Implicit and Explicit Stances in Logic

Abstract

We identify a pervasive contrast between implicit and explicit stances in logical analysis and system design. Implicit systems change received meanings of logical constants and sometimes also the notion of consequence, while explicit systems conservatively extend classical systems with new vocabulary. We illustrate the contrast for intuitionistic and epistemic logic, then take it further to information dynamics, default reasoning, and other areas, to show its wide scope. This gives a working understanding of the contrast, though we stop short of a formal definition, and acknowledge limitations and borderline cases. Throughout we show how awareness of the two stances suggests new logical systems and new issues about translations between implicit and explicit systems, linking up with foundational concerns about identity of logical systems. But we also show how a practical facility with these complementary working styles has philosophical consequences, as it throws doubt on strong philosophical claims made by just taking one design stance and ignoring alternative ones. We will illustrate the latter benefit for the case of logical pluralism and hyper-intensional semantics.

Compositionality in Context

“…[20] on algebraizability of logics), abstract recursion theory (compositionality as computability, [59]) and in particular, category theory, with Samson Abramsky's long-standing work as a prominent instance. Bits and pieces of relevant formal theory can also be found elsewhere, for instance in the extensive body of work on translations between logical systems, [24], [79], especially, when viewing giving a semantics as a form of translation between object and meta-language.…”

“…By contrast, epistemic logic has explicit operators for knowledge of agents, but with these in place, the base logic can remain classical, based on the traditional notion of truth. A general study of the 'implicit' versus the 'explicit' methodology, and the many new questions to which this gives rise, is made in [79].…”

“…A concrete case continues with the earlier topic of generalized first-order semantics in Section 6.3. As an analysis of dependence, the logic CRS is 'implicit', [79]. It merely reinterprets the existing language of first-order logic, and locates information about (in-)dependence of variables in failures of classical laws.…”

Inquisitive Logic as an Epistemic Logic of Knowing How

“…Combining this technique with some formalized BHK-interpretation of intuitionistic logic may allow us to turn intuitionistic logic and various intermediate logics into an epistemic logic of knowing how. The general method is to use a powerful epistemic language based on classical logic to unload the implicit epistemic content hidden behind the propositional language of propositional intuitionistic logic, foreseen by Hintikka and van Benthem viewing intuitionistic logic as an implicit epistemic logic [17,29]. For example, an intuitionistic logic formula α is first turned into a knowhow formula Khα in our setting; then depending on the structure of α and the BHK-interpretation, we can further decode α by reducing its complexity within the logical language, e.g., Kh(α ∨ β) can be decomposed to (Khα ∨ Khβ), where the connectives outside the scope of Kh are classical.…”

Modal Information Logics: Axiomatizations and Decidability

“…They do so by extending the language of propositional logic with a binary modality defined in terms of being the supremum of two states. First proposed in 1996, MILs have been around for some time, yet not much is known: (van Benthem 2017(van Benthem , 2019 pose two central open problems, namely (1) axiomatizing the two basic MILs of suprema on preorders and posets, respectively, and (2) proving (un)decidability. The main results of the first part of this paper are solving these two problems: (1) by providing an axiomatization [with a completeness proof entailing the two logics to be the same], and (2) by proving decidability.…”