Abstract
In 1989, Plaza introduced the “knowing value” operator \(\textsf{Kv}_id\) to characterize the “Mr. Sum and Mr. Product” puzzle in propositional modal logic. Previous research had primarily focused on the \(\textsf{Kv}d\) operator, which captures the idea of knowing the value of a designator d. This paper expands the scope of application for the \(\textsf{Kv}\) operator beyond designators to include predicates, interpreting the \(\textsf{Kv}P\) operator as denoting knowledge of the value of a predicate P. Additionally, we present two distinct semantics - MS (Mention-Some) semantics and MA (Mention-All) semantics - for the \(\textsf{Kv}P\) operator, and prove the strong completeness theorem for two axiom systems containing only the \(\textsf{Kv}P\) operator, as well as two axiom systems containing both the \(\textsf{Kv}P\) and \(\textsf{Kv}d\) operators.
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Notes
- 1.
However, the mention-all interpretation proposed by Groenendijk and Stokhof does not imply exhaustiveness, which means the respondent of a question may contain false belief. For example, suppose there are only two cafes in 2 km, Starbucks and Costa. The answer “Starbucks, Costa and Peet’s” may also count as an answer of the question “What is the value of the predicate ‘cafes within 2 km’?” under Groenendijk and Stokhof’s mention-all interpretation. To avoid this, in this article, when we refer to “mention-all interpretation”, we actually mean the “strongly exhaustive interpretation” proposed in [10]. In this situation, the formulation of mention-all interpretation should be \(\forall x(\hat{K}Px\rightarrow \textsf{K}_IPx)\).
- 2.
It should be particularly noted that for the 0-ary predicate \(P^0\), \(V(P^0,w)\subseteq O^0=\{\varnothing \}\).
- 3.
As for 0-ary predicate \(P^0\), the semantics for formulas with \(P^0\) are:
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\(\mathcal {M},w\models _{\text {MS}} P^0\) if and only if \(\mathcal {M},w\models _{\text {MA}} P^0\), if and only if \(\varnothing \in V(P,w)\),
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\(\mathcal {M},w\models _{\text {MS}}\textsf{Kv}P\) if and only if for every possible world \(t\sim w\), we have \(\varnothing \in V(P,t)\),
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\(\mathcal {M},w\models _{\text {MA}}\textsf{Kv}P\) if and only if for any possible world \(t\sim w\), we have \(V(P,t)=V(P,w)\).
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- 4.
However, this does not imply that one operator has stronger expressive power than the other in MS semantics. Although the \(\textsf{Kv}\) operator can be combined with 0-ary predicates, it cannot express combinations of predicates within its scope (such as conjunction \(\wedge \), disjunction \(\vee \), etc.). Meanwhile, although logical connectives can be used within the scope of the \(\textsf{K}\) operator, we cannot express sentences such as “knowing the value of an n-ary predicate P” only using \(\textsf{K}\).
- 5.
From a semantic perspective, if there exist \(d_1,...,d_n\) such that \(\mathcal {M},w\vDash _{\text {MS}}\textsf{K}Pd_1...d_n\), then since \(d_i\) is a nonrigid designator, we may not have \(\mathcal {M},w\vDash _{\text {MS}}\textsf{Kv}P\).
- 6.
It should be noted that we do not need to construct possible world copies for MAP-Henkin sets here. This is because the definition of MAP-Henkin set ensures that every \(\lnot \textsf{Kv}P\in s\) in an MAP-Henkin set s has a corresponding MAP witness \(\hat{\textsf{K}} \lnot Pd_1...d_n\wedge \hat{\textsf{K}} Pd_1...d_n\in s\), which naturally leads to at least two distinct successor possible worlds of s.
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Hong, B. (2023). Knowing the Value of a Predicate. In: Alechina, N., Herzig, A., Liang, F. (eds) Logic, Rationality, and Interaction. LORI 2023. Lecture Notes in Computer Science, vol 14329. Springer, Cham. https://doi.org/10.1007/978-3-031-45558-2_12
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