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.
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Notes
- 1.
- 2.
For other undefinability results in the setting of dependence logic, see also [12, 13]. It is worth noting that, in the dependence logic literature, the standard notion of definability is called uniform definability; since there seems to be no special reason to add the qualification uniform (the notion of definability is intrinsically “uniform” in the relevant sense) we prefer to stick with the standard terminology.
- 3.
Truth-conditional formulas are called flat formulas in the dependence logic literature.
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Ciardelli, I., Barbero, F. (2019). Undefinability in Inquisitive Logic with Tensor. In: Blackburn, P., Lorini, E., Guo, M. (eds) Logic, Rationality, and Interaction. LORI 2019. Lecture Notes in Computer Science(), vol 11813. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-60292-8_3
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