17
edits
User:Magnap/Inquisitive Semantics Proposal: Difference between revisions
User:Magnap/Inquisitive Semantics Proposal (view source)
Revision as of 16:46, 26 September 2024
, 16:46, 26 September 2024→TODO proposal details: Split completeness out into its own proposal
(More details on the implementation) |
(→TODO proposal details: Split completeness out into its own proposal) |
||
| (4 intermediate revisions by the same user not shown) | |||
| Line 22: | Line 22: | ||
An important thing to bear in mind about inquisitive semantics is that it does not give us a Boolean algebra, but only a Heyting algebra, meaning that <math>\neg\neg\text{P} = \text{P}</math> is not guaranteed. In fact, the non-inquisitive projection operator <math>\lambda \text{P}. !\text{P}</math>, which collapses all the alternatives of a proposition into just one (called <math>\text{info}\left(\text{P}\right)</math>) which contains them all, thus keeping the assertion the same but ensuring that no question is asked, ''is'' just <math>\lambda \text{P}. \neg\neg\text{P}</math>. Another important operator is the non-informative projection operator <math>\lambda \text{P}. ?\text{P} = \lambda \text{P}. \text{P} \lor \neg\text{P}</math>, which ensures that a proposition does not assert anything by adding an alternative which covers all worlds that would otherwise have been ruled out. Finally (TODO source! well tbf I made it up initially but it ''is'' attested, at least in one presentation by one of the inqsem guys) there's the presupposition operator <math>\lambda \text{P}. ;\text{P} = \lambda \text{P}. \left(\text{P} | \text{info}\left(\text{P}\right)\right)</math>. | An important thing to bear in mind about inquisitive semantics is that it does not give us a Boolean algebra, but only a Heyting algebra, meaning that <math>\neg\neg\text{P} = \text{P}</math> is not guaranteed. In fact, the non-inquisitive projection operator <math>\lambda \text{P}. !\text{P}</math>, which collapses all the alternatives of a proposition into just one (called <math>\text{info}\left(\text{P}\right)</math>) which contains them all, thus keeping the assertion the same but ensuring that no question is asked, ''is'' just <math>\lambda \text{P}. \neg\neg\text{P}</math>. Another important operator is the non-informative projection operator <math>\lambda \text{P}. ?\text{P} = \lambda \text{P}. \text{P} \lor \neg\text{P}</math>, which ensures that a proposition does not assert anything by adding an alternative which covers all worlds that would otherwise have been ruled out. Finally (TODO source! well tbf I made it up initially but it ''is'' attested, at least in one presentation by one of the inqsem guys) there's the presupposition operator <math>\lambda \text{P}. ;\text{P} = \lambda \text{P}. \left(\text{P} | \text{info}\left(\text{P}\right)\right)</math>. | ||
TODO explain how adding presupposition to InqB would work in more detail? | |||
Since this proposal is, after all, suggesting that we base the formal language of our semantics on InqB+presupposition. | |||
===Examples=== | ===Examples=== | ||
| Line 70: | Line 73: | ||
The complementizer {{t|ma}} will gain semantics, specifically <math>\lambda \text{P}. ?!P</math>. | The complementizer {{t|ma}} will gain semantics, specifically <math>\lambda \text{P}. ?!P</math>. | ||
We'll probably want prefixes that apply to quantifiers and apply one of the following to the result of applying the quantifier (in other words, applying them outside the quantifier): <math>!</math>, <math>;</math>, <math>?</math>, <math>?!</math>. | We'll probably want prefixes that apply to quantifiers and apply one of the following to the result of applying the quantifier (in other words, applying them outside the quantifier): <math>!</math>, <math>;</math>, <math>?</math>, <math>?!</math>. | ||
| Line 80: | Line 81: | ||
Finally, the {{t|hí}} problem: if we want to remain backwards compatible with existing Toaq Delta wh-questions, we must assign it a meaning. Unfortunately, whether or not wh-questions generate presuppositions (in our notation, whether {{t|hí}} means <math>;\exists</math> or <math>?\exists</math>) is a highly contested issue that we would be forced to pick a side on... | Finally, the {{t|hí}} problem: if we want to remain backwards compatible with existing Toaq Delta wh-questions, we must assign it a meaning. Unfortunately, whether or not wh-questions generate presuppositions (in our notation, whether {{t|hí}} means <math>;\exists</math> or <math>?\exists</math>) is a highly contested issue that we would be forced to pick a side on... | ||
We might also want new or different connectives, or at least to give specific denotations for them. For example, we probably want {{t|ró}} to create two alternatives, which means we'd want it to be <math>\left(P \land \neg\text{Q}\right) \lor \left(\neg\text{P} \land{} \text{Q}\right)</math>. Other potentially quite useful connectives are (sometimes writing them out in a more verbose way than necessary to make their alternatives clearer): | We might also want new or different connectives, or at least to give specific denotations for them. For example, we probably want {{t|ró}} to create two alternatives, which means we'd want it to be <math>\left(P \land \neg\text{Q}\right) \lor \left(\neg\text{P} \land{} \text{Q}\right)</math>. | ||
Other potentially quite useful connectives are (sometimes writing them out in a more verbose way than necessary to make their alternatives clearer): | |||
* <math>?\text{P} \land{} ?\text{Q}</math> (one or the other or both or neither) | * <math>?\text{P} \land{} ?\text{Q}</math> (one or the other or both or neither) | ||
* <math>\left(P \land \text{Q}\right) \lor \left(P \land \neg\text{Q}\right) \lor \left(\neg\text{P} \land \text{Q}\right)</math> (one or the other or both) | * <math>\left(P \land \text{Q}\right) \lor \left(P \land \neg\text{Q}\right) \lor \left(\neg\text{P} \land \text{Q}\right)</math> (one or the other or both) | ||
* <math>\left(P \land \neg\text{Q}\right) \lor \left(\neg\text{P} \land \text{Q}\right) \lor \left(\neg\text{P} \land \neg\text{Q}\right)</math> (one or the other or neither) | * <math>\left(P \land \neg\text{Q}\right) \lor \left(\neg\text{P} \land \text{Q}\right) \lor \left(\neg\text{P} \land \neg\text{Q}\right)</math> (one or the other or neither) | ||
And so many more, this is rich bikeshedding territory and luckily more can easily be added later. | And so many more, this is rich bikeshedding territory and luckily more can easily be added later. | ||
Another concern is {{t|rí}} which is weird in the refgram, allowing you to answer with a connective!? I think we should fix that one up somehow. We almost certainly need a denotation that generates a non-informative proposition and doesn't presuppose anything, but there are many of those. Two reasonably intuitive options would be <math>?\left(\text{P} \lor \text{Q}\right)</math> and <math>?\text{P} \land{} ?\text{Q}</math>. | |||