Strict conditional: Difference between revisions

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{{Short description|Formal statement in logic}}

In [[logic]], a '''strict conditional''' (symbol: <math>\Box</math>, or ⥽) is a conditional governed by a [[modal operator]], that is, a [[logical connective]] 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 [[proposition]]s ''p'' and ''q'', the [[well-formed formula|formula]] ''p'' → ''q'' says that ''p'' [[material conditional|materially implies]] ''q'' while <math>\Box (p \rightarrow q)</math> says that ''p'' [[logical consequence|strictly implies]] ''q''.<ref>[[Graham Priest]], ''[[An Introduction to Non-Classical Logic|An Introduction to Non-Classical Logic: From if to is]]'', 2nd ed, Cambridge University Press, 2008, {{ISBN|0-521-85433-4}}, [https://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>[[Cooper{{cite Hbook|last1=Lewis|first1=C. Langford]] and [[I.|author1-link=C. I. Lewis]],|last2=Langford|first2=C.H.|author2-link=Cooper ''Harold Langford|year=1959|orig-year=1932|title=Symbolic Logic'' (New York, 1932), p.|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|1-4051-0679-4}}, "strict implication," [https://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>

==Avoiding paradoxes==

The strict conditionals may avoid [[paradoxes of material implication]]. The following statement, for example, is not correctly formalized by material implication:

: If Bill Gates has 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 leads to:

: Bill Gates graduated in Medicinemedicine → Elvis never died.

This formula is true because whenever the antecedent ''A'' is false, a formula ''A'' → ''B'' is true. Hence, this formula is not an adequate translation of the original sentence. An encoding using the strict conditional is:

: <math>\Box</math> (Bill Gates graduated in Medicinemedicine → 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 [[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 → 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 conditional]]s,<ref>Jens S. Allwood, Lars-Gunnar Andersson, and Östen Dahl, ''Logic in Linguistics'', Cambridge University Press, 1977, {{ISBN|0-521-29174-7}}, [https://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|3-937202-24-2}}, [https://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>

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==Constructive logic==

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

| doi = 10.1016/j.indag.2017.10.003

| last2=Visser |first2 = Albert

| author1 = Tadeusz Litak

| year = 2018

| author2 = Albert Visser

| title = Lewis meets Brouwer: Constructive strict implication

| journal = [[Indagationes Mathematicae]]

| doi = 10.1016/j.indag.2017.10.003

| volume = 29

| issue arxiv = 1 1708.02143

| pages volume = 36–9029

| yearissue = 1 = 2018

| pages = 36–90

| url = https://arxiv.org/abs/1708.02143 | arxiv = 1708.02143

|s2cid = 12461587

}}</ref>

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* [[Counterfactual conditional]]

* [[Dynamic semantics]]

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

* [[Indicative conditional]]

* [[Logical consequence]]

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{{Logic}}

{{Formal semantics}}

[[Category:Conditionals]]

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[[Category:Modal logic]]

[[Category:Necessity]]

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