Strict conditional: Difference between revisions

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In [[logic]], a '''strict conditional''' is a [[material conditional]] that is acted upon by the necessity operator from [[modal logic]]. For any two propositions <math>p</math> and <math>q</math>, the formula <math>p \rightarrow q</math> says that <math>p</math> materially implies <math>q</math> while <math>\Box (p \rightarrow q)</math> says that <math>p</math> strictly implies <math>q</math>.<ref>Graham Priest, ''An Introduction to Non-Classical Logic: From if to is'', 2^nd ed, Cambridge University Press, 2008, ISBN 05218543340-521-85433-4, [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 14051067941-4051-0679-4, "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 01996965780-19-969657-8, [http://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 consequents that are 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 01951590390-19-515903-9, [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.

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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 conditionals]],<ref>Jens S. Allwood, Lars-Gunnar Andersson, and Östen Dahl, ''Logic in Linguistics'', Cambridge University Press, 1977, ISBN 05212917470-521-29174-7, [http://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 39372022423-937202-24-2, [http://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 05213902730-521-39027-3, [http://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.