Strict conditional: Difference between revisions

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{{Short description|Formal statement in logic}}

{{Expert-subject|Logic|date=December 2011}}

In [[logic]], a '''strict conditional''' (symbol: <math>\Box</math>, or ⥽) is a conditional governed by a [[materialmodal conditionaloperator]], that is, acteda upon[[logical byconnective]] of [[modal logic]]. It is [[logical equivalence|logically equivalent]] to the [[material conditional]] of [[classical logic]], combined with the [[Logical truth|necessity]] operator from [[modal logic]]. For any two propositions[[proposition]]s <math>''p</math>'' and <math>''q</math>'', the [[well-formed formula|formula]] <math>''p'' \rightarrow→ ''q</math>'' says that <math>''p</math>'' [[material conditional|materially implies]] <math>''q</math>'' while <math>\Box (p \rightarrow q)</math> says that <math>''p</math>'' [[logical consequence|strictly implies]] <math>''q</math>''.<ref>[[Graham Priest]], ''[[An Introduction to Non-Classical Logic|An Introduction to Non-Classical Logic: From if to is]]'', 2^nd2nd ed, Cambridge University Press, 2008, {{ISBN 0521854334|0-521-85433-4}}, [httphttps://books.google.com/books?id=rMXVbmAw3YwC&pg=PA72 p. 72.]</ref> Strict conditionals are the result of [[C. I. Lewis|Clarence Irving Lewis]]'s attempt to find a conditional for logic that can adequately express [[indicative conditional]]s in natural language.<ref>{{cite book|last1=Lewis|first1=C.I.|author1-link=C. I. Lewis|last2=Langford|first2=C.H.|author2-link=Cooper Harold Langford|year=1959|orig-year=1932|title=Symbolic Logic|edition=2|publisher=[[Dover Publications]]|isbn=0-486-60170-6|page=124}}</ref><ref>Nicholas Bunnin and Jiyuan Yu (eds), ''The Blackwell Dictionary of Western Philosophy'', Wiley, 2004, {{ISBN 1405106794|1-4051-0679-4}}, "strict implication," [httphttps://books.google.com/books?id=OskKWI1YA7AC&pg=PA660 p. 660].</ref> They have also been used in studying [[Molinism|Molinist]] theology.<ref>Jonathan L. Kvanvig, "Creation, Deliberation, and Molinism," in ''Destiny and Deliberation: Essays in Philosophical Theology'', Oxford University Press, 2011, {{ISBN|0-19-969657-8}}, [https://books.google.com/books?id=nQliRGPVpTwC&pg=PA127 p. 127–136].</ref>

{{Expert-subject|Philosophy|date=December 2011}}

In [[logic]], a '''strict conditional''' is a [[material conditional]] that is acted upon by the necessity operator from [[modal logic]]. For any two propositions <math>p</math> and <math>q</math>, the formula <math>p \rightarrow q</math> says that <math>p</math> materially implies <math>q</math> while <math>\Box (p \rightarrow q)</math> says that <math>p</math> strictly implies <math>q</math>.<ref>Graham Priest, ''An Introduction to Non-Classical Logic: From if to is'', 2^nd ed, Cambridge University Press, 2008, ISBN 0521854334, [http://books.google.com/books?id=rMXVbmAw3YwC&pg=PA72 p. 72.]</ref> Strict conditionals are the result of [[Clarence Irving Lewis]]'s attempt to find a conditional for logic that can adequately express [[indicative conditional]]s.<ref>Nicholas Bunnin and Jiyuan Yu (eds), ''The Blackwell Dictionary of Western Philosophy'', Wiley, 2004, ISBN 1405106794, "strict implication," [http://books.google.com/books?id=OskKWI1YA7AC&pg=PA660 p. 660.]</ref>

==Avoiding paradoxes==

The strict conditionalconditionals avoidsmay theavoid [[paradoxes of material implication]]. The following statement, for example, is not correctly formalized by material implication.:

: If Bill Gates had graduated in Medicinemedicine, then Elvis never died.

This condition should clearly be false: the degree of Bill Gates has nothing to do with whether Elvis is still alive. However, the direct encoding of this formula in [[classical logic]] using material implication leadleads to:

: Bill Gates graduated in Medicinemedicine <math>\rightarrow</math>→ Elvis never died.

This formula is true because awhenever formulathe antecedent <math>''A'' \rightarrowis B</math>false, isa trueformula whenever''A'' the→ antecedent <math>A</math>''B'' is falsetrue. Hence, this formula is not an adequate translation of the original sentence. StrictAn conditionsencoding areusing encodingsthe ofstrict implications in modal logic attempting A different encodingconditional is:

: <math>\Box</math> (Bill Gates graduated in Medicinemedicine <math>\rightarrow</math>→ Elvis never died.).

In modal logic, this formula means (roughly) that, in every possible world in which Bill Gates graduated in Medicinemedicine, Elvis never died. Since one can easily imagine a world where Bill Gates is a Medicinemedicine graduate and Elvis is dead, this formula is false. Hence, this formula seems to be a correct translation of the original sentence.

==Problems==

Although the strict conditional is much closer to being able to express natural language conditionals than the material conditional, it has its own problems with antecedents[[consequent]]s that are [[Logical truth|necessarily true]] (such as 2 + 2 = 4) or antecedents that are necessarily false.<ref>Roy A. Sorensen, ''A Brief History of the Paradox: Philosophy and the labyrinths of the mind'', Oxford University Press, 2003, {{ISBN 0195159039|0-19-515903-9}}, [httphttps://books.google.com/books?id=PB8I0kHeKy4C&pg=PA105 p. 105.].</ref> The following sentence, for example, is not correctly formalized by a strict conditional:

: If Bill Gates graduated in Medicinemedicine, then 2 + 2 = 4.

Using strict conditionals, this sentence is expressed as:

: <math>\Box</math> (Bill Gates graduated in Medicinemedicine <math>\rightarrow</math>→ 2 + 2 = 4)

In modal logic, this formula means that, in every possible world where Bill Gates graduated in medicine, it holds that 2 + 2 = 4. Since 2 + 2 is equal to 4 in all possible worlds, this formula is true, although it does not seem that the original sentence should be. A similar situation arises with 2 + 2 = 5, which is necessarily false:

: If 2 + 2 = 5, then Bill Gates graduated in Medicinemedicine.

Some logicians view this situation as indicating that the strict conditional is still unsatisfactory. Others have noted that the strict conditional cannot adequately express [[counterfactual conditionalsconditional]]s,<ref>Jens S. Allwood, Lars-Gunnar Andersson, and Östen Dahl, ''Logic in Linguistics'', Cambridge University Press, 1977, {{ISBN 0521291747|0-521-29174-7}}, [httphttps://books.google.com/books?id=hXIpFPttDjgC&pg=PA120 p. 120.].</ref> and that it does not satisfy certain logical properties.<ref>Hans Rott and Vítezslav Horák, ''Possibility and Reality: Metaphysics and Logic'', ontos verlag, 2003, {{ISBN 3937202242|3-937202-24-2}}, [httphttps://books.google.com/books?id=ov9kN3HyltAC&pg=PA271 p. 271].</ref> In particular, the strict conditional is [[Transitive relation|transitive]], while the counterfactual conditional is not.<ref>John Bigelow and Robert Pargetter, ''Science and Necessity'', Cambridge University Press, 1990, {{ISBN|0-521-39027-3}}, [https://books.google.com/books?id=O-onBdR7TPAC&pg=PA116 p. 116].</ref>

Some logicians, such as [[Paul Grice]], have used [[conversational implicature]] to argue that, despite apparent difficulties, the material conditional is just fine as a translation for the natural language 'if...then...'. Others still have turned to [[relevance logic]] to supply a connection between the antecedent and consequent of provable conditionals.

==Constructive logic==

==In mathematics==

In a [[Constructive logic|constructive]] setting, the symmetry between ⥽ and <math>\Box</math> is broken, and the two connectives can be studied independently. Constructive strict implication can be used to investigate [[interpretability]] of [[Heyting arithmetic]] and to model [[arrow (computer science)|arrows]] and guarded [[recursion (computer science)|recursion]] in computer science.<ref>{{cite journal

The rule of [[Modal logic#Axiomatic systems|necessitation]] in modal logic allows us to infer the necessity of any theorem which has been proved without requiring hypotheses, i.e. from <math>\vdash A</math>, infer <math>\vdash \Box A</math>.<ref>James W. Garson, ''Modal Logic for Philosophers'', Cambridge University Press, 2006, ISBN 0521682290, [http://books.google.com/books?id=xFNbDZPZERcC&pg=PA30 p. 30.]</ref> If the theorem has the form of a conditional, i.e. <math>\vdash P \rightarrow Q</math>, it follows that <math>\vdash \Box (P \rightarrow Q)</math>. Thus theorems having the form of a conditional are also strict conditionals.

| last1=Litak |first1 = Tadeusz

| last2=Visser |first2 = Albert

| year = 2018

| title = Lewis meets Brouwer: Constructive strict implication

| journal = [[Indagationes Mathematicae]]

| doi = 10.1016/j.indag.2017.10.003

| arxiv = 1708.02143

| volume = 29

| issue = 1

| pages = 36–90

|s2cid = 12461587

}}</ref>

==See also==

* [[Corresponding conditional]]

* [[Counterfactual conditional]]

* [[Dynamic semantics]]

* [[Import-Export (logic)|Import-Export]]

* [[Indicative conditional]]

* [[Logical implicationconsequence]]

* [[Material conditional]]

* [[Logical implication]]

* [[Corresponding conditional]]

==References==

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==Bibliography==

*Edgington, Dorothy, 2001, "Conditionals," in Goble, Lou, ed., ''The Blackwell Guide to Philosophical Logic''. Blackwell.

*For an introduction to non-classical logic as an attempt to find a better translation of the conditional, see:

**[[Graham Priest|Priest, Graham]], 2001. ''An Introduction to Non-Classical Logic''. Cambridge Univ. Press.

*For an extended philosophical discussion of the issues mentioned in this article, see:

**[[Mark Sainsbury (philosopher)|Mark Sainsbury]], 2001. ''Logical Forms''. Blackwell Publishers.

*[[Jonathan Bennett (philosopher)|Jonathan Bennett]], 2003. ''A Philosophical Guide to Conditionals''. Oxford Univ. Press.

{{Logic}}

{{Formal semantics}}

[[Category:Conditionals]]

[[Category:PropositionalLogical calculusconnectives]]

[[Category:Modal logic]]

[[Category:Necessity]]

[[Category:Formal semantics (natural language)]]

[[es:Condicional estricto]]

[[fr:Implication stricte]]

[[zh:严格条件]]