Strict conditional: Difference between revisions

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In [[logic]], a '''strict conditional''' is 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: From if to is'', 2nd ed, Cambridge University Press, 2008, ISBN 0-521-85433-4, [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 in natural language.<ref>Nicholas Bunnin and Jiyuan Yu (eds), ''The Blackwell Dictionary of Western Philosophy'', Wiley, 2004, ISBN 1-4051-0679-4, "strict implication," [http://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, [http://books.google.com/books?id=nQliRGPVpTwC&pg=PA127 p. 127–136].</ref>

==Avoiding paradoxes==

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: Bill Gates graduated in Medicine → Elvis never died.

This formula is true because a formula ''A''  →  ''B'' is true whenever the antecedent ''A'' is false. 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.)

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==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, [http://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:

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* [[Material conditional]]

* [[Logical consequence]]

* [[Corresponding conditional]]

==References==