937|title=Der Minimalkalkül, ein reduzierter intuitionistischer Formalismus|url=http://www.numdam.org/item/CM_1937__4__119_0|journal=[[Compositio Mathematica]]|volume=4|pages=119–136|language=de}}

*{{Cite book | last =Mendelson | first =Elliott | author-link =Elliott Mendelson |title=Introduction to Mathematical Logic | year=2015 | edition=6th | ___location=Boca Raton | publisher=CRC Press/Taylor & Francis Group (A Chapman & Hall Book) | isbn=978-1-4822-3778-8 | page=2 }}

*{{Cite web |url=https://github.com/mdnahas/Peano_Book/blob/46e27bdb5aed51c078ad99e5a78d134fd2a0c3ca/Peano.pdf |title=English Translation of 'Arithmetices Principia, Nova Methodo Exposita' |access-date=2022-08-10 |first=Michael |last=Nahas |date=25 Apr 2022 |publisher=GitHub}}

*{{cite journal |last1=Oaksford |first1=M. |last2=Chater |first2=N. |year=1994 |title=A rational analysis of the selection task as optimal data selection |journal=[[Psychological Review]] |volume=101 |issue=4 |pages=608–631 |doi=10.1037/0033-295X.101.4.608 |citeseerx=10.1.1.174.4085 |s2cid=2912209 }}

*{{cite book | last = Prawitz | first = Dag | author-link = Dag Prawitz | year = 1965 | title = Natural Deduction: A Proof-Theoretic Study | series = Acta Universitatis Stockholmiensis; Stockholm Studies in Philosophy, 3 | publisher = Almqvist & Wiksell | ___location = Stockholm, Göteborg, Uppsala | oclc = 912927896 }}

*{{cite encyclopedia |last=Starr |first=Will |editor-last1=Zalta |editor-first1=Edward N. |encyclopedia=The Stanford Encyclopedia of Philosophy |title=Counterfactuals |year=2019 |url=https://plato.stanford.edu/archives/fall2019/entries/counterfactuals}}

*{{cite journal |last1=Stenning |first1=K. |last2=van Lambalgen |first2=M. |year=2004 |title=A little logic goes a long way: basing experiment on semantic theory in the cognitive science of conditional reasoning |journal=Cognitive Science |volume=28 |issue=4 |pages=481–530 |doi=10.1016/j.cogsci.2004.02.002 |citeseerx=10.1.1.13.1854 }}

*{{cite thesis |last=Von Sydow |first=M. |title=Towards a Flexible Bayesian and Deontic Logic of Testing Descriptive and Prescriptive Rules |year=2006 |___location=Göttingen |publisher=Göttingen University Press |doi=10.53846/goediss-161 |s2cid=246924881 |url=https://ediss.uni-goettingen.de/handle/11858/00-1735-0000-0006-AC29-9|type=doctoralThesis |doi-access=free }}

*{{cite book | last = Tennant | first = Neil | title = Natural Logic | publisher = [[Edinburgh University Press]] | year = 1990 | orig-year = 1978 | edition = 1st, repr. with corrections | isbn = 0852245793 }}

== Further reading ==

* Brown, Frank Markham (2003), ''Boolean Reasoning: The Logic of Boolean Equations'', 1st edition, [[Kluwer]] Academic Publishers, [[Norwell, Massachusetts|Norwell]], MA. 2nd edition, [[Dover Publications]], [[Mineola, New York|Mineola]], NY, 2003.

* [[Dorothy Edgington|Edgington, Dorothy]] (2001), "Conditionals", in Lou Goble (ed.), ''The Blackwell Guide to Philosophical Logic'', [[Wiley-Blackwell|Blackwell]].

* [[W. V. Quine|Quine, W.V.]] (1982), ''Methods of Logic'', (1st ed. 1950), (2nd ed. 1959), (3rd ed. 1972), 4th edition, [[Harvard University Press]], [[Cambridge, Massachusetts|Cambridge]], MA.

* [[Robert Stalnaker|Stalnaker, Robert]], "Indicative Conditionals", ''[[Philosophia (journal)|Philosophia]]'', '''5''' (1975): 269–286.

== External links ==

* {{cite SEP |url-id=conditionals |title=Conditionals |last=Edgington, |first=Dorothy}}

{{Logical connectives}}

{{Common logical symbols}}

{{Mathematical logic}}

[[Category:Logical connectives]]

[[Category:Conditionals]]

[[Category:Logical consequence]]

[[Category:Semantics]]

[[am:ጥገኛ አምክንዮ]]

[[ar:قضية شرطية]]

[[cs:Implikace]]

[[eo:Implico]]

[[fa:شرطی منطقی]]

[[it:Implicazione logica]]

[[he:אם-אז]]

[[mk:Материјална импликација]]

[[nl:Logische implicatie]]

[[ja:論理包含]]

[[no:Subjunksjon (logikk)]]

[[pms:Amplicassion]]

[[kaa:İmplikatsiya]]

[[sk:Implikácia]]

[[th:เงื่อนไขเชิงตรรกศาสตร์]]

[[zh:实质条件]]