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Glivenko's theorems from an ecumenical perspective

This paper reexamines Glivenko's theorems, which link classical and intuitionistic logic, through an ecumenical lens by analyzing their historical context and extensions within three specific systems: Prawitz's NE, Krauss's NEK, and Barroso-Nascimento's ECI.

Original authors: Luiz Carlos Pereira, Victor Barroso-Nascimento, Elaine Pimentel

Published 2026-05-05

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CC BY 4.0

Original authors: Luiz Carlos Pereira, Victor Barroso-Nascimento, Elaine Pimentel

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 hosting a dinner party where two very different groups of guests are arriving: the Classical Logicians and the Intuitionistic Logicians.

The Classical Logicians are like people who believe that if you can prove something can't be false, then it must be true. They are comfortable with "double negatives" canceling each other out to make a positive statement. They are confident, decisive, and willing to say "It is true" even if they haven't built the object yet, as long as they know it's impossible for it not to exist.
The Intuitionistic Logicians are like careful builders. They only say "It is true" if they have actually constructed the proof or the object. To them, saying "It's not false" isn't enough; they need to see the thing itself.

For a long time, these two groups spoke different languages. But in 1929, a mathematician named Valery Glivenko discovered a fascinating translation trick. He found that if a Classical Logician proves a statement, an Intuitionistic Logician can prove that "It is not the case that the statement is false." In other words, you can translate a Classical victory into an Intuitionistic "double-negative" victory.

This paper, written by Pereira, Barroso-Nascimento, and Pimentel, takes Glivenko's old trick and asks: What happens if we put both groups in the same room, using a single, unified system? They call this an "Ecumenical" perspective (from the Greek for "universal" or "worldwide").

Here is how the paper breaks down this experiment using three different "dinner party" setups:

1. The "Two-Sided" Room (Prawitz's System NE)

Imagine a room where the guests share some furniture (like a table for "AND" or a chair for "NOT") but have their own distinct tools for other tasks.

In this setup, there is a Classical "OR" and an Intuitionistic "OR". They look similar but work differently.
The authors show that even in this shared room, Glivenko's trick still works internally. If you use the Classical "OR" to prove something, you can translate it into the Intuitionistic "OR" by wrapping it in a "double negative."
The Analogy: It's like having a red button and a blue button. If you press the red button (Classical), you can prove that pressing the blue button (Intuitionistic) twice in a row will also get the job done. The paper proves this relationship holds for "OR," "IMPLIES," and "EXISTS."

2. The "Labeling" Room (The ECI System)

This system is different. Instead of having two different buttons, there is only one set of buttons, but you can stick a special sticker (the label c) on them to say, "This one is being used Classically."

If you have a statement $A$ , it's Intuitionistic. If you have $A^c$ (A with a sticker), it's Classical.
In this system, Glivenko's trick becomes almost too easy. The paper shows that if you have a Classical statement $A^c$ , it is automatically equivalent to saying "It is not the case that A is false" ( $\neg\neg A$ ).
The Catch: The authors point out a weird glitch when you add Universal Quantifiers (statements about "everything"). In this "Labeling" room, the sticker trick makes it look like Glivenko's theorem works for "everything," but it's actually a trick of the labels. It's like saying, "If I label this box 'Classical,' it magically becomes 'Double-Negative Intuitionistic'." The paper argues this is a bit of a mirage because the sticker changes the meaning of the box in a way that doesn't quite match the real-world logic of "everything."

3. The "Hybrid" Room (The NEK System)

This setup is a mix. It starts with the "Two-Sided" room but adds a Classical "AND" and a Classical "Universal" (for "everything").

The authors compare this system to the "Labeling" room (ECI).
The Big Discovery: For simple statements (without "everything"), the "Labeling" room and the "Hybrid" room are essentially the same. You can translate back and forth perfectly.
The Divergence: However, once you introduce the word "Everything" (Universal Quantifier), the two systems split apart.
- In the Hybrid Room, the Classical "Everything" is a strong, distinct tool.
- In the Labeling Room, the "Classical Everything" is just a sticker on an Intuitionistic "Everything."
- The paper argues that the Hybrid Room (NEK) is the more honest representation of what a Classical Logician actually means when they say "Everything." The Labeling Room (ECI) is a clever shortcut that works for simple things but breaks down when you try to talk about the whole universe.

The Core Takeaway

The paper isn't just about math rules; it's about how we define meaning.

Approach A (ECI): Change the proof (the method) to change the meaning. "If I use a classical proof method, this statement becomes classical."
Approach B (NE/NEK): Change the tool (the connective) itself. "This 'AND' is built differently from the start."

The authors conclude that while both approaches work for simple logic, they are fundamentally different when dealing with complex concepts like "everything." The "Labeling" approach (ECI) makes Glivenko's theorem look trivial and universal, but it hides the fact that Classical and Intuitionistic logic are actually doing different things. The "Hybrid" approach (NEK) respects the distinct nature of Classical logic, showing that you can't just slap a sticker on a statement and expect it to behave exactly like the original Intuitionistic one wrapped in a double negative.

In short: You can translate Classical logic into Intuitionistic logic using Glivenko's double-negation trick, but if you try to merge them into one system, you have to decide: do you want to change the tools themselves (which keeps them distinct and honest), or do you want to change the rules of the game (which creates a clever but potentially misleading shortcut)? The paper suggests that for a deep understanding of logic, changing the tools is the more faithful path.

Technical Summary: Glivenko's Theorems from an Ecumenical Perspective

Problem Statement
The paper addresses the foundational relationship between classical logic (CL) and intuitionistic logic (IL), specifically revisiting Glivenko's theorems. These theorems establish that if a formula is classically provable, its double negation is intuitionistically provable, and if a negated formula is classically provable, it is also intuitionistically provable. While these results are well-understood in the context of separate logical systems, the paper investigates how these relationships manifest within ecumenical systems. Ecumenical systems are unified frameworks where classical and intuitionistic logics coexist peacefully without collapsing into one another, often by distinguishing operators (e.g., classical vs. intuitionistic disjunction) or by labeling formulas to indicate their logical nature. The central problem is to determine how Glivenko-type results translate when classical and intuitionistic reasoning are integrated into a single deductive system, and to analyze the differences between two primary approaches to ecumenism: the "inferentialist" approach (distinguishing operators) and the "generalist" approach (distinguishing formulas via labels).

Methodology
The authors employ a proof-theoretic analysis using Natural Deduction systems. They examine three specific ecumenical frameworks:

NE (Prawitz's System): An inferentialist approach where classical and intuitionistic logicians share neutral operators (conjunction, universal quantifier, negation, absurdity) but possess distinct versions of non-neutral operators (implication, disjunction, existential quantifier).
ECI (Barroso-Nascimento's System): A generalist approach where a single set of operators is used, but formulas are labeled with a constant $c$ (e.g., $A^c$ ) to denote classical behavior. This system is built upon intuitionistic natural deduction augmented with introduction ( $I_C$ ) and elimination ( $E_C$ ) rules for the label.
NEK (Krauss-inspired System): A variation of Prawitz's system that includes a classical conjunction ( $\land_c$ ) and a classical universal quantifier ( $\forall_c$ ), based on Krauss's unpublished work.

The methodology involves:

Defining internal Glivenko-type theorems within these systems (relating classical operators/formulas to their intuitionistic counterparts).
Constructing derivations to prove these internal relations.
Defining translation functions ( $t_{ECI}$ and $t_{NEK}$ ) to map formulas between systems.
Investigating the deductive equivalence of these systems, particularly in the propositional fragment versus the first-order fragment involving universal quantification.

Key Contributions and Results

Internal Glivenko Theorems in NE:
The authors demonstrate that in Prawitz's system (NE), where distinct operators exist, a "Glivenko-type" relationship holds internally. For any formula $C$ where the main operator is classical, $C \vdash_{NE} \neg\neg C^*$ , where $C^*$ is the formula with the classical operator replaced by its intuitionistic counterpart. This applies to classical disjunction ( $\lor_c$ ), implication ( $\to_c$ ), and existential quantification ( $\exists_c$ ). A corresponding result holds for negated formulas: $\neg C \vdash_{NE} \neg C^*$ .
Triviality of Glivenko Results in ECI:
In the ECI system, the relationship between a classical formula $A^c$ and its intuitionistic counterpart is structurally trivial. The system's rules immediately yield $A^c \vdash_{ECI} \neg\neg A$ . Consequently, the translation $t_{ECI}$ , which maps $A^c$ to $\neg\neg A$ , reduces the problem of normalization in ECI to that of intuitionistic logic. The authors show that while this translation works for propositional logic, it obscures the specific location of classical reasoning in a derivation unless the derivation is carefully reconstructed to track the "classical nature" of derived formulas.
The Case of Classical Conjunction and NEK:
The paper introduces NEK, which adds classical conjunction ( $\land_c$ ) to Prawitz's system. The authors prove that $\land_c$ satisfies the internal Glivenko theorems ( $B \land_c C \vdash_{NEK} \neg\neg(B \land_i C)$ ). They also show that in the propositional fragment (or first-order logic without universal quantifiers), NEK and ECI are deductively equivalent. This equivalence is established via a translation $t_{NEK}$ that maps classical operators in NEK to labeled formulas in ECI.
The Divergence with Universal Quantification:
A critical result is the failure of deductive equivalence between NEK and ECI when universal quantification is introduced.
- In NEK, the classical universal quantifier $\forall_c$ is defined via rules that are harmonic with intuitionistic logic. However, the internal Glivenko theorem fails for $\forall_c$ : $\vdash_{NEK} \forall_c x A(x)$ does not imply $\vdash_{NEK} \neg\neg \forall_i x A(x)$ .
- In ECI, the labeled formula $(\forall x A(x))^c$ behaves differently. Due to the system's rules, $(\forall x A(x))^c \vdash_{ECI} \neg\neg \forall x A(x)$ holds.
- The authors explain this discrepancy: In NEK, $\forall_c x A(x)$ is interpreted as $\neg \exists x \neg A(x)$ , whereas in ECI, $(\forall x A(x))^c$ is effectively interpreted as $\neg\neg \forall x A(x)$ . These are not intuitionistically equivalent. The paper argues that the "triviality" of Glivenko results in ECI for universal formulas arises because the label $c$ forces a double-negation interpretation that does not align with the semantic behavior of the classical universal quantifier in the inferentialist approach.

Significance and Claims
The paper claims that the ecumenical perspective offers a deeper understanding of the interplay between classical and intuitionistic reasoning. By comparing the "inferentialist" (NE/NEK) and "generalist" (ECI) approaches, the authors highlight a subtle but significant conceptual difference:

Inferentialist systems (Prawitz/Krauss) change the meaning of connectives directly by providing distinct introduction and elimination rules for classical operators.
Generalist systems (ECI) change the level of proof, equipping intuitionistic logic with classical means of proof (via the label $c$ ) that indirectly alter the meaning of operators.

The authors conclude that while these approaches are equivalent in propositional logic, they diverge in the presence of universal quantification. This divergence suggests that the choice of ecumenical framework is not merely a notational preference but affects the logical behavior of quantifiers. The paper posits that the ecumenical perspective helps identify exactly where classical reasoning is necessary and clarifies that classical and intuitionistic principles can coexist without one collapsing into the other, provided the system is constructed with attention to these distinct inferential principles.

1. The "Two-Sided" Room (Prawitz's System NE)

2. The "Labeling" Room (The ECI System)

3. The "Hybrid" Room (The NEK System)

The Core Takeaway

Technical Summary: Glivenko's Theorems from an Ecumenical Perspective

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