On intermediate inquisitive and dependence logics: An algebraic study

Algebraizable Weak Logics

“…However, for any DNA-logic L = CPC it is the case that ¬¬p → p ∈ L ¬ , but (¬¬(p ∨ ¬p) → p ∨ ¬p) / ∈ L ¬ , showing that DNA-logics are not standard logics. DNA-logics can be further generalised to χ-logics, defined in [30], which offer another non-trivial example of weak logics.…”

“…Inquisitive logic and dependence logic are two related propositional systems, which are usually both defined in terms of so-called team semantics, originally introduced by Hodges in [20]. Here we introduce both of them in syntactic terms, analogously as in [30], and we investigate whether they are algebraizable or not in the sense provided by this article. We refer the reader to [8,36] for the standard presentation of inquisitive and dependence logic by their team semantics.…”

“…An algebraic semantics for the classical version of inquisitive logic InqB was introduced in [1, 2, 29]. Such semantics was generalized in [30] to provide a sound and complete algebraic semantics to InqI, InqB ⊗ , InqI ⊗ and other intermediate versions of inquisitive and dependence logic as well. Since these logical systems do not satisfy the rule of uniform substitution, it has so far been an open question whether such semantics are in any sense unique.…”

“…However, it can be checked using the semantics from [10] or [30] that ¬¬x / ∈ InqI, ¬¬x → x / ∈ InqI, x / ∈ InqI, ¬x / ∈ InqI, ⊥ / ∈ InqI. Otherwise, if ρ j (x) = ⊤, it follows that core(A) = A for all A ∈ Q, contradicting the fact that InqI is not closed under uniform substitution.…”

“…The behaviour of these logical systems prevented so far a uniform and general study of these logics, and did not allow for immediately applying the facts and results from abstract algebraic logic, which hold only for logical systems in Tarski's strong sense. For example, algebraic semantics for several versions of inquisitive and dependence logics were introduced in [1,2,27,30], but it remained an open question whether these semantics are in any sense unique. In fact, it is a seminal result from abstract algebraic logic that the equivalent 1 algebraic semantics of a logic is unique, e.g.…”