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From Knowledge to Conjectures: A Modal Framework for Reasoning about Hypotheses

This paper introduces a new family of non-trivial modal logics, specifically $\mathbf{KC}$ and $\mathbf{KDC}$ , which formalize conjectural reasoning by preserving known facts through Axiom C within a non-bivalent semantic framework to avoid modal collapse, while also providing a unified account of cognitive states and a dynamic operator for transitioning conjectures into reality.

Original authors: Fabio Vitali

Published 2026-03-24

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Fabio Vitali

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: The "What If?" Engine

Imagine your mind is a library.

Facts are the books that are already written, printed, and sitting on the shelves. You know they are true.
Beliefs are the books you think are true, even if they might be wrong or made up.
Knowledge is the specific section of the library where you are 100% sure the books are true.

This paper introduces a new section of the library called Conjectures.

A Conjecture isn't just a wild guess. It's a "What if?" scenario that starts with the facts you already know and adds one new, unproven idea to see where it leads.

Example: You know it's raining (Fact). You conjecture: "If I forget my umbrella, I will get wet." You aren't saying you will forget it; you are just exploring the consequences of that possibility while keeping the fact (it's raining) firmly in place.

The author, Fabio Vitali, wants to build a mathematical system (a logic) that perfectly describes how we do this kind of thinking without breaking the rules of math.

The Problem: The "Collapse" Trap

For a long time, mathematicians were afraid of a specific rule called Axiom C.

The Rule: "If something is true, then it is necessarily true in all our 'What If' scenarios." (Written as: $\phi \to \Box\phi$ ).

Why were they scared?
Imagine a magic mirror. If you look in it, it shows you exactly what you are holding.

If the mirror says "You are holding an apple," and you are holding an apple, that's fine.
But if the mirror is too powerful, it starts saying: "You are holding an apple" only because the mirror says so. It blurs the line between "what is actually happening" and "what the mirror says is happening."

In logic, this is called Modal Collapse. It means the difference between "It is raining" (a fact) and "It must be raining" (a necessity) disappears. Everything becomes the same, and the logic stops being useful for exploring possibilities. It's like a map where every path leads to the same spot; you can't navigate anymore.

The Solution: The "Undefined" Zone

The paper argues that the "Collapse" only happens if you assume everything in the universe is already either True or False (like a light switch that is either ON or OFF).

But in real life, and in our minds, many things are Undefined.

Analogy: Imagine a puzzle. You have the border pieces (Facts). The middle is empty.
Bivalent Logic (The old way): Assumes every empty spot is secretly either a "Sun" piece or a "Cloud" piece, even if you haven't found it yet.
Non-Bivalent Logic (This paper's way): Admits that the empty spot is just... empty. It's neither Sun nor Cloud yet. It's Open.

By using a logic that allows for "Open" or "Undefined" spots, the author shows you can use Axiom C safely.

The Magic Trick: You can say, "Whatever is already settled (the border pieces) stays settled in all our 'What If' scenarios."
Because the middle of the puzzle is still open, the "What If" scenarios can still be different from each other. The "Collapse" never happens because there is still room for the unknown to be unknown.

The New Systems: KC and KDC

The author builds two new logical systems based on this idea:

KC: The system for pure conjecture. You start with facts, add a "What if," and see what happens.
KDC: A slightly stricter version that ensures your "What if" scenarios actually have a path forward (they aren't dead ends).

These systems prove that you can have a logical framework where:

Facts are preserved.
New ideas can be tested.
The system doesn't break or become trivial.

The "Inclusion" Map

The paper creates a beautiful map of how different types of thinking relate to each other using simple set theory (like Venn diagrams).

Imagine a circle representing Reality (what is actually true).

Doxastic (Belief): Any circle that contains the shared facts but might have extra stuff that isn't real. (You believe X, but X might be false).
Epistemic (Knowledge): Any circle that is inside Reality. (You only believe things that are actually true).
Conjectural (Hypothesis): Any circle that contains Reality and adds new, unproven layers on top. (You take the truth and build a "What if" tower on it).
Delusional: A circle that contradicts Reality. (You believe X, but X is definitely false).

The paper shows that Conjecture is the perfect middle ground. It respects the truth (it doesn't contradict reality) but isn't limited by it (it explores what could be).

The Dynamic Part: "Settling" the Score

Finally, the paper introduces a dynamic operator called Settle(p).

The Scenario: You are guessing who will win the soccer game tomorrow.
- Conjecture: "If Team A wins, they go to the finals."
- Conjecture: "If Team B wins, they go to the finals."
- Right now, both are just possibilities floating in the air.
The Event: The game happens. Team A wins.
The Action (Settle): The operator Settle(Team A wins) takes that floating possibility and turns it into Fact.

The "Settle" operator is the bridge between the "What If" world and the "What Is" world. It describes the moment a hypothesis becomes a fact. This is crucial for things like:

Science: A hypothesis becomes a theory, then a fact.
Law: A suspect is "guilty" only after the trial settles the evidence.
AI: A robot guesses the best path, then "settles" on it and moves.

Why This Matters (The "So What?")

This isn't just abstract math. It helps us build better AI and understand human disagreement.

Handling Disagreement: Two people can look at the same facts (the puzzle border) and have different "Conjectures" (different guesses for the middle). This logic lets us model that disagreement without saying one person is "crazy" (delusional) or "wrong" (false). They are just exploring different valid extensions of the same facts.
AI and Robotics: Robots often have to make decisions with incomplete information. This logic gives them a formal way to say, "I know X is true. I don't know Y yet, but let's assume Y is true for a moment to see if it helps us solve the problem."
Future Events: It helps us reason about things that haven't happened yet (like the soccer game) without pretending they have already happened.

Summary in One Sentence

This paper invents a new way of doing math that lets us safely explore "What If" scenarios by keeping our known facts solid while admitting that the rest of the world is still open, undefined, and waiting to be discovered.

1. Problem Statement

The paper addresses a gap in cognitive modal logics, which traditionally model doxastic states (belief, potentially false) and epistemic states (knowledge, true but partial). There is no formal system designed to model conjectural reasoning: a cognitive state where an agent possesses a set of established facts but adopts unverified assumptions to explore their consequences without contradicting the known facts.

A central technical challenge in formalizing this is Axiom C ( $\phi \to \Box\phi$ ). Historically, this axiom has been viewed with suspicion because, when combined with Axiom T ( $\Box\phi \to \phi$ ) in a bivalent (classical true/false) logic, it leads to modal collapse ( $\phi \leftrightarrow \Box\phi$ ). Modal collapse trivializes the logic by making every true proposition necessarily true, eliminating the distinction between actuality and possibility. The paper seeks to determine if Axiom C can be used constructively for conjectures without causing collapse.

2. Methodology

The author proposes a framework based on non-bivalent semantics and partial valuations.

Non-Bivalent Base: The framework rejects the assumption that every well-formed formula must be strictly true or false. Instead, it utilizes partial valuations where propositions can be undetermined (undefined). This is achieved through semantic frameworks such as:
- Supervaluation Semantics: Truth is defined by preservation across all admissible completions of a partial base.
- Weak Kleene Logic: A three-valued logic where undefinedness propagates through connectives.
- Description Logic (Open World Assumption): Where the absence of information does not imply falsity.
Conjectural Accessibility: The accessibility relation ( $R$ ) in Kripke models is constrained to be definedness-preserving. If a world $w$ accesses $w'$ , then $w'$ must be an extension of $w$ that preserves all truth values already determined in $w$ but may resolve previously undetermined propositions.
Axiom C Re-evaluated: In this non-bivalent setting, Axiom C ( $\phi \to \Box\phi$ ) is interpreted as a preservation constraint: if a proposition is determined as true in the current world, it must remain true in all accessible (conjectural) worlds. It does not force undetermined propositions to become true, nor does it equate truth with necessity unless Axiom T is also present.

3. Key Contributions

A. The Logics KC and KDC

The paper defines two new modal systems:

KC: The minimal system containing Axiom K and Axiom C. It models conjectures where established facts are preserved across extensions.
KDC: KC plus Axiom D ( $\Box\phi \to \neg\Box\neg\phi$ ), ensuring seriality (every world has at least one accessible extension), preventing vacuous modal truths.

B. Resolution of Modal Collapse

The paper proves that Axiom C alone does not cause modal collapse.

Theorem: In a system validating C, modal collapse ( $\phi \leftrightarrow \Box\phi$ ) occurs if and only if the system is concretely bivalent (every formula is either true or false) or Axiom T is adopted.
Mechanism: By adopting a non-bivalent base where some formulas are undetermined, the equivalence $\phi \leftrightarrow \Box\phi$ fails for those undetermined formulas. Thus, the modal operator remains distinct from the identity operator, preserving the logical space for conjecture.

C. The Inclusion Theorem

A major theoretical contribution is a set-theoretic characterization of cognitive states based on the inclusion relations between valuation sets:

Let $s$ be the shared knowledge base, $r$ be the designated reality, and $w$ be a cognitive world.
Doxastic ( $K$ ): $s \subseteq w$ (The world extends shared knowledge).
**Epistemic ($KT $):**$ s \subseteq w \subseteq r$ (The world is a subset of reality; it contains only truths).
**Conjectural ($KC $):**$ r \subseteq w$ (The world extends reality with new assumptions).
Delusional: $w$ contradicts $r$ (inconsistent with reality).
Opinion: $w$ is disjoint from $r$ (beyond shared knowledge).
This theorem unifies doxastic, epistemic, and conjectural logics into a single geometric framework based on set inclusion.

D. Conjectural Dynamic Logic (CDL)

The paper introduces a dynamic operator, $settle(p)$, which models the transition where a conjecture becomes reality.

Semantics: $settle(p)$ transforms the designated reality $r$ into $r_p$ , the "least definedness-preserving extension" where $p$ is true.
Properties: Settlements are iterable and commute (if compatible), allowing for the progressive determination of an initially incomplete reality. This creates a dynamic logic parallel to Dynamic Epistemic Logic (DEL) and Dynamic Doxastic Logic (DDL).

4. Key Results

Soundness and Completeness: The systems KC and KDC are proven to be sound and complete with respect to the class of Kripke models satisfying definedness-preserving accessibility and Axiom C.
Derivation of 4 and 5: The paper demonstrates that Axiom C directly implies Axiom 4 ( $\Box\phi \to \Box\Box\phi$ ) and Axiom 5 ( $\neg\Box\phi \to \Box\neg\Box\phi$ ) via uniform substitution, meaning these are not independent axioms in this framework but structural consequences of C.
Non-Triviality: The systems are proven non-trivial; they do not collapse into classical logic or identity logic as long as the base logic remains non-bivalent.
Formal Distinction: The framework successfully distinguishes between:
- Belief: $s \subseteq w$ (may be false).
- Knowledge: $w \subseteq r$ (must be true).
- Conjecture: $r \subseteq w$ (extends truth with assumptions).

5. Significance

Theoretical: It resolves the historical "suspicion" surrounding Axiom C by showing that modal collapse is a result of bivalence and Axiom T, not C itself. It provides a rigorous logical foundation for reasoning under uncertainty where facts are partial.
Philosophical: It offers a formal model for "future contingents," historical disagreements (e.g., art attribution), and scientific hypotheses where multiple incompatible extensions of a shared evidence base are possible.
Practical Applications:
- Semantic Web & Provenance: Modeling conflicting claims from different sources where the "truth" is undetermined but the shared facts are preserved.
- Neuro-Symbolic AI: Enabling hybrid systems where neural networks provide partial/uncertain inputs that symbolic reasoners can extend via conjectures, which are then "settled" as new data arrives.
- Multi-Agent Systems: Formalizing how agents with different partial views of reality can reason about hypothetical scenarios without contradicting their shared knowledge.

In summary, Vitali introduces a robust family of logics that formalize the human capacity to reason hypothetically while respecting established facts, utilizing non-bivalent semantics to prevent logical collapse and dynamic operators to model the evolution of conjecture into fact.