Positive Noncontingency Logic

Qianli Zeng

Munich Center for Mathematical Philosophy (MCMP), LMU Munich

We introduce the positive fragment of noncontingency logic, provide an axiomatic system for the minimal system of it, and prove its completeness. By exploring the positive fragment, we isolate the fundamental behavior of the noncontingency operator without the interference of classical negation. We adapt the algebraic and theory-pair methods in Dunn (1995) for positive modal logic, and we rely on the almost-definability schema proposed by Fan and Wang to formulate the completeness proof.

Bundled fragments of modal logic are systems that take such bundle operators as primitives. For instance, in several bundled fragments of first-order modal logic, quantifiers and modalities can occur only together and cannot appear independently. Bundles like ∃x${\displaystyle ◻}$are treated as individual operators, while the separate operators ${\displaystyle ◻}$ and ∃ are not part of the language (Wang2017). Languages with bundles can often be viewed as fragments of richer modal languages. The bundle phenomenon has close connections to other areas. For example, in multi-agent epistemic logic, the bundling of quantifier and modality is illustrated by the distributed knowledge operator S_Gφ, which is read as “someone} in the group G knows φ”. In Computation Tree Logic (CTL), the operator EGφ is read as “there is a path on which φ holds forever”.
Other Examples of bundled formulas include:
φ∧${\displaystyle ◻}$φ, ${\displaystyle ◻}$◊φ,${\displaystyle ◻}$_aφ∧${\displaystyle ◻}$_b¬φ, ∃x${\displaystyle ◻}$ ,${\displaystyle ◻}$∀x

The bundle phenomenon appears everywhere in logic. An excellent example of this is the logic of ignorance (Hoek 2004) and its close relative, noncontingency logic (Fan 2015). In the logic of non-contingency, the formula “φ is non-contingent”, written as Δφ, semantically ${\displaystyle ◻}$φ${\displaystyle \vee }$${\displaystyle ◻}$¬φ, is defined as a unified bundle operator. In an epistemic setting, this captures the concept of “knowing whether”.

The formal study of noncontingency logic has a rich history, dating back to Montgomery and Routley (Montgomery1966). Since then, a variety of axiomatizations over various frame classes have been developed by Humberstone (Humberstone 1995), Kuhn (Kuhn 1995), and Zolin (Zolin 1999). Recently, the literature has been extensively expanded to capture richer modalities, including the logic of temporal contingency (Fan 2022a), dyadic contingency (Fan 2022b), unified logics for contingency and accident (Fan 2022c), von Wright’s deontic necessity (Fan 2022d), and neighborhood contingency logics (Fan 2019).

Parallel to the study of bundled constraints is the development of positive modal logic. J. Michael Dunn (Dunn 1995) formulated positive modal logic as a logic lacking both negation and implication, evaluating the fundamental behavior of modal operators over distributive lattices.

Despite these advancements, restricting noncontingency logic to its positive fragment, a system devoid of negation, remains unexplored. This restriction is theoretically interesting for two main reasons. First, the original motivation of noncontingency logic is to focus purely on the metaphysical concept of “noncontingency”. By not redundantly defining it through other concepts (like necessity), we characterize its behavior strictly via axioms. In the absence of negation, the positive fragment provides a purer environment to reveal the fundamental relationships and interactions between the noncontingent and contingent operators. Second, without negation, the duality of the noncontingent and contingent operators, as well as their self-dual properties (Δφ${\displaystyle \equiv }$Δ¬φ$ and $∇φ${\displaystyle \equiv }$∇¬φ), cannot be expressed directly in the syntax. Consequently, our axiomatization must capture these structural properties implicitly and indirectly.

We bridge these frameworks. We introduce the minimal system of positive fragment of noncontingency logic, K₊^Δ . We provide an axiomatic system and prove its completeness over the class of all frames. To achieve this, we adapt the structural foundation of Dunn’s (Dunn 1995) disjoint theory-pairs. For the canonical model construction, our completeness proof relies on the “almost-definability” schema introduced by Fan and Wang (Fan 2014, Fan 2015).

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