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Guises and Perspectives: An Intentional and Hyperintensional Sketch

This paper presents a formal logic of guises rooted in Castañeda's internalist and Leibnizian philosophy, establishing a syntax, model theory, and proof theory where relations are treated as intentional perspectives encoded within property bundles rather than external causal links, thereby addressing hyperintensional phenomena and offering a distinct alternative to classical intensional and situation semantics.

Original authors: Juan J. Colomina-Alminana

Published 2026-05-15

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Juan J. Colomina-Alminana

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

The Big Idea: Seeing the World Through Different Lenses

Imagine you are looking at a statue. To a sculptor, it is a "bundle of clay and tools." To a historian, it is a "symbol of an empire." To a child, it is "a giant toy."

In the real world, it is the same statue. But in your mind, it is three completely different things depending on how you are looking at it.

This paper argues that everything in our minds works this way. We don't just think about "objects"; we think about guises. A "guise" is a specific bundle of properties that an object has under a specific perspective.

The author, Juan J. Colomina-Alminana, is trying to build a mathematical "rulebook" (a logic) to explain how these perspectives work. He combines two old-school ideas:

Leibniz's "Complete Concept": The idea that every object contains all the truths about itself inside its own "mind."
Castañeda's "Guise Theory": The idea that we can't swap one name for another (like "Superman" for "Clark Kent") if the perspective changes, even if they are the same person.

The Core Concepts (The "How It Works")

1. Guises are "Property Bundles"

Think of a guise like a backpack.

A backpack doesn't have to be a person; it's just a collection of items.
One backpack might contain: Red, Tall, Brave, Kryptonian.
Another backpack might contain: Glasses, Gentle, Reporter, Clark.
Even if both backpacks belong to the same person (Superman/Clark), they are different "guises" because the items inside are different.

2. Intention: The "Pointing" Finger

The paper introduces a special tool called Intention.

Imagine your mind is a flashlight. When you shine the light on a backpack, you are "intending" it.
The paper says that relations (how things connect) aren't like physical ropes tying two objects together. Instead, relations are like pointing.
If Guise A (the "Red/Brave" backpack) points to Guise B (the "Gentle/Reporter" backpack), they are related internally because of how the first one is looking at the second. They aren't tied by an external rope; they are connected by the act of looking.

3. The "Internal" vs. "External" Debate

Old View (External): Two things are related because there is a physical line connecting them (like a phone wire).
This Paper's View (Internal): Two things are related because one "contains" the idea of the other.
- Analogy: Imagine a mirror. The mirror doesn't need a wire to the object it reflects. The reflection is inside the mirror's surface. The paper argues that our thoughts work like mirrors: the whole universe is reflected inside our specific "guise" or perspective.

The Logic Puzzle: Why "Superman" isn't always "Clark Kent"

The paper solves a famous logic problem: Why can't we always swap names?

Scenario: Lois Lane loves Superman. Lois Lane does not know that Clark Kent is Superman.
The Mistake: If you say "Lois loves Clark Kent," you are wrong, even though Superman = Clark Kent.
The Paper's Solution: In this logic, "Superman" and "Clark Kent" are two different guises (different backpacks).
- Lois's mind has a "Superman Backpack" (Red, Brave, Flying).
- She does not have a "Clark Kent Backpack" (Glasses, Reporter) in the same way.
- Because the backpacks are different, you cannot swap them in her mind. The logic respects the perspective.

The Three "Systems" (The Blueprints)

The author builds three different versions of this logic to show how flexible it is:

System A (The Idealist): This is the "perfect" version. Every single thing you could think about a person, you do think about them. It's like a perfect mirror that reflects everything instantly. It's great for math, but maybe too perfect for real humans.
System B (The Template): This is more realistic. Imagine you have a limited set of "stamps" or "templates" (like "Hero," "Villain," "Friend"). You can only see people through these specific stamps. This makes the logic easier to calculate and more like how humans actually think (we have limited mental categories).
System C (The Lab): This is a small, concrete example using just four letters (a, b, c, d) to prove the math works. It's like a test kitchen where the author cooks a small meal to show the recipe works before serving a banquet.

Why This Matters (According to the Paper)

The paper claims to prove that relations are not external links, but internal perspectives.

Metaphor: Think of a dance. Two dancers aren't connected by a physical chain. They are connected because they are both looking at the same rhythm and moving in response to each other's internal steps.
The Result: The paper creates a formal language (a new kind of math) that allows us to talk about "self," "belief," and "identity" without breaking the rules of logic. It handles the messy parts of human thought (like "I believe I am me, but I don't know that my friend is me") by treating them as different guises rather than broken logic.

Summary in One Sentence

This paper builds a mathematical system where "things" are just bundles of properties viewed through specific lenses, proving that our connections to the world are created by how we look at things (our internal perspectives), not by physical chains tying us together.

Technical Summary: Guises and Perspectives: An Intentional and Hyperintensional Sketch

Problem Statement
The paper addresses the logical and metaphysical challenges of modeling intentional reference, de se attitudes, and quasi-indexicality (e.g., "he himself") where standard extensional substitution principles fail. It seeks to synthesize two historical traditions: Héctor-Neri Castañeda's theory of guises (property bundles presented under specific conceptual profiles) and Gottfried Wilhelm Leibniz's doctrine of internal relations and predicate containment. The core problem is to construct a formal system that treats relations not as external causal links but as internal structures encoded within the conceptual profiles (guises) of agents and objects, thereby preserving hyperintensional distinctions necessary for modeling cognitive significance while maintaining a rigorous semantic backbone.

Methodology
The author develops GL (Guise Logic), a many-sorted intensional language, and its extension GL⁺, a Hilbert-style proof system. The methodology proceeds through the following steps:

Syntax and Ontology: The language distinguishes between marks (properties), guises (bundles of marks), propositions (sets of marks), and worlds. It introduces a binary intentional operator $\mathcal{I}(g, \phi)$ ("guise $g$ intends proposition $\phi$ ") and an internal relation operator $R(g, h)$ .
Model Theory: The semantics are defined on structures $\mathcal{M} = \langle \mathcal{P}, G, W, T, \kappa, \mathcal{I} \rangle$ $M = ⟨ P, G, W, T, κ, I ⟩$ .
- Guises are sets of marks (properties).
- Closure ( $\kappa$ ): A Tarskian consequence operator induced by a background theory $T$ models "complete concepts" (Leibnizian containment).
- Worlds: Maximal $T$ -consistent sets of marks.
- Intention ( $\mathcal{I}$ ): A mapping assigning families of intended propositions to each guise.
- Truth Conditions: Singular predication is grounded in containment ( $P \in \kappa(g)$ ); internal relations are defined by the existence of an intended proposition contained in the target guise.
Axiomatization: The system includes axioms for Containment Truth, Leibnizian Containment, and Intentional Connection. Crucially, it incorporates principles for Intention–Closure Alignment (consequence closure, extension invariance, and commutation) and relational axioms (reflexivity, monotonicity, conditional transitivity).
Metatheoretic Analysis: The paper sketches proofs for soundness and canonical-model completeness. It analyzes the system's behavior regarding hyperintensionality, specifically how it handles substitution failures in intentional contexts.
System Variants: The author develops three specific instantiations (Systems A, B, and C) ranging from a canonical closure-intention system to template-restricted and finite explicit models to demonstrate decidability and tractability.

Key Contributions

Formal Synthesis: The paper provides the first rigorous formalization that integrates Castañeda's guise theory with Leibniz's containment semantics. It demonstrates that relational truths can be derived from conceptual containment and intentional structure rather than external ties.
Hyperintensional Semantics: The system explicitly models hyperintensional phenomena where coreferential guises (sharing the same complete concept $\kappa(g) = \kappa(h)$ ) may have distinct intention sets ( $\mathcal{I}(g) \neq \mathcal{I}(h)$ ). This allows for the failure of substitution in intentional contexts, capturing the cognitive significance of de se thought and quasi-indexicals.
Internal Relations as Perspectives: The framework redefines relations as "internal denominations." A relation $R(g, h)$ holds if $g$ intends a proposition that is contained within $h$ . This aligns with the Leibnizian view that monads represent the universe from their own point of view without external windows.
Modal Architecture: Possibility and necessity are modeled via consistency and inclusion across maximal $T$ -closed worlds, effectively creating a "Leibnizian ratiocination" layer within the guise logic.

Results

Soundness and Completeness: The paper establishes soundness by inspection of model clauses and sketches a canonical-model completeness proof. Completeness is shown to hold for the intended class of models satisfying the intention-closure alignment principles (CI1–CI3).
Decidability: For specific fragments—particularly those using Horn theories ( $T$ ) and finite template sets ( $\Theta$ )—the system possesses the finite model property. The propositional fragment is shown to be decidable (at worst PSPACE, often PTIME via forward chaining).
Handling of De Se Attitudes: The system successfully formalizes de se attitudes via a self-ascription operator ( $Self(g, \phi)$ ), defined as $Int(g, \phi) \land \phi \subseteq g$ , recovering the reflexivity of self-reference without positing exotic objects.
Failure Modes and Repairs: The paper identifies specific failure cases (e.g., non-commutation of closure and intention, non-transitive relations) and proposes axiomatic repairs or contextual restrictions (e.g., distinguishing ideal Leibnizian agents from cognitive variants) to handle them.

Significance and Claims
The paper claims to offer a "self-contained formal framework" that is both "metaphysically charitable to Leibniz" and "technically adequate for contemporary logic." Its primary significance lies in demonstrating that:

Relations are not external causal links but intentional internal structures encoded in guises (perspectives).
A logic of guises can preserve necessary intensional distinctions (hyperintensionality) while anchoring truth in extensional containment where safe.
Leibniz's dream of a "calculus of concepts" finds a renewed life in this architecture, where inclusion, intention, and necessity function as parts of a single intentional system.

The author positions this work as a methodological advancement, showing that a logic of guises can resolve the puzzles of indexicals and attitude ascriptions identified by Castañeda without abandoning the rigorous semantic foundations of predicate logic. The paper does not claim to solve all problems of intentionality but provides a specific, formalizable parameter set for modeling perspectives and internal relations.