Strict conditional: Difference between revisions

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In [[logic]], a '''strict conditional''' 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: From if to is'', 2nd 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 [[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," [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==

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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, [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 Medicine, then 2 + 2 = 4.

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: 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]]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.