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> SuchThey ahave conditionalalso would,been forused example,in avoid thestudying [[paradoxes of material implicationMolinism|Molinist]] theology.<ref>Jonathan TheL. followingKvanvig, statement"Creation, forDeliberation, exampleand Molinism," isin not''Destiny correctlyand formalizedDeliberation: byEssays materialin implicationPhilosophical 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> Such a conditional would, for example, avoid the [[paradoxes of material implication]]. The following statement, for example, is not correctly formalized by material implication.

==Avoiding paradoxes==

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

The strict conditionals may avoid [[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 lead to:

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 Medicine <math>\rightarrow</math> Elvis never died.

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

This formula is true because a formula <math>A \rightarrow B</math> is true whenever the antecedent <math>A</math> is false. Hence, this formula is not an adequate translation of the original sentence. Strict conditions are encodings of implications in modal logic attempting A different encoding is:

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 Medicine <math>\rightarrow</math> Elvis never died.)

: <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 Medicine, Elvis never died. Since one can easily imagine a world where Bill Gates is a Medicine graduate and Elvis is dead, this formula is false. Hence, this formula seems a correct translation of the original sentence.

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.

Although the strict conditional is much closer to being able to express natural language conditionals than the material conditional, it has its own problems. The following sentence, for example, is not correctly formalized by a strict conditional:

==Problems==

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

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 [[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|0-19-515903-9}}, [https://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 inference schemesproperties.<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==

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.

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

| 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]]