21 research outputs found
Undefinability in Inquisitive Logic with Tensor
Logics based on team semantics, such as inquisitive logic and dependence logic, are not closed under uniform substitution. This leads to an interesting separation between expressive power and definability: it may be that an operator O can be added to a language without a gain in expressive power, yet O is not definable in that language. For instance, even though propositional inquisitive logic and propositional dependence logic have the same expressive power, inquisitive disjunction and implication are not definable in propositional dependence logic. A question that has been open for some time in this area is whether the tensor disjunction used in propositional dependence logic is definable in inquisitive logic. We settle this question in the negative. In fact, we show that extending the logical repertoire of inquisitive logic by means of tensor disjunction leads to an independent set of connectives; that is, no connective in the resulting logic is definable in terms of the others.Peer reviewe
An Acceptance Semantics for Stable Modal Knowledge
We observe some puzzling linguistic data concerning ordinary knowledge
ascriptions that embed an epistemic (im)possibility claim. We conclude that it
is untenable to jointly endorse both classical logic and a pair of intuitively
attractive theses: the thesis that knowledge ascriptions are always veridical
and a `negative transparency' thesis that reduces knowledge of a simple negated
`might' claim to an epistemic claim without modal content. We motivate a
strategy for answering the trade-off: preserve veridicality and (generalized)
negative transparency, while abandoning the general validity of contraposition.
We survey and criticize various approaches for incorporating veridicality into
domain semantics, a paradigmatic `information-sensitive' framework for
capturing negative transparency and, more generally, the non-classical behavior
of sentences with epistemic modals. We then present a novel
information-sensitive semantics that successfully executes our favored
strategy: stable acceptance semantics.Comment: In Proceedings TARK 2023, arXiv:2307.0400
Proof Theory for Positive Logic with Weak Negation
Proof-theoretic methods are developed for subsystems of Johansson's logic
obtained by extending the positive fragment of intuitionistic logic with weak
negations. These methods are exploited to establish properties of the logical
systems. In particular, cut-free complete sequent calculi are introduced and
used to provide a proof of the fact that the systems satisfy the Craig
interpolation property. Alternative versions of the calculi are later obtained
by means of an appropriate loop-checking history mechanism. Termination of the
new calculi is proved, and used to conclude that the considered logical systems
are PSPACE-complete
An Inquisitive Perspective on Modals and Quantifiers
Inquisitive semantics enriches the standard truth-conditional notion of meaning, in order to facilitate an integrated semantic analysis of statements and questions. Taking this richer view on meaning as a starting point, this review presents a new perspective on modal operators and quantifiers, one that has the potential to address a number of challenges for standard semantic analyses of such operators. To illustrate the new perspective, we present an inquisitive take on the semantics of attitude verbs and on quantifiers taking scope out of questions
Revisiting Semilattice Semantics
The operational semantics of Urquhart is a deep and important part of the development of relevant logics. In this paper, I present an overview of work on Urquhartâs operational semantics. I then present the basics of collection frames. Finally, I show how one kind of collection frame, namely, functional set frames, is equivalent to Urquhartâs semilattice semantics
An Algebraic Approach to Inquisitive and DNA-Logics
This article provides an algebraic study of the propositional system InqB of inquisitive logic. We also investigate the wider class of DNA-logics, which are negative variants of intermediate logics, and the corresponding algebraic structures, DNA -varieties. We prove that the lattice of DNA-logics is dually isomorphic to the lattice of DNA -varieties. We characterise maximal and minimal intermediate logics with the same negative variant, and we prove a suitable version of Birkhoff's classic variety theorems. We also introduce locally finite DNA -varieties and show that these varieties are axiomatised by the analogues of Jankov formulas. Finally, we prove that the lattice of extensions of InqB is dually isomorphic to the ordinal omega + 1 and give an axiomatisation of these logics via Jankov DNA -formulas. This shows that these extensions coincide with the so-called inquisitive hierarchy of [9].(1)Peer reviewe
Stable Acceptance for Mighty Knowledge
Drawing on the puzzling behavior of ordinary knowledge ascriptions that embed an epistemic (im)possibility claim, we tentatively conclude that it is untenable to jointly endorse (i) an unfettered classical logic for epistemic language, (ii) the general veridicality of knowledge ascription, and (iii) an intuitive ânegative transparencyâ thesis that reduces knowledge of a simple negated âmightâ claim to an epistemic claim without modal content. We motivate a strategic trade-off: preserve veridicality and (generalized) negative transparency, while abandoning the general validity of contraposition. We criticize various approaches to incorporating veridicality into domain semantics, a paradigmatic âinformation-sensitiveâ framework for capturing negative transparency and, more generally, the non-classical behavior of sentences with epistemic modals. We then present a novel information-sensitive semantics that successfully executes our favored strategy: stable acceptance semantics, extending a vanilla bilateral state-based semantics for epistemic modals with a knowledge operator loosely inspired by the defeasibility theory of knowledge