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Hyperformalism for Relevant Modal Logics

This paper extends the concept of hyperformalism to relevant modal logics by introducing MPos-hyperformalism, proving that the weak logic B-Box possesses this property, investigating its closure under specific non-uniform substitutions, refining the variable sharing property, and defining K-MPos as the largest MPos-hyperformal sublogic of classical modal logic K.

Original authors: Thomas Macaulay Ferguson (Rensselaer Polytechnic Institute), Shay Allen Logan (Kansas State University)

Published 2026-07-01

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CC BY 4.0

Original authors: Thomas Macaulay Ferguson (Rensselaer Polytechnic Institute), Shay Allen Logan (Kansas State 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 strict librarian in a library of logic. In this library, every book (or formula) is made of sentences built from basic building blocks called "atoms" (like $p$ , $q$ , $r$ ).

The Old Way: The Uniform Rule

Traditionally, librarians followed a simple rule: Uniform Substitution.
If a book says, "If $p$ happens, then $p$ happens again," and you decide to swap the letter $p$ for the word "Rain," you must swap every single instance of $p$ with "Rain."

Before: If it rains, it rains.
After: If it rains, it rains.
You can't change just one $p$ to "Rain" and the other to "Snow." They are treated as the exact same thing, everywhere.

The New Idea: Hyperformalism

The authors of this paper introduce a much more flexible, "hyper" way of organizing the library called Hyperformalism.

Imagine a special kind of librarian who looks at where a word appears in a sentence. They realize that two instances of the same letter might actually be doing different jobs depending on their location.

The Analogy: Think of a word appearing in a sentence as a person wearing a different hat depending on where they stand in a room.
- If $p$ is standing alone, it wears a "Red Hat."
- If $p$ is standing inside a box (a conditional statement like "If... then..."), it wears a "Blue Hat."
- If $p$ is standing inside a box inside another box, it wears a "Green Hat."

In a Hyperformal logic, the librarian says: "Because the $p$ in the Red Hat is in a different spot than the $p$ in the Green Hat, they are actually different people." You can swap the Red-Hat $p$ with "Rain" and the Green-Hat $p$ with "Snow" without breaking the rules of the library.

This paper shows that this "different hats" approach works incredibly well for Relevant Logics (logics that demand the "if" part of a sentence actually has something to do with the "then" part).

Adding the "Box" (Modal Logic)

The paper takes this idea a step further by adding Modal Logic (the logic of "necessity" or "possibility," represented by a box symbol $\square$ ).

In standard logic, $\square p$ means "It is necessary that $p$ ."
The authors ask: Does the "hat" system work when we have these boxes?

They define a new system called MPos-hyperformalism. Here, the "hat" (or position) of a letter depends on:

How many boxes it is inside.
Whether it is on the left or right side of an "If/Then" statement.
Whether it is negated (inside a "Not" statement).

The Big Discovery (Theorem 2.1):
The authors prove that a specific, very weak logic called $B_\square$ is "MPos-hyperformal."

What this means: In this logic, you can treat every single instance of a letter as a unique individual based on its exact location in the sentence structure. If a sentence is a valid theorem, it will remain valid even if you swap different instances of the same letter with completely different words, as long as you respect their "hats" (positions).

The "Variable Sharing" Rule

Relevant logics have a golden rule: Variable Sharing.

The Rule: In a valid "If $A$ , then $B$ " statement, $A$ and $B$ must share at least one common ingredient (a variable). You can't say "If the moon is made of cheese, then I am a potato" because they share nothing.
The Twist: Because of the "hat" system, the authors found that in $B_\square$ $B_{□}$ , the shared ingredient must be in the same type of hat.
- If $p$ is shared, it must be in the same number of boxes in both the "If" part and the "Then" part.
- This creates a very strict, precise version of relevance.

The "Grand Champion" Logic: $K_{MPos}$

The paper also introduces a new logic called $K_{MPos}$ .

Think of $K$ as the "Classical" library, which is huge and allows almost anything.
The authors asked: "What is the largest possible section of the Classical library that still follows our strict 'Hat' rules (Hyperformalism)?"
They found it: $K_{MPos}$ .

Why is $K_{MPos}$ special?

It's the biggest: It contains every possible sentence that fits the "Hat" rules.
It's safe: Unlike some other "relevant" logics that are just classical logic with a "sieve" (a filter) slapped on top, $K_{MPos}$ is built from the ground up to be consistent.
It doesn't break: The authors prove that this logic is transitive.
- Analogy: If "If A then B" is true, and "If B then C" is true, then "If A then C" is definitely true. Some weird "relevant" logics break this chain, but $K_{MPos}$ keeps it intact.

Summary of the Authors' Conclusion

The authors are essentially saying:
"We have shown that the 'different hats' approach (MPos-hyperformalism) works perfectly for weak relevant logics like $B_\square$ . But if you want the strongest, most robust logic that still follows these rules, you shouldn't stick with $B_\square$ . You should look at $K_{MPos}$ ."

They challenge other logicians: "If you prefer the old, weaker logics, you need to give us a good reason why. If your reason isn't about 'variable sharing' or 'classicality,' then you might be missing out on the superior $K_{MPos}$ ."

In short: The paper builds a new, highly organized system for logic where the location of a word determines its identity, proves this system works for specific types of logic, and then finds the "ultimate" version of this system that is stronger and more reliable than previous attempts.

Technical Summary: Hyperformalism for Relevant Modal Logics

Problem and Context
The paper addresses the analysis of relevant logics through the lens of hyperformalism, a property wherein a logic remains closed under non-uniform substitutions. In standard formal logics, atomic formulas are treated as independent via uniform substitution. Hyperformalism extends this independence, allowing distinct occurrences of the same atom to be treated as independent if they appear in sufficiently different syntactic contexts (e.g., at different "depths" or under different modal operators). While hyperformalism has been successfully applied to non-modal relevant logics (such as $B$ and $DR$), revealing strong variable sharing properties and identifying new relevant logics, its application to relevant modal logics remains underexplored. The authors seek to extend these techniques to modal settings, specifically investigating how the presence of the modal operator $\square$ interacts with non-uniform substitutions and variable sharing.

Methodology
The authors employ a combination of proof-theoretic and semantic techniques to analyze relevant modal logics:

Syntactic Framework (MPos): The authors define a set of positions, $MPos = \{l, r, n, b\}^*c$ , where $l$ and $r$ denote left and right branches of conditionals, $n$ denotes negation, and $b$ denotes the scope of the modal operator $\square$ .
Substitutions: They define MPos-substitutions as functions $\sigma: MPos \times At \to L$ that respect the structure of formulas. Crucially, these substitutions treat atoms at different positions as independent, subject to specific "faithfulness" constraints (e.g., $\sigma_{xnn}(p) = \sigma_x(p)$ to preserve double negation properties).
Proof-Theoretic Analysis: The paper analyzes the base relevant modal logic $B^\square$ (formed by adding axioms $C$ and rule $RM$ to the relevant logic $B$ ). It proves closure under MPos-substitutions via induction on derivations.
Extensions: The authors define equivalence relations on $MPos$ corresponding to modal axioms $T, 4, K,$ and $RN$. They investigate the resulting logics $B^\square_S$ and their closure under corresponding $MPos_S$ -substitutions.
Semantic Characterization: To characterize the largest MPos-hyperformal sublogic of classical modal logic $K$ $K$ (denoted $KMPos$), the authors introduce:
- MPosK-assignments: Functions $\alpha: MPos \times W \times At \to \{0, 1\}$ over Kripke frames, where truth values vary based on the position index.
- Analytic Tableaux: A tableau system with nodes $\langle x, \omega, A, i \rangle$ tracking positions ( $x$ ), worlds ( $\omega$ ), formulas ( $A$ ), and truth values ( $i$ ).
Canonical Models: Completeness is established via the construction of maximal, witnessed, tableau-consistent branchsets and their corresponding canonical models.

Key Contributions and Results

MPos-Hyperformalism of $B^\square$ :
- Theorem 2.1: The base logic $B^\square$ is proven to be MPos-hyperformal. The proof demonstrates that the axioms ( $C$ ) and rules ($RM$) of the system preserve theoremhood under MPos-substitutions.
- Variable Sharing: As a consequence, $B^\square$ satisfies the MPos-variable sharing property. If $A \to B$ is a theorem, there exists a shared variable $p$ such that the position of $p$ in $A$ and $B$ are equivalent under the relation $\sim_{nn}$ (differing only by pairs of negations).
- Corollary 2.6: A specific consequence is that any theorem $A \to B$ in $B^\square$ shares a variable nested within the same number of $\square$ operators in both antecedent and consequent.
Extensions of $B^\square$ :
- Theorem 2.7: For any subset $S \subseteq \{T, 4, K, RN\}$ , the logic $B^\square_S$ is $MPos_S$ -hyperformal, where $MPos_S$ incorporates equivalence relations reflecting the specific modal axioms (e.g., $xby \sim_4 xbby$ for axiom 4).
- Refined Variable Sharing: These extensions yield refined variable sharing properties. For instance, in $B^\square_{\{4\}}$ , shared variables must have the same "mode" (extensional vs. intensional), effectively eliminating the count of $\square$ operators from the sharing condition.
Characterization of $KMPos$:
- The authors define $KMPos$ as the largest MPos-hyperformal sublogic of the classical modal logic $K$ .
- Semantic Characterization: $KMPos$ is characterized by validity in Kripke frames equipped with MPosK-assignments.
- Tableau Characterization: A sound and complete tableau system is provided for $KMPos$.
- Theorem 3.6: $KMPos$ is equivalent to the set of formulas $A$ such that for some atomic, injective-up-to-faithfulness substitution $\iota$ , $\iota(A)$ is a theorem of $K$ . This confirms $KMPos$ is the maximal subset of $K$ closed under MPos-substitutions.
- Variable Sharing: Theorem 3.9 establishes that $KMPos$ possesses the variable sharing property.
Transitivity and Nature of $KMPos$:
- Theorem 4.1: $KMPos$ is transitive. The authors argue that unlike "syntactic sieve" approaches to relevance (which often fail transitivity), $KMPos$ avoids such pathologies because it is defined by a closure property (hyperformalism) rather than a static filter on classical theorems.

Significance and Claims
The paper claims that hyperformalism is a powerful tool for analyzing relevant modal logics, capable of revealing strong variable sharing properties even in weak systems like $B^\square$ . However, the authors argue that $B^\square$ is not the strongest logic with these features.

The primary significance lies in the introduction and characterization of $KMPos$. The authors posit that $KMPos$ represents the "largest" relevant modal logic in the sense of being the maximal sublogic of classical $K$ that is closed under MPos-substitutions. They claim that $KMPos$ offers semantic and proof-theoretic advantages over $B^\square$ and other traditional relevant modal logics. Specifically, they argue that $KMPos$ is not merely a "syntactic sieve" (a pejorative term for logics defined by simply filtering classical theorems) because it is transitive and defined by a robust closure property.

The paper concludes by challenging the preference for "traditional" relevant modal logics (like $B^\square$ ) over $KMPos$. The authors suggest that unless there are specific reasons to prefer the former, researchers should consider $KMPos$, as it retains the desirable variable sharing features while being a maximal sublogic of classical modal logic. The paper acknowledges that an axiomatic characterization of $KMPos$ remains an open problem, noting that such logics have historically resisted recursive axiomatization.