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Modal Extensions of CLoN with Bi-neighborhood Semantics

This paper introduces a bi-neighborhood semantics for non-normal modal extensions of the sublogic CLoN of FDE, demonstrating how to validate non-trivial axioms involving weak negation to construct deontic logics that accommodate both standard principles and moral dilemmas without trivialization.

Original authors: Mahan Vaz, Daniel Skurt

Published 2026-06-30

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CC BY 4.0

Original authors: Mahan Vaz, Daniel Skurt

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to build a rulebook for a very strange, chaotic city. In this city, the usual rules of logic don't always work. Sometimes, a statement can be both true and false at the same time (like a traffic light that is red and green simultaneously), or it might be neither true nor false (like a traffic light that is broken and unlit).

This paper is about creating a new, flexible rulebook for this chaotic city, specifically for handling obligations (what people must do) and possibilities (what people can do), even when the city is full of contradictions.

Here is the breakdown of the paper's ideas using simple analogies:

1. The Foundation: A City with "Glitchy" Logic

The authors start with a base logic called CLoN. Think of this as a city where the rules for "AND," "OR," and "IF... THEN" work perfectly, just like in our normal world. However, the rule for NOT (negation) is broken or "glitchy."

The Problem: In normal logic, if you know "It is raining," you automatically know "It is NOT sunny." In this glitchy city, knowing "It is raining" doesn't automatically tell you anything about "It is NOT sunny." The "NOT" button is loose; it doesn't have a fixed rule for what happens when you press it.
The Goal: The authors want to add "Modal" operators (like Must and Can) to this glitchy city without breaking it further. They want to say things like "You must stop" or "It is possible to go," even when the logic of "stopping" or "going" is messy.

2. The Solution: Two Separate Neighborhoods

To handle this mess, the authors invent a new way of looking at the city called Bi-neighborhood Semantics.

Imagine every person in the city has two separate lists of neighbors:

The Verification List (The "Yes" Neighborhood): This list contains all the places where a statement is considered True.
The Rejection List (The "No" Neighborhood): This list contains all the places where a statement is considered False (or rejected).

In normal logic, these two lists are perfect opposites. If a place is on the "Yes" list, it cannot be on the "No" list. But in this paper's city, these lists are independent.

A place can be on the "Yes" list but not on the "No" list (True but not False).
A place can be on the "No" list but not on the "Yes" list (False but not True).
A place can be on both lists (True AND False).
A place can be on neither list (Neither True nor False).

This independence is the key. Because the "NOT" operator is glitchy, the authors treat the "Yes" and "No" lists as separate rooms. They don't force them to be mirror images of each other. This allows them to define "Must" and "Can" without needing to know exactly how "NOT" behaves.

3. Building the Rules (The Axioms)

The authors show that even with this glitchy "NOT," they can build a solid system for obligations and possibilities.

The "Must" Operator (Necessity): They define "You must do X" by looking at the "Yes" neighborhood. If all the neighbors in your "Yes" list agree that X is happening, then you "Must" do X.
The "Can" Operator (Possibility): They define "You can do X" independently, using a separate set of rules for the "Yes" neighborhood.

Crucially, they show that you can add standard rules (like "If you must do A and must do B, you must do A and B") without causing the whole system to collapse into nonsense, even if the "NOT" part is messy.

4. The Real-World Use: Moral Dilemmas

The paper argues that this specific type of logic is perfect for solving Moral Dilemmas.

The Scenario: Imagine a person, let's call her Sarah, who faces a terrible choice.

She has a moral duty to save her child (Obligation A).
She has a moral duty to save her spouse (Obligation B).
But, she physically cannot save both at the same time.

In normal logic, this creates a paradox. If she must save the child and must save the spouse, but can't do both, the system breaks down. It leads to "trivialization," meaning the logic says, "Since you failed, you might as well commit a crime, or the universe explodes."

The Paper's Solution:
Using their "Bi-neighborhood" system, Sarah's situation doesn't break the logic.

The system accepts that she has a conflict of obligations.
It allows the statement "Sarah must save the child" to be True AND "Sarah must save the spouse" to be True at the same time, even though they contradict each other in reality.
Because the logic is designed to handle "glitches" (contradictions) without exploding, it can model this moral tragedy without saying that Sarah is a criminal or that the universe ends. It simply acknowledges the dilemma exists and is unsolvable, without forcing a "wrong" answer.

Summary

The authors have built a new mathematical toolkit (a logic system) that treats "True" and "False" as two separate, independent lists. This allows them to create rules for "Must" and "Can" that work even when the concept of "Not" is broken. They prove this works mathematically and suggest it is the best way to formally understand moral dilemmas, where people are forced to choose between two conflicting duties, without the logic system falling apart.

Technical Summary: Modal Extensions for CLoN with Bi-neighborhood Semantics

Problem Statement
The paper addresses the challenge of constructing non-normal modal logics based on CLoN (Classical Logic both gappy and glutty with respect to Negation), a subsystem of FDE (First Degree Entailment). CLoN is characterized as positive classical logic augmented with a negation operator ( $\neg$ ) that lacks specific axioms or semantic falsity conditions. Consequently, $\neg$ in CLoN is not congruential (it does not satisfy the replacement property), and the logic is both paraconsistent and paracomplete.

Existing approaches to modal extensions of CLoN-based systems often assume strong axioms (such as Axiom K) from the outset or utilize Kripke semantics, which may be too restrictive for specific philosophical applications like deontic logic. Furthermore, standard duality between necessity ( $\Box$ ) and possibility ( $\Diamond$ ) cannot be defined in CLoN due to the lack of falsity conditions for negation. The authors identify a gap in the literature regarding a systematic, minimal neighborhood semantics for CLoN that accommodates its weak negation without trivializing the system, particularly for applications involving moral dilemmas where conflicting obligations must be handled without explosion.

Methodology
The authors employ bi-neighborhood semantics, a framework adapted from previous work on FDE-based non-normal modal logics (Dalmonte et al., 2018; Drobyshevich and Skurt, 2021). The methodology involves the following key components:

Base Logic (CLoN, CLuN, CLaN): The propositional base is defined via a Hilbert calculus containing positive classical axioms but no rules for negation. Semantically, the authors utilize swap structures (non-deterministic matrices) where truth values are pairs $(v_1, v_2)$ representing verification and rejection conditions.
- CLoN: The base system with no constraints on the relationship between verification and rejection.
- CLuN: Obtained by adding the Law of Excluded Middle ( $\phi \lor \neg \phi$ ), enforcing that every world verifies or rejects a formula (paraconsistent).
- CLaN: Obtained by adding the Principle of Explosion ( $\phi \to (\neg \phi \to \psi)$ ), enforcing that no world both verifies and rejects a formula (paracomplete).
Bi-neighborhood Frames: The modal logic CLoNE is defined using frames $\mathcal{W} = \langle W, N^+_2, N^+_3 \rangle$ (and later extended with $N^-_2, N^-_3$ ).
- Verification Sets ( $[\phi]^+$ ): Defined via a neighborhood function $N^+$ , where $w \in [\Box \phi]^+$ iff $[\phi]^+ \in N^+_2(w)$ .
- Rejection Sets ( $[\phi]^-$ ): Defined independently via a separate neighborhood function $N^-$ . Crucially, the rejection set of a modal formula is not derived from the verification set of its negation (due to the weak negation). Instead, $[\Box \phi]^-$ is defined as $\{w \in W \mid [\phi]^- \in N^-_2(w)\}$ .
- Independence: The operators $\Box$ and $\Diamond$ are defined independently rather than as duals. This allows the logic to handle negated modalities (e.g., $\neg \Box \phi$ ) without requiring standard duality axioms.
Extensionality Rules: The system adopts the extensionality rule ( $\phi \leftrightarrow \psi / \Box \phi \leftrightarrow \Box \psi$ ) for the positive fragment but acknowledges that weak negation prevents congruence for negated formulas. To handle negated modalities, the system introduces a specific extensionality rule for negation: $\neg \phi \leftrightarrow \neg \psi / \neg \Box \phi \leftrightarrow \neg \Box \psi$ .
Proof Systems: The authors establish soundness and completeness for CLoNE and its extensions with respect to the class of bi-neighborhood frames. They utilize canonical models constructed from prime theories to prove completeness.

Key Contributions and Results

Formalization of CLoNE: The paper presents CLoNE, the minimal non-normal modal expansion of CLoN, equipped with a sound and complete bi-neighborhood semantics.
Handling Weak Negation: The framework successfully validates non-trivial axioms containing the weak negation operator by defining independent rejection sets for modal operators. This allows for the expression of negated modalities without collapsing into classical duality.
Axiomatic Extensions: The authors characterize various axiomatic extensions (M, C, K, T, 4, D, etc.) in terms of frame properties.
- Positive Extensions: Standard axioms like M (Augmentation) and C (Closure) correspond to properties of the verification neighborhood functions.
- Negated Extensions: New axioms involving negation (e.g., $M^-$ , $C^-$ , $T^-$ ) are shown to correspond to specific interactions between verification and rejection neighborhood functions.
Multimodal Capabilities: The framework supports multimodal systems where different modalities (e.g., obligation and possibility) are governed by independent neighborhood functions, connected only by specific bridge principles (e.g., interaction axioms).
Completeness: The paper proves that CLoNE and its canonical extensions are sound and complete with respect to their respective classes of frames.

Significance and Claims
The authors claim that this work provides the necessary semantic foundation for deontic logics capable of accommodating moral dilemmas without trivialization.

Moral Dilemmas: In standard deontic logics, the existence of genuine moral dilemmas (conflicting obligations that cannot be resolved) often leads to logical explosion (triviality) when combined with standard principles like Aggregation (Axiom C) or the Principle of Explosion.
Paraconsistency: By basing the system on CLuN (an extension of CLoN), the framework allows for the coexistence of conflicting obligations ( $O\phi$ and $O\neg\phi$ ) without deriving arbitrary conclusions.
Independence of Modalities: The bi-neighborhood approach allows for the modeling of distinct modal concepts (e.g., obligation vs. metaphysical possibility) that do not rely on classical duality. This is crucial for formalizing scenarios where obligations are not defeasible and standard logical resolutions fail.
Philosophical Utility: The paper argues that while the weak negation limits certain classical inferences, it is precisely this feature that allows the system to model "unsolvable" deontic situations naturally, avoiding the "artificial" introduction of consistency operators found in other paraconsistent deontic logics (like LFIs).

The paper concludes that the proposed framework offers a flexible, mathematically rigorous tool for exploring non-standard philosophical positions, particularly those involving contradictions in moral reasoning, while maintaining a minimal commitment to modal axioms.