Undefinability in Inquisitive Logic with Tensor

Abstract

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.

Characterizing counterfactuals and dependencies over causal and generalized causal teams

“…This proposition implies that causal discourse cannot be entirely reduced to the (propositional) pure team setting, in analogy with the motto that causation cannot be explained in terms of correlation. 14 A further simple consequence of the above proposition is the undefinability of in terms of the other connectives (see [29,8] for details on the notion of definability of connectives in the context of team semantics).…”

“…It is easy to verify that α ⊃ ϕ ≡ ¬α ∨ ϕ in the languages considered in this paper. 8 We thus treat the selective implication α ⊃ ϕ as a shorthand in our logics. The selective implication generalizes material implication in the sense that it behaves in the same way on singleton causal teams; i.e., ({s},…”

An Epistemic Interpretation of Tensor Disjunction

“…Following [6], we introduce the language and semantics of Inquisitive Logic with Tensor Disjunction (InqB ⊗ ). In contrast with [6], we use the symbol ∨ for the inquisitive disjunction and adopt the model-based semantics as in [4]. Throughout the paper, we fix a countable set P of proposition letters.…”

“…The initial idea is based on an unexpected connection between the tensor disjunction and the so-called weak disjunction in Medvedev's early work [13] on the problem semantics of intuitionistic logic, following Kolmogorov's problem-solving interpretation [12]. This connection is best exposed in the setting of inquisitive logic with tensor disjunction discussed in [6], since inquisitive logic has intimate connections with both the propositional dependence logic [22] and Medvedev's logic [8]. More specifically, various versions of propositional dependence logic can be viewed as the disguised inquisitive logic, e.g., the dependence atom =(p, q) becomes (p ∨ ¬p) → (q ∨ ¬q) [20,22,5].…”

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This paper aims to give an epistemic interpretation to the tensor disjunction in dependence logic, through a rather surprising connection to the so-called weak disjunction in Medvedev's early work on intermediate logic under the Brouwer-Heyting-Kolmogorov (BHK)-interpretation. We expose this connection in the setting of inquisitive logic with tensor disjunction InqB ⊗ discussed by [6], but from an epistemic perspective. More specifically, we translate the propositional formulae of InqB ⊗ into modal formulae in a powerful epistemic language of knowing how following the proposal by [19,16].

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