Strict conditional: Difference between revisions

Revision as of 07:27, 14 January 2012

Revision as of 14:29, 14 January 2012

Line 4:

==Avoiding paradoxes==

The strict conditionalconditionals avoidsmay theavoid [[paradoxes of material implication]]. The following statement, for example, is not correctly formalized by material implication:

: If Bill Gates had graduated in Medicine, then Elvis never died.

Line 19:

==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 antecedents that are necessarily true or false.<ref>Roy A. Sorensen, ''A Brief History of the Paradox: Philosophy and the labyrinths of the mind'', Oxford University Press, 2003, ISBN 0195159039, [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.

Line 27:

: <math>\Box</math> (Bill Gates graduated in Medicine <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. Similarly, problems can arise with antecedents that are necessarily true or false.<ref>Roy A. Sorensen, ''A Brief History of the Paradox: Philosophy and the labyrinths of the mind'', Oxford University Press, 2003, ISBN 0195159039, [http://books.google.com/books?id=PB8I0kHeKy4C&pg=PA105 p. 105.]</ref> A similar situation arises with:

: If 2 + 2 = 5, then Bill Gates graduated in Medicine.

Line 35:

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.

The rule of [[Modal logic#Axiomatic systems|necessitationAxiomatic systems]] in modal logic allows us to infer the necessity of any theorem which has been proved without requiring hypotheses, i.e. from <math>\vdash A</math>, infer <math>\vdash \Box A</math>.<ref>James W. Garson, ''Modal Logic for Philosophers'', Cambridge University Press, 2006, ISBN 0521682290, [http://books.google.com/books?id=xFNbDZPZERcC&pg=PA30 p. 30.]</ref> If the theorem has the form of a conditional, i.e. <math>\vdash P \rightarrow Q</math>, it follows that <math>\vdash \Box (P \rightarrow Q)</math>. Thus theorems having the form of a conditional are also strict conditionals.

==In mathematics==

The rule of [[Modal logic#Axiomatic systems|necessitation]] in modal logic allows us to infer the necessity of any theorem which has been proved without requiring hypotheses, i.e. from <math>\vdash A</math>, infer <math>\vdash \Box A</math>.<ref>James W. Garson, ''Modal Logic for Philosophers'', Cambridge University Press, 2006, ISBN 0521682290, [http://books.google.com/books?id=xFNbDZPZERcC&pg=PA30 p. 30.]</ref> If the theorem has the form of a conditional, i.e. <math>\vdash P \rightarrow Q</math>, it follows that <math>\vdash \Box (P \rightarrow Q)</math>. Thus theorems having the form of a conditional are also strict conditionals.

==See also==