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 <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 |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 |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}}, [httphttps://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 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 leads to:

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

This formula is true because awhenever formulathe <math>antecedent ''A \rightarrow B</math>'' is truefalse, whenevera theformula antecedent <math>''A</math>'' → ''B'' is falsetrue. 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 <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 [[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}}, [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 |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 |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}}, [httphttps://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==

== More about problems evoked in the preceding paragraph. Nothing is solved by defining strict implication as material implication acted upon by necessity L ==

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

If one uses the Polish notation where the square, symbol of necessity, is replaced by the symbol L, if one adopts the initial definition given by this article of wikipedia and consequently describes the strict implication of q by p as material implication p → q acted upon by necessity L , one may represent the said strict implication of q by p thus: L (p → q) or ~M (p & ~q).Conformably to De Morgan's laws, p → q says the same thing as ~(p & ~q). To say that p materially implies q is to say that p is incompatible with not-q. Therefore to say that necessarily p implies q is to say that necessarily p is incompatible with not-q.

| last1=Litak |first1 = Tadeusz

| last2=Visser |first2 = Albert

L (p → q) is tantamount to L ~(p & ~q). If, of necessity, one excludes the conjunction p & ~q, another way to express the fact is to say that the conjunction p & ~q is im-possible. Hence the equivanent expression ~M (p & ~q) ''It is impossible to have the conjunction of p and not-q''

| year = 2018

| title = Lewis meets Brouwer: Constructive strict implication

Now, let us suppose that one has necessarily not-p : L~p or in other terms that p is im-possible: ~Mp, it is obvious that the conjunctions p & q and p & ~q are both impossible. If one has ~Mp, if p is im-possible, one can write ~M (p & ~q) as well as ~M (p & q) . Let us conclude: L (p → q), that is to say, ~M (p & ~q) does not represent the strict implication of q by p because ~M (p & ~q) may come from the fact that p is im-possible and not at all from the fact that p entails q.

| 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 consequence]]

* [[Material conditional]]

* [[Logical consequence]]

* [[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]]

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[[Category:Modal logic]]

[[Category:Necessity]]

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