💻 computer science

Stratified Counterpossible Logic

This paper introduces the stratified logic systems SCP and SCP1, which distinguish between logical and metaphysical impossibility to avoid vacuous counterpossible truths while proving their soundness, completeness, and decidability.

Original authors: Chen Huang (Institute of Logic,Cognition, Department of Philosophy, Sun Yat-sen University), Xuefeng Wen (Institute of Logic,Cognition, Department of Philosophy, Sun Yat-sen University)

Published 2026-07-01

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CC BY 4.0

Original authors: Chen Huang (Institute of Logic,Cognition, Department of Philosophy, Sun Yat-sen University), Xuefeng Wen (Institute of Logic,Cognition, Department of Philosophy, Sun Yat-sen University)

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 a judge presiding over a courtroom of "What Ifs." Usually, these cases are easy. If someone asks, "What if I missed the bus?" you can easily imagine a world where you missed the bus and calculate the consequences (you're late, you're hungry, etc.).

But what happens when someone asks, "What if 2 + 2 = 5?" or "What if Hobbes squared the circle?" (a mathematically impossible task)?

For a long time, logicians argued that because these starting points are impossible, the answer to any question following them should be "True" by default. It's like saying, "Since the premise is nonsense, the whole sentence is automatically true." This is called Vacuism.

The authors of this paper, Chen Huang and Xuefeng Wen, say, "Wait a minute. That doesn't feel right."

Intuition 1: "If Hobbes had squared the circle, sick children in Afghanistan would have been cured." This feels False. Squaring a circle has nothing to do with curing children.
Intuition 2: "If Hobbes had squared the circle, we would have been surprised." This feels True. If someone did the impossible, we'd be shocked.

If the "Vacuism" rule were true, both sentences would be true, which ignores the difference between a silly connection and a logical one. The authors want to build a system that respects these differences.

The Problem: Mixing Up "Hard" and "Soft" Impossibilities

The authors argue that previous systems treated all "impossible" things as the same mess. They didn't distinguish between:

Metaphysical Impossibility: Things that break the rules of our universe but not the rules of math. (e.g., "Gold has atomic number 50" or "Water is not H2O").
Logical Impossibility: Things that break the fundamental rules of logic itself. (e.g., "A square circle" or "P and not-P").

The authors propose a Stratified Counterpossible Logic (SCP). Think of this as a three-story building where the rules change on each floor.

The Three Floors of the Building

The Top Floor (Metaphysical Possibility): This is our normal world and worlds that are slightly different but still make sense. Here, gold is gold, water is water, and the laws of physics hold.
The Middle Floor (Logic-Normal but Metaphysically Impossible): This is the "What If" zone. Here, we can imagine gold having atomic number 50. It's weird and breaks the laws of nature, but the laws of logic still work. We can still do math, and we can still say "If gold is 50, it's still a metal." The logic is intact; the chemistry is just different.
The Basement (Anti-Logical Worlds): This is the zone of pure chaos. Here, the rules of logic themselves break down. A statement can be both true and false at the same time. This is where "squaring the circle" lives.

The Key Innovation: The authors' system allows us to step down from the Top Floor to the Middle Floor to discuss "anti-essential" scenarios (like gold having a different number) without losing our logical minds. We can still reason clearly there. But if we go all the way to the Basement, the rules of logic collapse, and we have to be very careful about how we judge the results.

The Two Systems: SCP and SCP1

The paper introduces two versions of this logic:

1. SCP (The Basic System):
This system successfully separates the floors. It allows us to say that "If water were XYZ, it would still be wet" is a valid thought experiment because we are on the Middle Floor where logic still works. However, the authors noticed a tiny loophole: if you ask about a logical impossibility (like squaring the circle), the system could technically say, "Oh, there are no worlds in the Basement where this happens," and declare the statement "True" just because the list of worlds is empty. This is a form of the "Vacuism" they wanted to avoid.

2. SCP1 (The Upgraded System):
To fix the loophole, the authors created SCP1. They added a strict rule: The selection function cannot be empty.
Imagine a judge saying, "Even if the premise is impossible, you must find at least one scenario in the Basement to evaluate the consequence."

If you ask, "If Hobbes squared the circle, would children be cured?" SCP1 forces the system to look for a world where Hobbes squared the circle.
In that chaotic Basement world, since there is no logical link between "squaring circles" and "curing children," the system concludes the statement is False.
This prevents the system from cheating by saying "It's true because nothing exists."

Why This Matters

The paper proves that this new way of thinking is mathematically sound (it doesn't break), complete (it can answer all valid questions), and decidable (we can write a computer program to solve it).

The Takeaway:
The authors have built a better map for "Impossible Worlds."

Old maps said: "All impossible places are the same; nothing makes sense there."
This paper says: "No, there are different kinds of impossible places. Some are just weird (Middle Floor) where we can still think clearly. Others are chaotic (Basement) where logic breaks. And even in the chaotic basement, we shouldn't just give up and say everything is true; we should check if the 'What If' actually connects to the result."

By separating these layers, the authors allow us to have serious philosophical conversations about impossible scenarios without losing our logical footing.

Technical Summary: Stratified Counterpossible Logic (SCP and SCP1)

Problem Statement
The paper addresses the limitations of standard counterfactual semantics (e.g., Stalnaker-Lewis) when applied to counterpossibles—conditionals with impossible antecedents. Standard semantics often yield "vacuism," where all counterpossibles are trivially true because their antecedents are unsatisfiable. This conflicts with intuitive distinctions, such as the difference between "If Hobbes had squared the circle, children would be cured" (intuitively false) and "If Hobbes had squared the circle, we would be surprised" (intuitively true).

Furthermore, existing non-vacuist approaches (e.g., Nolan's impossible worlds) often treat all impossibilities as a monolithic category. This conflation of metaphysical impossibilities (e.g., "water is not H2O," which violates essential properties but obeys classical logic) with logical contradictions (e.g., $P \land \neg P$ ) creates a dilemma: systems that accommodate logical contradictions often sacrifice the inference rules necessary for reasoning about metaphysical hypotheses. The paper argues that current frameworks fail to sufficiently model the internal structure of the impossible domain, leading to a trade-off between logical strength and the ability to evaluate non-trivial counterpossibles.

Methodology
The authors propose a stratified semantic framework that partitions the logical space into three distinct layers:

Logic-Normal Worlds ( $N$ ): Worlds consistent with classical logic. This region is further subdivided by a metaphysical accessibility relation ( $R_M$ ) to distinguish between metaphysically possible and metaphysically impossible (but logically possible) scenarios.
Anti-Logical Worlds ( $I$ ): Worlds where the norms of classical logic fail (e.g., paraconsistent worlds where $\phi \land \neg \phi$ holds).
Selection Semantics: The system employs a selection function $f$ $f$ to evaluate conditionals. The function is constrained by the type of antecedent:
- If the antecedent is metaphysically possible, selection is restricted to $R_M$ -accessible worlds.
- If the antecedent is metaphysically impossible but logically possible, selection extends to $N$ but remains within $R_L$ (logical accessibility).
- If the antecedent is logically impossible, selection extends to $I$ .

The paper defines two systems:

SCP (Stratified Counterpossible Logic): The base system establishing the $N/R_M/I$ hierarchy. It validates standard classical logic within $N$ and allows for non-vacuous evaluation of metaphysical impossibilities.
SCP1: An extension of SCP designed to resolve the "vacuism" issue in the anti-logical domain. It imposes a non-empty constraint on the selection function ( $f(w, \phi) \neq \emptyset$ ) and treats logical accessibility ( $R_L$ ) as a universal relation over $N$ .

Key Contributions and Results

Stratified Semantics: The paper formalizes a three-layered model where metaphysical necessity is nested within logical necessity. This allows the system to evaluate "anti-essential" conditionals (e.g., "If gold had atomic number 50...") using classical logic within $N$ , while reserving $I$ for genuine logical contradictions.
Axiomatic Systems:
- SCP is defined with axioms for classical propositional logic, S5 modal operators for logical ( $\Box_L$ ) and metaphysical ( $\Box_M$ ) necessity, and specific conditional axioms (ID, SOC-M, SIC).
- SCP1 extends SCP with axioms (NE) and (SOC-L) to enforce non-vacuousness and universal logical accessibility.
Meta-Logical Properties: The authors provide rigorous proofs for both systems demonstrating:
- Soundness: The axiomatic systems are sound with respect to their respective semantic models.
- Completeness: Both systems are complete; any valid inference is derivable.
- Decidability: Both systems possess the finite model property, ensuring they are decidable. This is achieved through filtration techniques that construct finite models from canonical models.
Resolution of Vacuism: In SCP1, the non-empty constraint ensures that counterpossibles with impossible antecedents are not trivially true. Instead, their truth value depends on the substantive connection between the antecedent and consequent within the selected anti-logical worlds.

Significance
The paper claims that this stratified approach successfully distinguishes between different types of impossibility without sacrificing logical inference rules for metaphysical reasoning. By separating the domain of logical contradiction from metaphysical impossibility, the system allows for meaningful deduction in scenarios where essential properties are violated (e.g., scientific thought experiments) while maintaining a robust framework for handling logical contradictions.

The authors suggest that this hierarchical framework is generalizable to other modal domains, such as distinguishing primary and secondary obligations in deontic logic or modeling cognitive limitations in epistemic logic. The work provides a formal semantic path for resolving modal conflicts and characterizing the structure of impossibility, moving beyond the binary "possible vs. impossible" dichotomy.

The Problem: Mixing Up "Hard" and "Soft" Impossibilities

The Three Floors of the Building

The Two Systems: SCP and SCP1

Why This Matters

Technical Summary: Stratified Counterpossible Logic (SCP and SCP1)

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