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Classical Logic as Intuitionistic Logic with Duality

This paper proposes a novel proof-theoretic semantics for classical logic that extends intuitionistic logic by operating over primitive dual literals rather than atomic propositions, thereby demonstrating that classical logic can be understood as intuitionistic logic supplemented by an inferentially encoded principle of duality.

Original authors: Alexander V. Gheorghiu, Yll Buzoku

Published 2026-06-03

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CC BY 4.0

Original authors: Alexander V. Gheorghiu, Yll Buzoku

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: Logic as a Game of "Yes" and "No"

Imagine you are trying to explain how logic works to a friend. Usually, we think of logic as a set of rules for determining what is True and what is False.

This paper proposes a different way to look at it. Instead of asking "Is this statement true?", the authors ask: "Can I assert this?" (say "Yes") or "Can I deny this?" (say "No").

Their main slogan is:

Classical Logic = Intuitionistic Logic + Duality

In plain English: They are saying that the complex, "classical" logic we use in math and science is actually just the simpler, "intuitionistic" logic (which is very strict about proof) plus a special rule that treats "Yes" and "No" as two equal, opposite forces right from the very start.

The Cast of Characters

To understand their argument, we need to meet the three main characters in their story:

The Content (The "What"): This is the raw idea, like "The concert is good."
The Force (The "How"): This is how you deliver the idea. You can Assert it (say "The concert is good!") or Deny it (say "The concert is bad!").
The Formula (The "Structure"): This is the complex sentence built from those ideas, like "If the concert is good, then I will stay."

The Old Way vs. The New Way

The Old Way (Traditional Logic):
Imagine you have a light switch. It's either ON (True) or OFF (False).

If you want to say "No," you just flip the switch to OFF.
In this view, "Denying" something is just the same as "Asserting its opposite."
The Problem: The authors say this is too simple. It assumes we already know the answer is either yes or no before we even start arguing. It's like assuming the light switch must work before you even plug it in.

The New Way (The Authors' Approach):
Imagine a conversation between two people, a "Yes-Sayer" and a "No-Sayer."

They don't start with a light switch. They start with Literals.
A Literal is a basic speech act.
- Positive Literal: "I assert: The concert is good."
- Negative Literal: "I deny: The concert is good" (which is like saying "The concert is bad").
Crucial Point: These two are not defined by each other. You don't say "Denial is just the opposite of Assertion." They are two separate, primitive tools, like a hammer and a screwdriver. You can't turn a hammer into a screwdriver just by flipping it over; they are fundamentally different tools.

The "Duality" Secret Sauce

The paper argues that to get from simple logic (Intuitionistic) to complex logic (Classical), you just need to add a rule about how these "Yes" and "No" tools interact at the very bottom level (the atomic level).

They introduce two simple rules for these basic tools:

The "Exclusion" Rule: You cannot say "Yes" and "No" to the exact same thing at the same time. If you try to assert "The concert is good" AND deny "The concert is good," you crash the system (you get a contradiction, or "absurdity").
The "Case Analysis" Rule: If you can prove a result whether you assume "Yes" OR whether you assume "No," then you don't need to know which one is true to get the result. You can just say, "It's proven either way."

The Analogy:
Imagine you are trying to get into a club.

Intuitionistic Logic: You need a specific invitation (proof) to get in. If you don't have it, you can't enter.
Classical Logic (with Duality): You can get in if you have a "Yes" pass OR if you have a "No" pass (which acts like a "No, I'm not staying out" pass).
The authors show that if you have these two passes (Assertion and Denial) and the two rules above, you can build the entire structure of Classical Logic without ever needing to assume a pre-existing "True/False" universe.

Why This Matters (The "Why Should I Care?")

The authors are solving a philosophical puzzle.

The Puzzle: How can we use Classical Logic (which assumes everything is either True or False) if we are "Anti-Realists" (people who believe truth is something we construct through proof, not something that exists floating in the sky)?
The Solution: By treating "Yes" and "No" as two equal, primitive speech acts, they show that Classical Logic doesn't need to assume a "True/False" world exists beforehand. It just needs the rules of how we argue with "Yes" and "No."

The Technical Achievement

The paper does something very clever mathematically:

They take the standard, simple rules for Intuitionistic Logic (which are already well-understood).
They add the "Duality" rules (Exclusion and Case Analysis) only at the very bottom level (the literals).
They prove that this simple addition creates the full, powerful system of Classical Logic.

They call this Base-Extension Semantics. Think of it like building a house:

Intuitionistic Logic is the foundation and the frame.
Classical Logic is the same house, but with a special "Dual Door" installed at the very bottom.
Once that door is there, the whole house functions differently, allowing for all the complex reasoning we use in classical math, but it was built using only the simple, constructive bricks of intuitionistic logic.

Summary

The paper says: Classical Logic isn't a mysterious, pre-existing truth. It's just Intuitionistic Logic (logic based on proof) plus a simple agreement that "Yes" and "No" are two equal, opposing forces that we can use to build complex arguments.

By treating "Denial" as a primitive tool rather than just "Asserting the opposite," they provide a fresh, constructive way to understand how classical reasoning works without needing to believe in a magical "True/False" universe.

Technical Summary: Classical Logic as Intuitionistic Logic with Duality

1. The Problem
The paper addresses a foundational challenge in the theory of meaning: providing a proof-theoretic semantics (P-tS) for classical logic that adheres to anti-realist (constructivist) principles without presupposing the principle of bivalence or verification-transcendent truth. While Sandqvist has successfully developed base-extension semantics (B-eS) for intuitionistic logic (IPL) and a version for classical logic (CPL), limitations remain. Sandqvist's CPL semantics is restricted to specific connectives (initially $\to, \bot, \forall$ ) and relies on a technically complex completeness proof involving bar induction. Furthermore, existing approaches often struggle to treat the full propositional language (including $\lor$ and $\exists$ ) constructively. The authors aim to resolve this by offering a B-eS for full classical propositional logic that maintains an elementary, constructive completeness proof, thereby answering Dummett's challenge regarding the reconciliation of classical inference with anti-realist meaning.

2. Methodology and Philosophical Framework
The authors propose a bilateralist framework where classical logic is conceptualized as "Intuitionistic Logic + Duality." This approach departs from traditional views by distinguishing between content and illocutionary force at the atomic level, while treating complex formulae as logico-grammatical structures rather than propositions bearing force themselves.

Literals as Propositions: The system introduces "literals" ( $L$ ) as the only objects of assertion and denial. A content $c$ is paired with a force to form a positive literal ( $c^+$ , assertion) or a negative literal ( $c^-$ , denial). Crucially, duality is primitive: $c^+$ and $c^-$ are independent atomic speech acts, not defined via negation ( $\neg$ ).
Logico-Grammatical Structures: Complex formulae (e.g., $\phi \land \psi$ ) are not propositions to be asserted or denied but are structures organizing reasoning. Consequently, the bilateral distinction (assertion vs. denial) is restricted to the atomic level (literals).
Inferential Duality: Classicality is recovered not by adding a truth-value semantics, but by introducing two harmonious inferential principles at the atomic level (following Negri and von Plato):
1. Exclusion (EXC1): Asserting and denying the same content leads to absurdity ( $l, l^\perp \vdash \bot$ ).
2. Case Analysis (EXC2): If a conclusion follows from both an assertion and its dual denial, it follows outright ( $l \vdash \phi, l^\perp \vdash \phi \implies \vdash \phi$ ).

3. Technical Development

Natural Deduction System ( $NK^\pm$ ): The authors define a natural deduction system comprising standard intuitionistic rules ($NJ$) plus the atomic duality rules (EXC1 and EXC2). This system treats literals as the base, with complex formulae built via standard connectives.
Base-Extension Semantics (B-eS):
- Bases ( $B$ ): A base is a set of atomic rules (inferential relationships between literals).
- Derivability ( $\vdash_B$ ): Derivability in a base is defined inductively. It includes standard structural rules (reflexivity, application) and the two new clauses (exc1, exc2) that enforce duality at the literal level.
- Support ( $\Vdash_B$ ): The validity of a formula is defined inductively on the structure of formulae. Crucially, the support clauses for connectives ( $\to, \land, \lor$ ) are identical to Sandqvist's intuitionistic definitions. The classical character of the semantics is carried entirely by the duality clauses in the definition of derivability in the base, not by the support clauses for complex formulae.
- Validity ( $\Gamma \Vdash \phi$ ): A formula is valid if it is supported in every base.

4. Key Results

Soundness (Theorem 5.1): The system is sound: if $\Gamma \vdash_{NK^\pm} \phi$ , then $\Gamma \Vdash \phi$ . The proof proceeds by induction on the derivation, showing that the classical rules (EXC1, EXC2) preserve support.
Completeness (Theorem 6.2): The system is complete: if $\Gamma \Vdash \phi$ $Γ ⊩ ϕ$ , then $\Gamma \vdash_{NK^\pm} \phi$ $Γ ⊢_{N K^{\pm}} ϕ$ .
- The proof adapts Sandqvist's elementary method for intuitionistic logic.
- It utilizes a "simulation base" ( $\mathcal{N}$ ) constructed from the subformulae of the sequent.
- A "flattening" operator maps complex formulae to atomic literals within the simulation base.
- The "Naturalizing" lemma demonstrates that derivability in the simulation base can be translated back into derivability in $NK^\pm$ .
- Unlike previous approaches, this proof handles the full propositional language (including $\lor$ ) and relies on elementary mathematical resources, avoiding bar induction or canonical models.

5. Significance and Claims
The paper claims to provide a novel proof-theoretic semantics for classical logic that:

Unifies IPL and CPL: It demonstrates that classical logic can be understood as intuitionistic logic supplemented by a primitive principle of duality at the atomic level.
Respects Anti-Realism: By avoiding truth values and bivalence, and by using a constructive completeness proof, it offers a genuinely anti-realist foundation for classical logic, addressing Dummett's foundational challenge.
Simplifies Technical Complexity: By restricting bilateralism to literals and treating complex formulae as structural, the authors achieve a completeness proof of "strikingly elementary character" comparable to Sandqvist's work on intuitionistic logic, overcoming the technical difficulties of previous classical B-eS attempts.
Offers an Alternative to Model Theory: The authors position this work as an alternative to model-theoretic semantics and existing P-tS approaches that rely on model-theoretic concepts (like those of Makinson or Eckhardt and Pym), aiming to establish a purely inferentialist account of classical logic.

The paper concludes that this framework opens the possibility for a systematic treatment of classical logics analogous to the systematic treatment already established for intuitionistic logics within the P-tS tradition.