MoL-2009-11: Inquisitive Semantics and Intermediate Logics

MoL-2009-11: Ciardelli, Ivano A. (2009) Inquisitive Semantics and Intermediate Logics. [Report]

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This thesis has been concerned with the development of inquisitive semantics for both a propositional and a first-order language, and with the investigation of the logical systems they give rise to. In the first place, we discussed the features of the system arising from the semantics proposed by (Groenendijk, 2008a) and (Ciardelli, 2008), explored the associated logic and its connections with intermediate logics and established a whole range of sound and complete axiomatizations; these are obtained by expanding certain intermediate logics, among which the Kreisel-Putnam and Medvedev logics, with the double negation axiom for atoms. We showed that the schematic fragment of inquisitive logic coincides with Medvedev's logic of finite problems, thus establishing interesting connections between the latter and other well-understood intermediate logics: in the first-place, ML is the set of schematic validities of a recursively axiomatized derivation system, obtained (for instance) by expanding the Kreisel-Putnam logic with atomic double negation axioms; in the second place, a formula phi is provable in Medvedev's logic if and only if any instance of it obtained by replacing an atom with a disjunction of negated atoms is provable in the Kreisel-Putnam logic (or indeed in any logic within a particular range). These results also prompted us to undertake a more general investigation of intermediate logics whose atoms satisfy the double negation law. Furthermore, we showed how the original `pair' version of inquisitive semantics can be understood as one of a hierarchy of specializations of the `generalized' semantics we discussed, and argued in favour of the generalized system. Finally, we turned to the task of extending inquisitive semantics to a firstorder language and found that a straightforward generalization of our propositional approach was not viable due to the absence of certain maximal states. In order to overcome this difficulty, we proposed a variant of the semantics, which we called inquisitive possibility semantics, based on an inductive definition of possibilities. We examined the resulting system, arguing that it retains most of the properties of the semantics discussed in the previous chapters, including the logic, and we proposed a possible way to interpret the additional aspects of meaning that appear in the new semantics. We discussed the distinction between entailment and strong entailment and gave a sound and complete axiomatization of the latter notion as well. We showed that possibility semantics can be extended naturally to the predicate case, tested the predictions of the resulting system and found them satisfactory, especially in regard to the treatment of issues and information, and we saw that the system comprehends Groenendijk's logic of interrogation as a special case. We concluded sketching some features of the associated predicate logic. The semantics we have discussed are new, in fact completely new in the case of possibility semantics and its first-order counterpart. I hope to have managed to provide some evidence of their great potential for linguistic applications: in a very simple system and without any ad-hoc arrangement, we can deal with phenomena such as polar, conditional and who questions, inquisitive usage of indefinites and disjunction, perhaps even might statements, all of this in symbiosis with the classical treatment of information. For obvious reasons, in this thesis we have limited ourselves to remarking that these phenomena can be modelled: of course, a great deal of work remains to be done in order to understand what account each of them is given in inquisitive semantics. Aspects that may be worth particular consideration are the notions of answerhood and compliance that the semantics gives rise to, as well as the type of pragmatic inferences it justifies. Also, the role of suggestive possibilities (if any) and their relations to natural language constructions such as might and perhaps has to be clarified. From the logical point of view, natural directions of research are a more in-depth study of inquisitive logic and strong entailment in the first-order case, possibly leading up to a syntactic characterization. Beyond the borders of inquisitive semantics, a further possible stream of research along the lines of chapter 5 would be to study the behaviour of intermediate logics with atoms satisfying special classical properties (say, the Scott formula or the G�odel-Dummett formula) or perhaps even arbitrary properties. The resulting objects would be weak logics (weak intermediate logics in the case of classical properties) and may therefore be studied by means of constructions analogous to those devised in chapters 3 and 5 for the particular case of the double negation property. Finally, we hope that the connections established here between Medvedev's logic and decidable logics such as ND and KP may serve as a useful tool to cast some light on this ever-mysterious intermediate logic and on the long-standing issue of its decidability.

Item Type: Report
Report Nr: MoL-2009-11
Series Name: Master of Logic Thesis (MoL) Series
Year: 2009
Date Deposited: 12 Oct 2016 14:38
Last Modified: 12 Oct 2016 14:38

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