Paper III · revised 2026.04

Paraconsensus

Paraconsistent four-valued logic for federated meta-learning

By Larry Klosowski, Kurt McKelvey

Abstract

Standard consensus protocols treat disagreement as failure: when nodes produce contradictory outputs, the protocol picks one and discards the others. Paraconsensus extends consensus to a paraconsistent four-valued logic — true, false, both, neither — that preserves disagreement as information. The paper develops the formal model, proves liveness and safety in the BFT setting, and demonstrates the protocol's performance on a federated meta-learning workload where contradictory model outputs carry epistemic content. The Citrate Network's inference verification layer is the production implementation.

PDF coming soon — the LaTeX pipeline lands in a forthcoming sprint

1. The problem with discarded disagreement

Classical Byzantine fault-tolerant consensus produces a single agreed-upon value. When nodes produce contradictory outputs, the protocol selects one output (by quorum, by leader-election, by ordering) and discards the rest. The discarded outputs are treated as failures.

For inference workloads, this is a category error. When two correctly-running inference nodes produce contradictory outputs, the contradiction is itself information about the input — typically that the input is at the edge of the model’s training distribution, or that the model has internal instability at this input. Discarding the contradiction discards the epistemic signal.¹

2. Four-valued logic

Paraconsistent logic admits four truth values: true, false, both, neither. The both value represents simultaneous evidence for and against a proposition; the neither value represents absence of evidence in either direction. Classical logic collapses these into true or false; paraconsistent logic preserves them. We denote the four-valued truth set as \(V_4\):

\[V_4 = \{\top,\ \bot,\ \mathbf{B},\ \mathbf{N}\}\]

where \(\top\) is true, \(\bot\) is false, \(\mathbf{B}\) is both, and \(\mathbf{N}\) is neither. The lattice on \(V_4\) is the Belnap-Dunn bilattice; both meet/join operators on the truth-order and the information-order are well defined.

Paraconsensus extends BFT consensus to the four-valued setting: when nodes produce contradictory outputs \(o_i, o_j\) on the same input \(x\), the protocol records the contradiction as the both value rather than discarding either output. The recording rule is given by (·):

\[\text{commit}(x) = \begin{cases} \top & \text{if } o_i = \top\ \forall i \\ \bot & \text{if } o_i = \bot\ \forall i \\ \mathbf{B} & \text{if } \exists i, j : o_i = \top \land o_j = \bot \\ \mathbf{N} & \text{otherwise} \end{cases}\]

Downstream consumers read the four-valued state and decide how to handle disagreement on a per-application basis. The \(\mathbf{B}\) commit is not a failure mode; it is a primary protocol output.

3. Safety and liveness

The paper proves that Paraconsensus retains BFT’s safety property — no two correct nodes commit different values for the same proposition at the same height — extended to the four-valued lattice. Formally, for any height \(h\) and any two correct nodes \(i, j\):

\[\forall h \in \mathbb{N},\ \forall i, j \in C : \text{commit}_i(h) = \text{commit}_j(h)\]

The liveness property — every proposition eventually reaches a four-valued commit — follows from BFT liveness applied component-wise to the four possible commit values. The combined safety and liveness statement is (·) with the existence claim for \(\text{commit}_i(h)\) for every height. The proofs are constructive: the protocol is implementable on the same network primitives BFT uses, with one additional message type for the both commit.

4. Production implementation

The Citrate Network’s inference verification layer is the production implementation. Each verified inference produces a four-valued state; the verification API exposes the full state to consumers, and the default consumer-facing render collapses both to a "disagreement detected" indicator that links to the underlying contradicting outputs.

The implementation has run continuously since 2026.02 with no reported consistency violations and a measured liveness characteristic within 1.4× the classical BFT baseline.