Bilateral Natural Deduction System

• Bilateral Natural Deduction System is a logical framework that formalizes both proofs and refutations with distinct rules for assertion and denial.
• It employs dual introduction and elimination rules along with coordination strategies to prevent simultaneous derivations, ensuring normalization and the subformula property.
• Its robust structure supports computational applications such as proof assistants and theorem provers, aligning constructive falsity with formal epistemic principles.

A bilateral natural deduction system is a proof-theoretic framework in which both the assertion (provability) and denial (refutability) of formulas are treated as distinct but fundamentally related inferential acts. Unlike standard natural deduction systems, which encode only proof-theoretic rules for assertion, a bilateral system formalizes rules for refutation in parallel, enforcing a rigorous opposition between what is provable and what is refutable. Recent developments have produced systems in which explicit normalization, semantic consistency, and computational tractability are achieved by forbidding the simultaneous provability and refutability of any formula, aligning constructive interpretations of negation and falsity with epistemic exclusion principles (Barroso-Nascimento et al., 19 Oct 2025).

1. Formal Structure of Bilateral Natural Deduction

A bilateral system comprises derivations annotated with two sets of assumptions: one for asserted formulas and one for denied formulas, customarily written as sequents

$\Gamma ; \Delta \vdash^{*} \phi$

where $\Gamma$ denotes the set of proven/assumed formulas, $\Delta$ the set of refuted/denied formulas, and the marker $*$ is $+$ for a proof and $-$ for a refutation. Each connective in the language (such as $\land$, $\lor$, $\rightarrow$, and $\mapsfrom$) admits both introduction and elimination rules for assertion (thin inference lines) and for refutation (double inference lines). Coordination rules—typically labeled $\mathrm{PR}(+)$ and $\mathrm{PR}(-)$—operate to assert the incompatibility between proof and refutation for a given formula, so that deriving both $\Gamma;\Delta \vdash^+ \phi$ and $\Gamma;\Delta \vdash^- \phi$ forces inconsistency.

This architecture builds on, and further develops, frameworks such as Wansing’s N2Int and the bi-intuitionistic logic 2Int, but in the constructive variant advanced by (Barroso-Nascimento et al., 19 Oct 2025), explicit mechanisms prevent the simultaneous derivability of a formula and its denial. Moreover, the system extends intuitionistic logic by incorporating a Nelson-style notion of constructive falsity for refutation, rather than classical negation.

2. Proof-Theoretic Normalization and Subformula Property

The main proof-theoretic achievement is a normalization theorem: every derivation in the bilateral natural deduction system admits a reduction to normal form via elimination of detours (e.g., consecutive introduction followed by elimination of the same connective, or redundant use of coordination rules). The technical heart of normalization is a complexity metric:

• Degree $d[\phi]$ is computed recursively, for an atomic $p$, $d[p] = 0$; for a compound $\phi \circ \psi$, $d[\phi \circ \psi] = d[\phi] + d[\psi] + 1$ for each $\circ \in \{\to, \land, \lor, \mapsfrom\}$.
• Pair $\langle d, n \rangle$, where $n$ counts occurrences at maximal degree involved in critical rule instances ($\mathrm{PR}(+),\mathrm{PR}(-),\bot(+),\top(-)$).

Normalized derivations satisfy the subformula property: every formula occurring is a subformula of the conclusion or of an undischarged assumption. This analytic character prohibits “wild” inferential jumps, constrains proof complexity, and guarantees consistency (no derivations of $\bot$ or refutations of $\top$ from the empty context).

3. Base-Extension Semantics and Epistemic Adequacy

The semantics for the bilateral system is given by base-extension semantics (BeS). Here, meaning is determined not by truth in a classical model, but by explicit constructions from a bilateral atomic base $\mathcal{B}$—a set of atomic formulas (including designated symbols for $\bot,\top$).

For atoms, conditions read:

$$\mathcal{B} \Vdash_+ p \quad \Leftrightarrow \quad \mathcal{B} \vdash_+ p$$

$$\mathcal{B} \Vdash_- p \quad \Leftrightarrow \quad \mathcal{B} \vdash_- p$$

For compound formulas, proof and refutation conditions respect their logical constructors. For instance, positive validity for conjunction is:

$$\mathcal{B} \Vdash_+ (\phi \land \psi) \quad \Leftrightarrow \quad \mathcal{B} \Vdash_+ \phi \;\text{and}\; \mathcal{B} \Vdash_+ \psi$$

and, for refutation (negative validity):

$$\mathcal{B} \Vdash_- (\phi \land \psi) \quad \Leftrightarrow \quad \forall \mathcal{C} \supseteq \mathcal{B},\, \forall p,\, \text{if}\; \emptyset; \phi \Vdash_{\mathcal{C}}^+ p \;\text{and}\; \emptyset; \psi \Vdash_{\mathcal{C}}^+ p \;\text{then}\; \mathcal{C} \Vdash_{+} p$$

Critically, epistemic adequacy requires that, for any $\mathcal{B}$, provability and refutability of any $\phi$ are mutually exclusive:

$$\mathcal{B} \Vdash_+ \phi \implies \lnot \mathcal{B} \Vdash_- \phi,\quad \mathcal{B} \Vdash_- \phi \implies \lnot \mathcal{B} \Vdash_+ \phi$$

thus formally modeling the epistemic exclusion principle.

4. Soundness, Completeness, and Simulation Bases

Soundness asserts: if $\Gamma;\Delta \vdash^* \phi$ is derivable, then $\Gamma;\Delta \Vdash^* \phi$ semantically holds in every epistemically adequate base. Completeness is shown constructively by mapping formulas to atomic representatives (maintaining $\bot,\top$), and building simulation bases so that semantic validity is witnessed by syntactic derivability. The key lemma formalizes:

$$\mathcal{B} \Vdash^* \phi \iff \mathcal{B} \vdash^* p^{\phi}$$

where $p^{\phi}$ is the atom associated to $\phi$.

5. Relation to Constructive Falsity and Nelson Logic

The refutation mechanism in bilateral natural deduction aligns with David Nelson's constructive falsity rather than classical negation. In Nelson’s logic, refuting $\phi$ (establishing “strong negation”) requires explicit counter-construction rather than merely failing to construct a proof of $\phi$, i.e., $\sim\phi$ is not defined as $\phi\to\bot$, but as an explicit dual with its own introduction and elimination rules. In BPR, refutation rules admit dual elimination principles and coordination rules, ensuring that constructing a refutation differs fundamentally from merely inferring $\phi\to\bot$ intuitionistically. The system allows both intuitionistic (weak) negation and constructive (strong) refutation, but preserves their inferential separation via semantic and syntactic constraints.

6. Semantic Harmony, Epistemic Applications, and Computational Interpretations

The bilateral system offers a modular and semantic-harmonious framework for proof-theoretic semantics: connective meanings are given “by use”—via their proof and refutation rules. This enables the rigorous modeling of constructive epistemic attitudes: both positive proofs (evidence) and counterexamples (refutations) form first-class inferential objects. This has direct computational applications: proof assistants, automated theorem provers, and systems for formal epistemology benefit from a logic where evidence and counterevidence are symmetrically formalized and computationally tractable. The normalization theorem and the subformula property also endow the system with strong computational properties (e.g., proof search complexity, reducibility).

Example Table: Core Elements of BPR System

Component	Proof Rule	Refutation Rule
Sequent Form	$\Gamma;\Delta\vdash^+\phi$	$\Gamma;\Delta\vdash^-\phi$
Negation	$\phi\to\bot$	Explicit counterexample
Coordination	PR(+): asserts $\phi$ and refutes $\phi \Rightarrow \bot$	PR(-): refutes and asserts $\phi \Rightarrow \bot$

Conclusion

The bilateral natural deduction system formalizes a logic in which proofs and refutations are syntactically and semantically incompatible, yet equally fundamental. By introducing explicit coordination rules and base-extension semantics that enforce the mutual exclusion of assertion and denial, the system strengthens constructive logic with a Nelson-style notion of falsity, underpins normalization and semantic harmony, and provides a robust foundational mechanism for applications in proof theory, epistemology, and computational logic (Barroso-Nascimento et al., 19 Oct 2025).