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In [[Logic|logic]], a '''strict conditional''' (symbol: <math>\Box</math>, or ⥽) is a conditional governed by a [[Modal operator|modal operator]], that is, a [[Logical connective|logical connective]] of [[Modal logic|modal logic]]. It is [[Logical equivalence|logically equivalent]] to the [[Material conditional|material conditional]] of [[Philosophy:Classical logic|classical logic]], combined with the [[Logical truth|necessity]] operator from [[Modal logic|modal logic]]. For any two [[Proposition|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>[[Biography:Graham Priest|Graham Priest]], ''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 [[Biography:C. I. Lewis|Clarence Irving Lewis]]'s attempt to find a conditional for logic that can adequately express [[Indicative conditional|indicative conditional]]s in natural language.<ref>{{cite book|last1=Lewis|first1=C.I.|last2=Langford|first2=C.H.|year=1959|orig-year=1932|title=Symbolic Logic|edition=2|publisher=[[Social:Dover Publications|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 [[Philosophy: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|paradoxes of material implication]]. The following statement, for example, is not correctly formalized by material implication:

: If Bill Gates graduated in medicine, 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 [[Philosophy:Classical logic|classical logic]] using material implication leads to:

: Bill Gates graduated in medicine → 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 medicine → 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 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|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 medicine, then 2 + 2 = 4.

Using strict conditionals, this sentence is expressed as:

: <math>\Box</math> (Bill Gates graduated in medicine → 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 medicine.

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|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>

Some logicians, such as [[Biography:Paul Grice|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 [[Philosophy:Relevance logic|relevance logic]] to supply a connection between the antecedent and consequent of provable conditionals.

==Constructive logic==
In a 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 [[Philosophy:Interpretability|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]]
* [[Philosophy:Import-Export (logic)|Import-Export]]
* [[Indicative conditional]]
* [[Logical consequence]]
* [[Material conditional]]

==References==
{{reflist}}

==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:
**[[Biography: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:
**[[Biography:Mark Sainsbury (philosopher)|Mark Sainsbury]], 2001. ''Logical Forms''. Blackwell Publishers.
*[[Biography:Jonathan Bennett (philosopher)|Jonathan Bennett]], 2003. ''A Philosophical Guide to Conditionals''. Oxford Univ. Press.

{{Logic}}

[[Category:Conditionals]]
[[Category:Logical connectives]]
[[Category:Modal logic]]
[[Category:Necessity]]

{{Sourceattribution|Strict conditional}}