Inquisitive Semantics and Intermediate Logics
Ivano A. Ciardelli
Abstract:
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.