Abstract
AbstractInquisitive first-order logic, InqBQ, is a system which extends classical first-order logic with formulas expressing questions. From a mathematical point of view, formulas in this logic express properties of sets of relational structures. This paper makes two contributions to the study of this logic. First, we describe an Ehrenfeucht–Fraïssé game for InqBQ and show that it characterizes the distinguishing power of the logic. Second, we use the game to study cardinality quantifiers in the inquisitive setting. That is, we study what statements and questions can be expressed in InqBQ about the number of individuals satisfying a given predicate. As special cases, we show that several variants of the question how many individuals satisfy $\alpha (x)$ are not expressible in InqBQ, both in the general case and in restriction to finite models.
Highlights
According to the traditional view, the semantics of a logical system specifies truth-conditions for the sentences in the language
The meaning of a sentence is laid out not by specifying when the sentence is true relative to a state of affairs, but rather by specifying when it is supported by a given state of information. This view allows us to interpret in a uniform way both statements and questions: for instance, the statement it rains will be supported by an information state s if the information available in s implies that it rains, while the question whether it rains will be supported by s if the information available in s determines whether or not it rains
In addition to standard first-order formulas like Pa and ∀x.Px, we have formulas like ?Pa (“does a have property P ?”), ∃ x.Px
Summary
According to the traditional view, the semantics of a logical system specifies truth-conditions for the sentences in the language. For all classical formulas α ∈ Lc, all models M, assignments g, and information states s: M,s |=g α ⇐⇒ ∀w ∈ s : Mw |=g α holds in first-order logic where g is the assignment mapping x to the ∼w-equivalence class of g(x).
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