general setting to inquisitive logic. This semantics allows to apply methods of uni- versal algebra to studyDNA-logics and inquisitive logic from a novel perspective. Let
us briefly summarize our main results. In Chapter 3 we introducedDNA-logics and
their algebraic semantics and we gave two different proofs of the dual isomorphism
DNAL∼=opDNAVbetweenDNA-logics and DNA-varieties. In Chapter 4 we studied
closer the relation betweenDNA-logics and intermediate logics and we proved a suit-
able version of some classical results for the setting of DNA-varieties. In particular,
we showed that everyDNA-variety is generated by its regular subdirectly irreducible
members and that theDNA-logic of all Heyitng algebrasIPC¬is not locally finite. We
introduced a suitable version of Jankov formulas and we showed that this provides an axiomatisation of locally finite DNA-varieties. Finally, in Chapter 5 we used the
algebraic semantics of DNA-logics to study the inquisitive logic InqB. In particular,
we showed that the sublattice of its extensions is dually isomorphic to ω+ 1 and that it actually coincides with the inquisitive hierarchy studied in [10].
In addition to these results, we think that one of the main contributions of this thesis is that it provides a new setting for the study of inquisitive logic. The system InqB had so far been considered as the logic of the evaluation states or as
the negative variant of the logics between ND and ML – here we showed that one
can also consider InqB as the logic of a specific class of Heyting algebras, under
a suitable semantics. Most importantly, this new perspective at the propositional system of inquisitive logic allows us to raise new questions and consider new issues. We mention here some possible directions for future work, both concerning InqB
and the general theory ofDNA-logics.
From Negative Variants to Propositional VariantsIn this thesis we in-
troducedDNA-logics as the negative variant of some intermediate logicL. Every
DNA-logic Λ is thus such that Λ ={ϕ∈ LP :ϕ[¬p/p]∈L}for some intermediate logicL. A possible direction of future work is to study what happens if, instead of the negative substitution p 7→ ¬p, we consider the substitution p 7→ χ(p) for an arbitrary polynomial χ ∈ LP. In fact, it seems possible to extend at least part of the theory ofDNA-logics to this extended framework. In the case
of negative variants we rely on the fact that in intuitionistic logic¬¬¬p=¬p. This property however is shared in a more general form by every polynomial
χ. Ruitenberg’s Theorem [46, 47, 23] states that for any polynomial χ we can find a number n∈N such that χn =χn+2. This allows to introduce the χ-variant of an intermediate logicL asLχ={ϕ∈ LP :ϕ[χn(p)/p]∈L} and to generalize our study ofDNA-logics to arbitraryχ-variants. See for instance the upcoming [25].
From Inquisitive Logic to Dependence LogicIt was noticed recently that
there is a close relation between inquisitive logic and dependence logic. This connection has been studied e.g. in [9, 11] and suggests further directions of research. Similarly to inquisitive logic, the semantics of propositional de- pendence logic [51] consists of a set of possible valuations instead of a single valuation. Is it possible to adapt the algebraic semantics ofDNA-logics to obtain
an algebraic semantics for propositional dependence logic? A related question which is considered in [30] is what happens, both in inquisitive and dependence logic, if instead of starting with classical valuations we start with intuitionistic valuations. Is it possible to adapt the algebraic semantics developed in this thesis to this alternative setting?
From Jankov Formulas to Canonical Formulas In Section 4.3 we in-
troduced Jankov formulas for DNA-models and we showed that locally finite DNA-logics are axiomatised by these formulas. Is it possible to extend to the
setting ofDNA-logics other applications of Jankov formulas? For example, can
we prove using Jankov formulas that the lattice ofDNA-logic has the cardinality
of the continuum? Or can we extend Jankov formulas to subframe formulas, and in general to canonical formulas, as it is the case both for intermediate [1] and modal logics [2]? There are many ways in which one can use Jankov formulas to study Heyting algebras and it seems natural to extend them to the setting ofDNA-logics.
From Algebraic to Topological Semantics It is a well-known fact [19]
that Heyting algebras are dual to order-topological spaces known as Esakia spaces. This allows us to have both an algebraic and topological semantics for intermediate logics. In this thesis we did not look at possible connections to topology and we restricted our analysis to the algebraic setting. However, already in [3] a topological semantics forInqB is providedvia UV-spaces. Is it possible to generalize this semantics to arbitraryDNA-logics? Similarly, can we
define a suitable class of topological models forDNA-logics to obtain a general
duality betweenDNA-models based on Heyting algebras andDNA-models based
on Esakia spaces? A related issue concerns the characterisation of finite regular subdirectly irreducible Heyting algebras. We know by duality that a finite subdirectly irreducible Heyting algebra is the upset algebra of a finite rooted frame. Can we obtain a similar characterisation for regular finite subdirectly irreducible Heyting algebras? What properties should a rooted frame satisfy in order for its dual Heyting algebra to be regular?
From InqB to a Theory of DNA-Logics Finally, it is worth mentioning that
there are still many open questions concerning DNA-logics and their relations
them. First, in Section 4.1 we have studied the connections betweenDNA-logics
and intermediate logics which they are negative variants of. It is an important result proved in [10] that IPC is a DNA-maximal logic. Therefore, since IPC
is obviously also a DNA-minimal logic it follows that it is both DNA-maximal
and DNA-minimal. Is IPC the only intermediate logic to be DNA-maximal and DNA-minimal? Can we find other logics with this property? Secondly, we have
seen in Section 5.2 that the extensions of InqB are linearly ordered. Is this
a feature shared by otherDNA-logics or is this a property which is specific for InqB? Finally, the example ofInqBalso shows that aDNA-logic Λ can be locally
finite even if all the intermediate logics inI(Λ) are not. The locally finiteness
and the finite model property ofDNA-logics are thus interesting properties that
can be investigated further. For instance, one could try to define suitable notions of filtrations for these logics and to introduce suitable classes of stable logics [31]. We leave these and possibly other interesting questions for future work.