Editing Material conditional

{{short description|Logical connective}}
{{Redirect|Logical conditional|other related meanings|Conditional statement (disambiguation){{!}}Conditional statement}}
{{distinguish|Material inference|Material implication (rule of inference)}}
{{Infobox logical connective
| 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]].

== Notation ==
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>.

== History ==
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}}{{sfn|Allegranza|2015}} [[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 \to B</math> in 1954.{{sfn|Bourbaki|1954|page=14}}

==Semantics==
===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".
Examples are ...
* ... 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."''

===Analytic tableaux===
{{further|Method of analytic tableaux}}
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
:{| style="border: none; border-spacing: 1px; border-collapse: separate;"
|-
| style="vertical-align: top;" | <math>\frac{\boldsymbol{\mathsf{T}}(A \to B)}{\boldsymbol{\mathsf{F}}(A) 
\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>
|-
|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)
|}

<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 \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;">
{{collapse top
| 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>
| bg=#ffffff | fg=#000000
}}
<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 ==
{{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.

{| class="wikitable"
|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>.

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

This rule corresponds to [[modus ponens]].

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


<math>\frac{A \quad A \to B}{B}</math> (<math>\to</math>E)
|-
|valign="top"| '''[[Double negation|Double Negation Elimination]]''' (<math>\neg\neg</math>E)


<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)

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

* '''[[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}}
<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}}
:<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)==
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 ==

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}}}}

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}}

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}}

==See also==
{{Div col|colwidth=20em}}
* [[Boolean domain]]
* [[Boolean function]]
* [[Boolean logic]]
* [[Conditional quantifier]]
* [[Implicational propositional calculus]]
* ''[[Laws of Form]]''
* [[Logical graph]]
* [[Logical equivalence]]
* [[Material implication (rule of inference)]]
* [[Peirce's law]]
* [[Propositional calculus]]
* [[Sole sufficient operator]]
{{Div col end}}

===Conditionals===
* [[Corresponding conditional]]
* [[Counterfactual conditional]]
* [[Indicative conditional]]
* [[Strict conditional]]

== Notes ==
{{Reflist}}

== Bibliography ==
*{{Cite web |url=https://math.stackexchange.com/a/1146502/186330 |title=elementary set theory – Is there any connection between the symbol &sup; when it means implication and its meaning as superset? |access-date=2022-08-10 |first=Mauro |last=Allegranza |date=2015-02-13 |website=Mathematics Stack Exchange |publisher=Stack Exchange Inc |language=en |at=Answer}}

* {{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]]