Algebraic foundations for the semantic treatment of inquisitive content

Abstract

In classical logic, the proposition expressed by a sentence is construed as a set of possible worlds, capturing the informative content of the sentence. However, sentences in natural language are not only used to provide information, but also to request information. Thus, natural language semantics requires a logical framework whose notion of meaning does not only embody informative content, but also inquisitive content. This paper develops the algebraic foundations for such a framework. We argue that propositions, in order to embody both informative and inquisitive content in a satisfactory way, should be defined as non-empty, downward closed sets of possibilities, where each possibility in turn is a set of possible worlds. We define a natural entailment order over such propositions, capturing when one proposition is at least as informative and inquisitive as another, and we show that this entailment order gives rise to a complete Heyting algebra, with meet, join, and relative pseudo-complement operators. Just as in classical logic, these semantic operators are then associated with the logical constants in a first-order language. We explore the logical properties of the resulting system and discuss its significance for natural language semantics. We show that the system essentially coincides with the simplest and most well-understood existing implementation of inquisitive semantics, and that its treatment of disjunction and existentials also concurs with recent work in alternative semantics. Thus, our algebraic considerations do not lead to a wholly new treatment of the logical constants, but rather provide more solid foundations for some of the existing proposals.

Information, Issues, and Attention

“…At that time, the framework of inquisitive semantics that the paper builds on still stood in its infancy. In the meantime, the framework has become more mature, and our general perspective on it has changed in certain respects (see, e.g., Ciardelli et al, 2012Ciardelli et al, , 2013aRoelofsen, 2013). However, these changes in perspective are largely orthogonal to the main proposal that is made in the present paper, which is to generalize the basic notion of meaning that is provided by inquisitive semantics in order to capture attentive content as well as informative and inquisitive content.…”

“…See Groenendijk (2009);Mascarenhas (2009) for a different formulation, and Ciardelli (2009) ;Ciardelli and Roelofsen (2011);Ciardelli et al (2013b) for arguments in favor of the former. Also, see Ciardelli et al (2012Ciardelli et al ( , 2013a; Roelofsen (2013) for a more recent perspective on the system proposed in Ciardelli (2009) ;Groenendijk and Roelofsen (2009). 2 The idea that a semantic analysis of might sentences should capture their potential to draw attention to certain possibilities is not new.…”

Composing alternatives

“…In particular, it is anti-symmetric, which means that any two expressions that are logically equivalent express the same meaning, as desired. Furthermore, as shown in Roelofsen (2013), the space of inquisitive meanings ordered by entailment forms a complete Heyting algebra, just like the space of classical propositions ordered by classical entailment. This means in particular that two inquisitive meanings P and Q always have:…”

“…However, as discussed in Roelofsen (2013), there is a fundamental problem with this notion. Namely, entailment defined in this way does not amount to a partial order on the space of meanings.…”

Lifting conditionals to inquisitive semantics

“…In our approach, disjunction is not treated as a special connective; rather, all connectives are taken to operate on inquisitive meanings, rather than on truth conditions. This allows us to retain a logically well-behaved theory of propositional connectives (Roelofsen 2013), avoiding some thorny issues that arise in the theory underlying Alonso-Ovalle's account (see Ciardelli & Roelofsen 2015a).…”