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Material conditional
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<h2 id="notation">Notation</h2> <p>In logic and related fields, the material conditional is customarily notated with an infix operator → {\displaystyle \to } .<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a> The material conditional is also notated using the infixes ⊃ {\displaystyle \supset } and ⇒ {\displaystyle \Rightarrow } .<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> In the prefixed <a href="/facts/Polish_notation/dUzpf2Ys">Polish notation</a>, 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 <i><a href="/facts/Antecedent_(logic)/8VVVqa06">antecedent</a></i> and q {\displaystyle q} is termed the <i><a href="/facts/Consequent/UIzcuaKF">consequent</a></i> 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)} . </p> <h3>History</h3> <p>In <i><a href="/facts/Arithmetices_principia,_nova_methodo_exposita/IjaKUSMM">Arithmetices Principia: Nova Methodo Exposita</a></i> (1889), <a href="/facts/Giuseppe_Peano/VT2Wsmu7">Peano</a> 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.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> He also expressed the proposition A ⊃ B {\displaystyle A\supset B} as A {\displaystyle A} Ɔ B {\displaystyle B} .<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a><a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a><a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> <a href="/facts/David_Hilbert/1vhlHn4b">Hilbert</a> expressed the proposition "If <i>A</i>, then <i>B</i>" as A → B {\displaystyle A\to B} in 1918.<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> <a href="/facts/Bertrand_Russell/SRpwNX2U">Russell</a> followed Peano in his <i><a href="/facts/Principia_Mathematica/7ufeceV4">Principia Mathematica</a></i> (1910–1913), in which he expressed the proposition "If <i>A</i>, then <i>B</i>" as A ⊃ B {\displaystyle A\supset B} . Following Russell, <a href="/facts/Gerhard_Gentzen/9ujN0ssI">Gentzen</a> expressed the proposition "If <i>A</i>, then <i>B</i>" as A ⊃ B {\displaystyle A\supset B} . <a href="/facts/Arend_Heyting/9sIh63gt">Heyting</a> expressed the proposition "If <i>A</i>, then <i>B</i>" 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. <a href="/facts/Nicolas_Bourbaki/qRJhs9nr">Bourbaki</a> expressed the proposition "If <i>A</i>, then <i>B</i>" as A ⇒ B {\displaystyle A\Rightarrow B} in 1954.<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> </p> <h2 id="definitions">Definitions</h2> <h3>Semantics</h3> <p>From a <a href="/facts/Classical_logic/ls62MAms">classical</a> <a href="/facts/Semantics_of_logic/VdAKHszK">semantic perspective</a>, material implication is the <a href="/facts/Binary_operator/CYtow0bz">binary</a> <a href="/facts/Truth_function/aH5kBhQ7">truth functional</a> operator which returns "true" unless its first argument is true and its second argument is false. This semantics can be shown graphically in a <a href="/facts/Truth_table/4KsmArDW">truth table</a> 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} . </p> <h3>Truth table</h3> <p>The <a href="/facts/Truth_table/4KsmArDW">truth table</a> of A → B {\displaystyle A\rightarrow B} : </p> <table><tbody><tr><th> A {\displaystyle A} </th><th> B {\displaystyle B} </th><th> A → B {\displaystyle A\rightarrow B} </th></tr><tr><td>F</td><td>F</td><td>T</td></tr><tr><td>F</td><td>T</td><td>T</td></tr><tr><td>T</td><td>F</td><td>F</td></tr><tr><td>T</td><td>T</td><td>T</td></tr></tbody></table> <p>The conditionals ( A → B ) {\displaystyle (A\to B)} where the antecedent A {\displaystyle A} is false, are called "<a href="/facts/Vacuous_truth/QBsue3pP">vacuous truths</a>". Examples are ... </p> <ul><li>... with B {\displaystyle B} false: <i>"If <a href="/facts/Marie_Curie/EGjeuJ14">Marie Curie</a> is a sister of <a href="/facts/Galileo_Galilei/98VT2z3v">Galileo Galilei</a>, then Galileo Galilei is a brother of Marie Curie."</i></li> <li>... with B {\displaystyle B} true: <i>"If Marie Curie is a sister of Galileo Galilei, then Marie Curie has a sibling."</i></li></ul> <h3>Syntactical properties</h3> <p>The semantic definition by truth tables does not permit the examination of structurally identical propositional forms in various <a href="/facts/Formal_system/KaeDAjK7">logical systems</a>, where different properties may be demonstrated. The system considered here is <a href="/facts/Implicational_propositional_calculus/Mbjk0Imj">implicational propositional calculus</a> to which a nullary connective ⊥ {\displaystyle \bot } for <a href="/facts/False_(logic)/H6v5aztR">falsity</a> is added. </p> <ul><li><a href="/facts/Minimal_logic/bIzUKg1A">Minimal logic</a>: By limiting the <a href="/facts/Natural_deduction/PPtgwoaJ">natural deduction</a> rules of this logic to <i>Implication Introduction</i> ( → {\displaystyle \to } I) and <i>Implication Elimination</i> ( → {\displaystyle \to } E), one obtains minimal logic (as discussed by Johansson).<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a> See below.</li> <li><a href="/facts/Intuitionistic_logic/zI7ulCnM">Intuitionistic logic</a>: By adding the rule <i>Falsum Elimination</i> ( ⊥ {\displaystyle \bot } E), one obtains intuitionistic logic. See below.</li> <li><a href="/facts/Classical_logic/ls62MAms">Classical logic</a>: If <i>Reductio ad Absurdum</i> (RAA) is also permitted, the result is classical logic. See below.</li></ul> <p>The <a href="/facts/Well-formed_formula/YpTVlzuj">well-formed formulas</a> are: </p> <ol><li>Each <a href="/facts/Propositional_variable/NtAz1rfI">sentence-letter</a> is a formula.</li> <li>" ⊥ {\displaystyle \bot } " is a formula.</li> <li>If A {\displaystyle A} and B {\displaystyle B} are formulas, so is ( A → B ) {\displaystyle (A\to B)} .</li> <li>Nothing else is a formula.</li></ol> <p>Formulas over the set of connectives { → , ⊥ } {\displaystyle \{\to ,\bot \}} are called f-implicational.<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a> All other connectives, such as ¬ {\displaystyle \neg } (<a href="/facts/Negation/q5AQBvBj">negation</a>), ∧ {\displaystyle \land } (<a href="/facts/Logical_conjunction/62uyyn1d">conjunction</a>) and ∨ {\displaystyle \lor } (<a href="/facts/Disjunction/jbI2EGlM">disjunction</a>), 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}}} </p><p>Consider the following (candidate) <a href="/facts/Natural_deduction/PPtgwoaJ">natural deduction</a> rules. </p> <table><tbody><tr><td>Implication Introduction ( → {\displaystyle \to } I)<p>If assuming A {\displaystyle A} one can derive B {\displaystyle B} , then one can conclude A → B {\displaystyle A\to B} .</p><p> [ A ] ⋮ B A → B {\displaystyle {\frac {\begin{array}{c}[A]\\\vdots \\B\end{array}}{A\to B}}} ( → {\displaystyle \to } I)</p><p> [ A ] {\displaystyle [A]} is an assumption that is discharged when applying the rule.</p></td><td>Implication Elimination ( → {\displaystyle \to } E)<p>This rule corresponds to <a href="/facts/Modus_ponens/Wq4NcmrG">modus ponens</a>.</p><p> A → B A B {\displaystyle {\frac {A\to B\quad A}{B}}} ( → {\displaystyle \to } E)</p><p> A A → B B {\displaystyle {\frac {A\quad A\to B}{B}}} ( → {\displaystyle \to } E)</p></td></tr><tr><td><a href="/facts/Reductio_ad_absurdum/5AJUtVmL">Reductio ad absurdum</a> (RAA)<p> [ A → ⊥ ] ⋮ ⊥ A {\displaystyle {\frac {\begin{array}{c}[A\to \bot ]\\\vdots \\\bot \end{array}}{A}}} (RAA)</p><p> [ A → ⊥ ] {\displaystyle [A\to \bot ]} is an assumption that is discharged when applying the rule.</p></td><td>Falsum Elimination ( ⊥ {\displaystyle \bot } E)<p>From falsum ( ⊥ {\displaystyle \bot } ) one can derive any formula.(ex falso quodlibet)</p><p> ⊥ A {\displaystyle {\frac {\bot }{A}}} ( ⊥ {\displaystyle \bot } E)</p></td></tr></tbody></table> <ul><li>When only → {\displaystyle \to } I and → {\displaystyle \to } E are admitted as deduction rules, the system corresponds to <a href="/facts/Minimal_logic/bIzUKg1A">minimal logic</a> as defined by <a href="/facts/Ingebrigt_Johansson/wfFed9bg">Johansson</a>.<a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a></li> <li>When ⊥ {\displaystyle \bot } E is added to the rules, the system defines <a href="/facts/Intuitionistic_logic/zI7ulCnM">intuitionistic logic</a>.</li></ul> ⊥ {\displaystyle \bot } E allows to prove A → ¬ ¬ A {\displaystyle A\to \neg \neg A} , but not the reverse implication which would entail the <a href="/facts/Law_of_excluded_middle/tDhbugCN">law of excluded middle</a>. <table><tbody><tr><th>Proof of p → ¬ ¬ p {\displaystyle p\to \neg \neg p\quad } , using ⊥ {\displaystyle \bot } E</th></tr><tr><td><table><tbody><tr><td>1. </td><td>[ p ]</td><td> // Assume</td></tr><tr><td>2. </td><td>[ p → ⊥ ]</td><td> // Assume</td></tr><tr><td>3. </td><td>⊥</td><td> // → {\displaystyle \to } E (1, 2)</td></tr><tr><td>4. </td><td>(p → ⊥) → ⊥)</td><td> // → {\displaystyle \to } I (2, 3), discharging 2</td></tr><tr><td>5. </td><td>p → ((p → ⊥) → ⊥)</td><td> // → {\displaystyle \to } I (1, 4), discharging 1</td></tr></tbody></table></td></tr></tbody></table> <ul><li>When all four natural deduction rules are admitted, the system defines <a href="/facts/Classical_logic/ls62MAms">classical logic</a>.</li></ul> The extra RAA allows to prove ¬ ¬ A → A {\displaystyle \neg \neg A\to A} . <table><tbody><tr><th>Proof of ¬ ¬ p → p {\displaystyle \neg \neg p\to p\quad } , using RAA</th></tr><tr><td><table><tbody><tr><td>1. </td><td>[ (p → ⊥) → ⊥ ]</td><td> // Assume</td></tr><tr><td>2. </td><td>[ p → ⊥ ]</td><td> // Assume</td></tr><tr><td>3. </td><td>⊥</td><td> // → {\displaystyle \to } E (1, 2)</td></tr><tr><td>4. </td><td>p</td><td> // RAA (2, 3), discharging 2</td></tr><tr><td>5. </td><td>((p → ⊥) → ⊥) → p</td><td> // → {\displaystyle \to } I (1, 4), discharging 1</td></tr></tbody></table></td></tr></tbody></table> <h2 id="a-selection-of-theorems-classical-logic">A selection of theorems (classical logic)</h2> <p>In <a href="/facts/Classical_logic/ls62MAms">classical logic</a> material implication validates the following equivalences: </p> <ul><li>Contraposition: P → Q ≡ ¬ Q → ¬ P {\displaystyle P\to Q\equiv \neg Q\to \neg P} </li> <li><a href="/facts/Import-Export_(logic)/vINr9nX1">Import-export</a>: P → ( Q → R ) ≡ ( P ∧ Q ) → R {\displaystyle P\to (Q\to R)\equiv (P\land Q)\to R} </li> <li>Negated conditionals: ¬ ( P → Q ) ≡ P ∧ ¬ Q {\displaystyle \neg (P\to Q)\equiv P\land \neg Q} </li> <li>Or-and-if: P → Q ≡ ¬ P ∨ Q {\displaystyle P\to Q\equiv \neg P\lor Q} </li> <li>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 )}} </li> <li><a href="/facts/Left_distributivity/0LNBrVJl">Left distributivity</a>: ( 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 )}} </li></ul> <p>Similarly, on classical interpretations of the other connectives, material implication validates the following <a href="/facts/Logical_consequence/AyDXp4dn">entailments</a>: </p> <ul><li>Antecedent strengthening: P → Q ⊨ ( P ∧ R ) → Q {\displaystyle P\to Q\models (P\land R)\to Q} </li> <li><a href="/facts/Vacuous_truth/QBsue3pP">Vacuous conditional</a>: ¬ P ⊨ P → Q {\displaystyle \neg P\models P\to Q} </li> <li><a href="/facts/Transitive_relation/9r90y50r">Transitivity</a>: ( P → Q ) ∧ ( Q → R ) ⊨ P → R {\displaystyle (P\to Q)\land (Q\to R)\models P\to R} </li> <li><a href="/facts/Simplification_of_disjunctive_antecedents/1xgUDRq8">Simplification of disjunctive antecedents</a>: ( P ∨ Q ) → R ⊨ ( P → R ) ∧ ( Q → R ) {\displaystyle (P\lor Q)\to R\models (P\to R)\land (Q\to R)} </li></ul> <p><a href="/facts/Tautology_(logic)/p82wWMaN">Tautologies</a> involving material implication include: </p> <ul><li><a href="/facts/Reflexive_relation/NzTytEg9">Reflexivity</a>: ⊨ P → P {\displaystyle \models P\to P} </li> <li><a href="/facts/Connex_relation/UpwopAts">Totality</a>: ⊨ ( P → Q ) ∨ ( Q → P ) {\displaystyle \models (P\to Q)\lor (Q\to P)} </li> <li><a href="/facts/Law_of_excluded_middle/tDhbugCN">Conditional excluded middle</a>: ⊨ ( P → Q ) ∨ ( P → ¬ Q ) {\displaystyle \models (P\to Q)\lor (P\to \neg Q)} </li></ul> <h2 id="discrepancies-with-natural-language">Discrepancies with natural language</h2> <p>Material implication does not closely match the usage of <a href="/facts/Conditional_sentence/qJvIK5y4">conditional sentences</a> in <a href="/facts/Natural_language/B4gLS7Kd">natural language</a>. For example, even though material conditionals with false antecedents are <a href="/facts/Vacuous_truth/QBsue3pP">vacuously true</a>, 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 <a href="/facts/Paradoxes_of_material_implication/kIBSEESa">paradoxes of material implication</a>.<a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a> In addition to the paradoxes, a variety of other arguments have been given against a material implication analysis. For instance, <a href="/facts/Counterfactual_conditional/GdfYSv1k">counterfactual conditionals</a> would all be vacuously true on such an account, when in fact some are false.<a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a> </p><p>In the mid-20th century, a number of researchers including <a href="/facts/Paul_Grice/IYrQGQNd">H. P. Grice</a> and <a href="/facts/Frank_Cameron_Jackson/EqUhKrEt">Frank Jackson</a> proposed that <a href="/facts/Pragmatics/KkK08yyI">pragmatic</a> principles could explain the discrepancies between natural language conditionals and the material conditional. On their accounts, conditionals <a href="/facts/Denotation/oABEtLtA">denote</a> material implication but end up conveying additional information when they interact with conversational norms such as <a href="/facts/Cooperative_principle/qSHtCJgq">Grice's maxims</a>.<a class="footnote-ref" id="fnref:14" href="#fn:14"><sup>14</sup></a><a class="footnote-ref" id="fnref:15" href="#fn:15"><sup>15</sup></a></ref> Recent work in <a href="/facts/Formal_semantics_(natural_language)/FlWb7Hxp">formal semantics</a> and <a href="/facts/Philosophy_of_language/qwNCwF4W">philosophy of language</a> has generally eschewed material implication as an analysis for natural-language conditionals.<a class="footnote-ref" id="fnref:16" href="#fn:16"><sup>16</sup></a> In particular, such work has often rejected the assumption that natural-language conditionals are <a href="/facts/Truth_function/aH5kBhQ7">truth functional</a> in the sense that the truth value of "If <i>P</i>, then <i>Q</i>" is determined solely by the truth values of <i>P</i> and <i>Q</i>.<a class="footnote-ref" id="fnref:17" href="#fn:17"><sup>17</sup></a> Thus semantic analyses of conditionals typically propose alternative interpretations built on foundations such as <a href="/facts/Modal_logic/I7uXvYYT">modal logic</a>, <a href="/facts/Relevance_logic/rVQufU1b">relevance logic</a>, <a href="/facts/Probability_theory/mFBn51rE">probability theory</a>, and <a href="/facts/Causal_graph/CSFCLNSA">causal models</a>.<a class="footnote-ref" id="fnref:18" href="#fn:18"><sup>18</sup></a><a class="footnote-ref" id="fnref:19" href="#fn:19"><sup>19</sup></a><a class="footnote-ref" id="fnref:20" href="#fn:20"><sup>20</sup></a> </p><p>Similar discrepancies have been observed by psychologists studying conditional reasoning, for instance, by the notorious <a href="/facts/Wason_selection_task/HUK1jqx9">Wason selection task</a> 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.<a class="footnote-ref" id="fnref:21" href="#fn:21"><sup>21</sup></a><a class="footnote-ref" id="fnref:22" href="#fn:22"><sup>22</sup></a><a class="footnote-ref" id="fnref:23" href="#fn:23"><sup>23</sup></a> </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Boolean_domain/9cYq9TC1">Boolean domain</a></li> <li><a href="/facts/Boolean_function/uT7E8pqC">Boolean function</a></li> <li><a href="/facts/Boolean_logic/VkCToAmg">Boolean logic</a></li> <li><a href="/facts/Conditional_quantifier/phaAC17A">Conditional quantifier</a></li> <li><a href="/facts/Implicational_propositional_calculus/Mbjk0Imj">Implicational propositional calculus</a></li> <li><i><a href="/facts/Laws_of_Form/rLNKFWqD">Laws of Form</a></i></li> <li><a href="/facts/Logical_graph/IMH9yIvX">Logical graph</a></li> <li><a href="/facts/Logical_equivalence/ysCWmsxA">Logical equivalence</a></li> <li><a href="/facts/Material_implication_(rule_of_inference)/1WHxa8Se">Material implication (rule of inference)</a></li> <li><a href="/facts/Peirce%27s_law/GxfEII3T">Peirce's law</a></li> <li><a href="/facts/Propositional_calculus/MNt5v31V">Propositional calculus</a></li> <li><a href="/facts/Sole_sufficient_operator/ncyHAsHH">Sole sufficient operator</a></li></ul> <h3>Conditionals</h3> <ul><li><a href="/facts/Corresponding_conditional/y4tFyKPY">Corresponding conditional</a></li> <li><a href="/facts/Counterfactual_conditional/GdfYSv1k">Counterfactual conditional</a></li> <li><a href="/facts/Indicative_conditional/MUzp5aL0">Indicative conditional</a></li> <li><a href="/facts/Strict_conditional/HEgeQBHq">Strict conditional</a></li></ul> <h2 id="notes">Notes</h2> <h2 id="bibliography">Bibliography</h2> <ul><li>Allegranza, Mauro (2015-02-13). <a href="https://math.stackexchange.com/a/1146502/186330">"elementary set theory – Is there any connection between the symbol ⊃ when it means implication and its meaning as superset?"</a>. <i>Mathematics Stack Exchange</i>. Stack Exchange Inc. Answer. Retrieved 2022-08-10.</li></ul> <ul><li>Bourbaki, N. (1954). <i>Théorie des ensembles</i>. Paris: Hermann & Cie, Éditeurs. p. 14.</li></ul> <ul><li>Edgington, Dorothy (2008). <a href="http://plato.stanford.edu/archives/win2008/entries/conditionals/">"Conditionals"</a>. In Edward N. Zalta (ed.). <i>The Stanford Encyclopedia of Philosophy</i> (Winter 2008 ed.).</li></ul> <ul><li>Von Fintel, Kai (2011). <a href="http://mit.edu/fintel/fintel-2011-hsk-conditionals.pdf">"Conditionals"</a> (PDF). In von Heusinger, Klaus; Maienborn, Claudia; Portner, Paul (eds.). <i>Semantics: An international handbook of meaning</i>. de Gruyter Mouton. pp. 1515–1538. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1515%2F9783110255072.1515">10.1515/9783110255072.1515</a>. <a href="/facts/Hdl_(identifier)/rdebSxmC">hdl</a>:<a href="https://hdl.handle.net/1721.1%2F95781">1721.1/95781</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-11-018523-2.</li></ul> <ul><li>Franco, John; Goldsmith, Judy; Schlipf, John; Speckenmeyer, Ewald; Swaminathan, R.P. (1999). <a href="https://doi.org/10.1016%2FS0166-218X%2899%2900038-4">"An algorithm for the class of pure implicational formulas"</a>. <i>Discrete Applied Mathematics</i>. 96–97: 89–106. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1016%2FS0166-218X%2899%2900038-4">10.1016/S0166-218X(99)00038-4</a>.</li></ul> <ul><li>Gillies, Thony (2017). <a href="http://www.thonygillies.org/wp-content/uploads/2015/11/gillies-conditionals-handbook.pdf">"Conditionals"</a> (PDF). In Hale, B.; Wright, C.; Miller, A. (eds.). <i>A Companion to the Philosophy of Language</i>. Wiley Blackwell. pp. 401–436. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1002%2F9781118972090.ch17">10.1002/9781118972090.ch17</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 9781118972090.</li></ul> <ul><li>Van Heijenoort, Jean, ed. (1967). <i>From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931</i>. Harvard University Press. pp. 84–87. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-674-32449-8.</li></ul> <ul><li>Hilbert, D. (1918). <i>Prinzipien der Mathematik (Lecture Notes edited by Bernays, P.)</i>.</li></ul> <ul><li><a href="/facts/Ingebrigt_Johansson/wfFed9bg">Johansson, Ingebrigt</a> (1937). <a href="http://www.numdam.org/item/CM_1937__4__119_0">"Der Minimalkalkül, ein reduzierter intuitionistischer Formalismus"</a>. <i><a href="/facts/Compositio_Mathematica/Igc7zupB">Compositio Mathematica</a></i> (in German). 4: 119–136.</li></ul> <ul><li><a href="/facts/Elliott_Mendelson/S6PpLYGJ">Mendelson, Elliott</a> (2015). <i>Introduction to Mathematical Logic</i> (6th ed.). Boca Raton: CRC Press/Taylor & Francis Group (A Chapman & Hall Book). p. 2. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-4822-3778-8.</li></ul> <ul><li>Nahas, Michael (25 Apr 2022). <a href="https://github.com/mdnahas/Peano_Book/blob/46e27bdb5aed51c078ad99e5a78d134fd2a0c3ca/Peano.pdf">"English Translation of 'Arithmetices Principia, Nova Methodo Exposita'"</a> (PDF). GitHub. Retrieved 2022-08-10.</li></ul> <ul><li>Oaksford, M.; Chater, N. (1994). "A rational analysis of the selection task as optimal data selection". <i><a href="/facts/Psychological_Review/mMgWVxEH">Psychological Review</a></i>. 101 (4): 608–631. <a href="/facts/CiteSeerX_(identifier)/SceDmd3c">CiteSeerX</a> <a href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.174.4085">10.1.1.174.4085</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1037%2F0033-295X.101.4.608">10.1037/0033-295X.101.4.608</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:2912209">2912209</a>.</li></ul> <ul><li>Starr, Will (2019). <a href="https://plato.stanford.edu/archives/fall2019/entries/counterfactuals">"Counterfactuals"</a>. In Zalta, Edward N. (ed.). <i>The Stanford Encyclopedia of Philosophy</i>.</li></ul> <ul><li>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". <i>Cognitive Science</i>. 28 (4): 481–530. <a href="/facts/CiteSeerX_(identifier)/SceDmd3c">CiteSeerX</a> <a href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.13.1854">10.1.1.13.1854</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1016%2Fj.cogsci.2004.02.002">10.1016/j.cogsci.2004.02.002</a>.</li></ul> <ul><li>Von Sydow, M. (2006). <a href="https://ediss.uni-goettingen.de/handle/11858/00-1735-0000-0006-AC29-9"><i>Towards a Flexible Bayesian and Deontic Logic of Testing Descriptive and Prescriptive Rules</i></a> (doctoralThesis). Göttingen: Göttingen University Press. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.53846%2Fgoediss-161">10.53846/goediss-161</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:246924881">246924881</a>.</li></ul> <h2 id="further-reading">Further reading</h2> <ul><li>Brown, Frank Markham (2003), <i>Boolean Reasoning: The Logic of Boolean Equations</i>, 1st edition, <a href="/facts/Kluwer/dlp5j9Jr">Kluwer</a> Academic Publishers, <a href="/facts/Norwell,_Massachusetts/q7EteUzI">Norwell</a>, MA. 2nd edition, <a href="/facts/Dover_Publications/RwjvlSFv">Dover Publications</a>, <a href="/facts/Mineola,_New_York/GWYvEokQ">Mineola</a>, NY, 2003.</li> <li><a href="/facts/Dorothy_Edgington/TKodF4c4">Edgington, Dorothy</a> (2001), "Conditionals", in Lou Goble (ed.), <i>The Blackwell Guide to Philosophical Logic</i>, <a href="/facts/Wiley-Blackwell/XUkBW8kE">Blackwell</a>.</li> <li><a href="/facts/W._V._Quine/2RrhATxa">Quine, W.V.</a> (1982), <i>Methods of Logic</i>, (1st ed. 1950), (2nd ed. 1959), (3rd ed. 1972), 4th edition, <a href="/facts/Harvard_University_Press/YluAnq74">Harvard University Press</a>, <a href="/facts/Cambridge,_Massachusetts/k9UMIJ7U">Cambridge</a>, MA.</li> <li><a href="/facts/Robert_Stalnaker/q0wY3nuL">Stalnaker, Robert</a>, "Indicative Conditionals", <i><a href="/facts/Philosophia_(journal)/0dHG3tCy">Philosophia</a></i>, 5 (1975): 269–286.</li></ul> <h2 id="external-links">External links</h2> <ul><li> Media related to Material conditional at Wikimedia Commons</li> <li>Edgington, Dorothy. <a href="https://plato.stanford.edu/entries/conditionals/">"Conditionals"</a>. In <a href="/facts/Edward_N._Zalta/U8JxHaaA">Zalta, Edward N.</a> (ed.). <i><a href="/facts/Stanford_Encyclopedia_of_Philosophy/LDuL5HW4">Stanford Encyclopedia of Philosophy</a></i>.</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Hilbert 1918. - Hilbert, D. (1918). Prinzipien der Mathematik (Lecture Notes edited by Bernays, P.). <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>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. <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>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. <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Note that the horseshoe symbol Ɔ has been flipped to become a subset symbol ⊂. <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Nahas 2022, p. VI. - Nahas, Michael (25 Apr 2022). "English Translation of 'Arithmetices Principia, Nova Methodo Exposita'" (PDF). GitHub. Retrieved 2022-08-10. <a href="https://github.com/mdnahas/Peano_Book/blob/46e27bdb5aed51c078ad99e5a78d134fd2a0c3ca/Peano.pdf" target="_blank">https://github.com/mdnahas/Peano_Book/blob/46e27bdb5aed51c078ad99e5a78d134fd2a0c3ca/Peano.pdf</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>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. <a href="https://math.stackexchange.com/a/1146502/186330" target="_blank">https://math.stackexchange.com/a/1146502/186330</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Hilbert 1918. - Hilbert, D. (1918). Prinzipien der Mathematik (Lecture Notes edited by Bernays, P.). <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>Bourbaki 1954, p. 14. - Bourbaki, N. (1954). Théorie des ensembles. Paris: Hermann & Cie, Éditeurs. p. 14. <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>Johansson 1937. - Johansson, Ingebrigt (1937). "Der Minimalkalkül, ein reduzierter intuitionistischer Formalismus". Compositio Mathematica (in German). 4: 119–136. <a href="http://www.numdam.org/item/CM_1937__4__119_0" target="_blank">http://www.numdam.org/item/CM_1937__4__119_0</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> <li id="fn:10"><p>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. <a href="https://doi.org/10.1016%2FS0166-218X%2899%2900038-4" target="_blank">https://doi.org/10.1016%2FS0166-218X%2899%2900038-4</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li> <li id="fn:11"><p>Johansson 1937. - Johansson, Ingebrigt (1937). "Der Minimalkalkül, ein reduzierter intuitionistischer Formalismus". Compositio Mathematica (in German). 4: 119–136. <a href="http://www.numdam.org/item/CM_1937__4__119_0" target="_blank">http://www.numdam.org/item/CM_1937__4__119_0</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li> <li id="fn:12"><p>Edgington 2008. - Edgington, Dorothy (2008). "Conditionals". In Edward N. Zalta (ed.). The Stanford Encyclopedia of Philosophy (Winter 2008 ed.). <a href="http://plato.stanford.edu/archives/win2008/entries/conditionals/" target="_blank">http://plato.stanford.edu/archives/win2008/entries/conditionals/</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li> <li id="fn:13"><p>For example, "If Janis Joplin were alive today, she would drive a Mercedes-Benz", see Starr (2019) <a href="/wiki/Janis_Joplin" target="_blank">/wiki/Janis_Joplin</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></p></li> <li id="fn:14"><p>Edgington 2008. - Edgington, Dorothy (2008). "Conditionals". In Edward N. Zalta (ed.). The Stanford Encyclopedia of Philosophy (Winter 2008 ed.). <a href="http://plato.stanford.edu/archives/win2008/entries/conditionals/" target="_blank">http://plato.stanford.edu/archives/win2008/entries/conditionals/</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></p></li> <li id="fn:15"><p>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. <a href="http://www.thonygillies.org/wp-content/uploads/2015/11/gillies-conditionals-handbook.pdf" target="_blank">http://www.thonygillies.org/wp-content/uploads/2015/11/gillies-conditionals-handbook.pdf</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></p></li> <li id="fn:16"><p>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. <a href="http://www.thonygillies.org/wp-content/uploads/2015/11/gillies-conditionals-handbook.pdf" target="_blank">http://www.thonygillies.org/wp-content/uploads/2015/11/gillies-conditionals-handbook.pdf</a> <a href="#fnref:16" class="footnote-back-ref">↩</a></p></li> <li id="fn:17"><p>Edgington 2008. - Edgington, Dorothy (2008). "Conditionals". In Edward N. Zalta (ed.). The Stanford Encyclopedia of Philosophy (Winter 2008 ed.). <a href="http://plato.stanford.edu/archives/win2008/entries/conditionals/" target="_blank">http://plato.stanford.edu/archives/win2008/entries/conditionals/</a> <a href="#fnref:17" class="footnote-back-ref">↩</a></p></li> <li id="fn:18"><p>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. <a href="http://www.thonygillies.org/wp-content/uploads/2015/11/gillies-conditionals-handbook.pdf" target="_blank">http://www.thonygillies.org/wp-content/uploads/2015/11/gillies-conditionals-handbook.pdf</a> <a href="#fnref:18" class="footnote-back-ref">↩</a></p></li> <li id="fn:19"><p>Edgington 2008. - Edgington, Dorothy (2008). "Conditionals". In Edward N. Zalta (ed.). The Stanford Encyclopedia of Philosophy (Winter 2008 ed.). <a href="http://plato.stanford.edu/archives/win2008/entries/conditionals/" target="_blank">http://plato.stanford.edu/archives/win2008/entries/conditionals/</a> <a href="#fnref:19" class="footnote-back-ref">↩</a></p></li> <li id="fn:20"><p>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. <a href="http://mit.edu/fintel/fintel-2011-hsk-conditionals.pdf" target="_blank">http://mit.edu/fintel/fintel-2011-hsk-conditionals.pdf</a> <a href="#fnref:20" class="footnote-back-ref">↩</a></p></li> <li id="fn:21"><p>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. <a href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.174.4085" target="_blank">https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.174.4085</a> <a href="#fnref:21" class="footnote-back-ref">↩</a></p></li> <li id="fn:22"><p>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. <a href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.13.1854" target="_blank">https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.13.1854</a> <a href="#fnref:22" class="footnote-back-ref">↩</a></p></li> <li id="fn:23"><p>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. <a href="https://ediss.uni-goettingen.de/handle/11858/00-1735-0000-0006-AC29-9" target="_blank">https://ediss.uni-goettingen.de/handle/11858/00-1735-0000-0006-AC29-9</a> <a href="#fnref:23" class="footnote-back-ref">↩</a></p></li> </ol>