Import–export (logic)

In propositional logic, import-export is a name given to the propositional form of Exportation:

: $(P \rightarrow ( Q \rightarrow R )) \leftrightarrow ((P \land Q) \rightarrow R)$ .

This already holds in minimal logic, and thus also in classical logic, where the conditional operator " $\rightarrow$ " is taken as material implication.

In the Curry-Howard correspondence for intuitionistic logics, it can be realized through currying and uncurrying.

Discussion

Import-export expresses a deductive argument form. In natural language terms, the formula states that the following English sentences are logically equivalent:{{cite book |last= Sider |first= Theodore |year=2010 |title= Logic for philosophy |url= |location= |publisher= Oxford University Press |pages=130,276-277}}{{cite encyclopedia |last1= Égré |first1= Paul |last2=Rott |first2=Hans |editor-last1=Zalta |editor-first1=Edward N.|encyclopedia=The Stanford Encyclopedia of Philosophy |title= The Logic of Conditionals |url= https://plato.stanford.edu/entries/logic-conditionals/ |year=2021 |publisher= Metaphysics Research Lab, Stanford University}}{{cite journal |last1=Gillies |first1=Anthony |year=2009 |title=On truth-conditions for if (but not quite only if) |journal=Philosophical Review |volume=118 |issue=3|doi=10.1215/00318108-2009-002}}

If Mary isn't at home, then if Sally isn't at home, then the house is empty.
If Mary isn't home and Sally isn't home, then the house is empty.

There are logics where it does not hold and its status as a true principle of logic is a matter of debate. Controversy over the principle arises from the fact that any conditional operator that satisfies it will collapse to material implication when combined with certain other principles. This conclusion would be problematic given the paradoxes of material implication, which are commonly taken to show that natural language conditionals are not material implication.{{cite encyclopedia |last=Gibbard |first=Allan|editor-last1= Harper |editor-last2=Stalnaker |editor-last3=Pearce |encyclopedia=Ifs |title= Two Recent Theories of Conditionals |year=1980 |publisher=Springer|doi=10.1007/978-94-009-9117-0_10}}

This problematic conclusion can be avoided within the framework of dynamic semantics, whose expressive power allows one to define a non-material conditional operator which nonetheless satisfies import-export along with the other principles.{{cite encyclopedia |last1=Kaufmann |first1=Magdalena |last2=Kaufmann |first2=Stefan |editor-last1=Lappin |editor-first1=Shalom | editor-last2=Fox |editor-first2=Chris |encyclopedia=The handbook of contemporary semantic theory |title= Conditionals and modality|year=2015 |publisher=Wiley }} However, other approaches reject import-export as a general principle, motivated by cases such as the following, uttered in a context where it is most likely that the match will be lit by throwing it into a campfire, but where it is possible that it could be lit by striking it. In this context, the first sentence is intuitively true but the second is intuitively false.{{cite encyclopedia |last1=Starr |first1=Will |editor-last1=Zalta |editor-first1=Edward N.|encyclopedia=The Stanford Encyclopedia of Philosophy|title=Counterfactuals|year=2021 |url=https://plato.stanford.edu/entries/counterfactuals}} {{cite journal |last1=Kaufmann |first1=Stefan|year=2005 |title=Conditional predictions |url= |journal= Linguistics and Philosophy |volume=28 |issue=2}}

If you strike the match and it lights, it will light.
If the match lights, it will light if you strike it.

References

Category:Theorems in propositional logic

Category:Classical logic

Category:Formal semantics (natural language)

Import–export (logic)

Discussion

See also

References