Continuous Modal Logical Neural Networks: Modal Reasoning via Stochastic Accessibility

Antonin Sulc

cs.LOMar 4, 2026

Continuous Modal Logical Neural Networks: Modal Reasoning via Stochastic Accessibility

Authors: Antonin Sulc

Organizations: Lawrence Berkeley National Laboratory, Berkeley, CA, USA

Abstract

We propose Fluid Logic, a paradigm in which modal logical reasoning, temporal, epistemic, doxastic, deontic, is lifted from discrete Kripke structures to continuous manifolds via Neural Stochastic Differential Equations (Neural SDEs). Each type of modal operator is backed by a dedicated Neural SDE, and nested formulas compose these SDEs in a single differentiable graph. A key instantiation is Logic-Informed Neural Networks (LINNs): analogous to Physics-Informed Neural Networks (PINNs), LINNs embed modal logical formulas such as ( $\Box$ bounded) and ( $\Diamond$ visits_lobe) directly into the training loss, guiding neural networks to produce solutions that are structurally consistent with prescribed logical properties, without requiring knowledge of the governing equations. The resulting framework, Continuous Modal Logical Neural Networks (CMLNNs), yields several key properties: (i) stochastic diffusion prevents quantifier collapse ( $\Box$ and $\Diamond$ differ), unlike deterministic ODEs; (ii) modal operators are entropic risk measures, sound with respect to risk-based semantics with explicit Monte Carlo concentration guarantees; (iii)SDE-induced accessibility provides structural correspondence with classical modal axioms; (iv) parameterizing accessibility through dynamics reduces memory from quadratic in world count to linear in parameters. Three case studies demonstrate that Fluid Logic and LINNs can guide neural networks to produce consistent solutions across diverse domains: epistemic/doxastic logic (multi-robot hallucination detection), temporal logic (recovering the Lorenz attractor geometry from logical constraints alone), and deontic logic (learning safe confinement dynamics from a logical specification).

Figures & tables

Panel of Figure 1 — Figure 1: The transition from Graph Logic to Fluid Logic. (A) MLNNs operate over finite world sets with static, enumerable accessibility relations. (B) CMLNNs embed worlds in a continuous manifold $\mathcal{W}$ where each modal operator $\Box_{i},\Diamond_{i}$ is defined by its own Neural SDE. The stochastic diffusion coefficient $\sigma^{(i)}_{\theta}$ generates genuine branching, giving $\Box$ (all sample paths) and $\Diamond$ (exists a sample path) semantically distinct meanings. Gradients flow through the SDE solver via the stochastic adjoint method.

Figure 2: Case Study 1: three multi-agent scenarios. (a) Hallucination detection (epistemic + doxastic SDEs): Rover 3 (faulty sensor) believes safety where danger exists. Solid lines show true trajectories; dashed red is the doxastic path; dotted purple is the epistemic path. The hallucination flag triggers when

L_{B_{3}}(\Box\,\text{safe}){>}0.8

and

U_{K_{\text{sw}}}(\Diamond\,\text{coll.}){>}0.3

. (b) Sustained escort (temporal SDE): a rescuer (blue) learns to shield Rover 3 (orange) around the true chasm toward the goal (star); final actual safety

0.82

, trust gap

W{=}1.53

. (c) Swarm defense (5 temporal formulas): the VIP (yellow) is escorted by scouts S1/S2 while evading enemies E1/E2; five logic objectives are jointly optimised. Additional detail in Appendix E .

Figure 3: Case Study 2: 3D attractor comparison on Lorenz-63. Ground truth shows the canonical two-lobe butterfly; the Pure MSE SDE collapses to one lobe; SDE+LINN recovers both lobes via

\Box(\text{bounded})\land\Diamond(\text{visits\_lobe})

Figure 4: Case Study 2: LINN as a logic regularizer. Left: Adding LINN to any base model reduces multi-step MSE and escape rate. Right: Lobe visitation frequency. Baseline models appear to have a balanced average Lobe Error, but suffer from structural collapse (individual trajectories rarely switch lobes). LINN forces explicit exploration of both lobes, recovering the global geometry at the cost of slightly perturbing the natural switching frequency.

Model	1-Step	Multi-S	$\Box$ MAE	$\Diamond$ MAE	$\Delta_{Q}$ MAE	Lobe Err	Esc.%
Neural ODE	0.013	132.2	0.133 †	0.063 †	0.124 †	0.012	0
Pure MSE SDE	0.015	68.7	0.112	0.074	0.121	0.006	0
PINN (det.)	0.814	16084	0.400 †	0.478 †	0.124 †	0.010	100
PINN + noise	0.816	12523	0.400	0.477	0.123	0.001	100
Ensemble ODE	0.013	134.4	0.119	0.071	0.119	0.016	0
ODE + LINN	0.064	111.3	0.091	0.079	0.130	0.771	0

Table 1: Case Study 2 results (Lorenz-63).

\dagger

: quantifier collapse (

\Box=\Diamond

Figure 5: Case Study 3: Deontic safe confinement. Left: 3D toroidal trajectories (poloidal cross-section wrapped onto torus geometry). Red paths show the temporal SDE (unconstrained physics, drifting outward and exiting the vessel). Blue paths show the deontic SDE (trained on

O(\Box_{[0,T]}\text{safe})

, staying confined with 0% exit). The confinement structure emerges from the logical objective alone, without a hand-crafted control law. Right:

L_{\Box}

and

U_{\Diamond}

\phi_{\text{safe}}

over time. The deontic SDE (blue) maintains

L_{\Box\text{safe}}=0.456

(6.1

\times

above temporal 0.075); the shaded quantifier gap

U_{\Diamond}-L_{\Box}>0

confirms non-collapsed quantifiers in both SDEs. Drift magnitude heatmaps showing the learned restoring force appear in Appendix E .

SDE	$L_{\Box\text{safe}}$	$U_{\Diamond\text{safe}}$	Exit %	$\Delta_{Q}$
Temporal	0.075	0.797	61.7	0.722
Deontic	0.456	0.857	0.0	0.401
Trivial baseline	0.148	—	24.2	—

Table 2: Case Study 3 results (deontic confinement).

Appendix figures & tables19 assets

Supplementary material from the paper’s appendix.

Appendix

Panel of Figure 6 — Figure 6: Quantifier non-collapse visualised. From the same starting world $w$ , four SDE sample paths fan out with different per-path truth scores $g_{n}$ (soft minima of $L_{\phi}$ along the path). Left ( $\Box$ ): necessity aggregates via soft min, so the worst-case path (red, $g=0.2$ ) dominates. Right ( $\Diamond$ ): possibility aggregates via soft max, so the best-case path (blue, $h=0.9$ ) dominates. Because the SDE produces a distribution of paths, $L_{\Box}<U_{\Diamond}$ (the gap is $0.7$ here). For a deterministic ODE, all four paths would coincide and $L_{\Box}=U_{\Diamond}$ (quantifier collapse).

Axiom	Property	SDE Mechanism	Status
T : $\Box\phi\to\phi$	Veridicality	$z(0)=x_{\text{true}}$	Encouraged (gap)
D : $\Box\phi\to\Diamond\phi$	Seriality	$\geq 1$ path	Free
4 : $\Box\phi\to\Box\Box\phi$	Introspection	Semigroup	Approx.
$\neg$ T	Fallibility	$z(0)=\hat{x}\neq x$	Enforced

Appendix

Table 3: Modal axioms and SDE structural counterparts.

#	Experiment	Modal Type	Key Formula	SDE	Domain
1	Swarm Hallucination	Epist.+Dox.	$B_{\text{r3}}(\Box\text{safe})\land K_{\text{sw}}(\Diamond\text{collision})$	Epist.+Dox.	Custom 2D Swarm
2	Lorenz Chaotic	Temporal	$\Box\text{(bounded)}\land\Diamond\text{(visits\_lobe)}$	Temporal	Lorenz-63
3	Safe Confinement	Deontic	$O(\Box\text{safe})$	Temp.+Deon.	2D Vessel

Appendix

Table 4: Experimental design: each case study exercises a different modal logic.

Figure 7: Training dynamics (5 seeds, mean

\pm

std). Left: doxastic bound

L_{B_{3}}(\Box\,\text{safe})

rises above the 0.8 threshold. Centre: epistemic bound

U_{K_{\text{sw}}}(\Diamond\,\text{collision})

rises above 0.3. Right: Wasserstein belief–reality gap stabilises at

W=0.39

Figure 8: Safety field over the arena. Left: true danger gradient centred on the real chasm. Right: danger field as believed by Rover 3, centred on the hallucinated (fake) chasm.

Figure 9: Spatial modal truth field ( $50\times 50$ grid; raw robustness $L$ , $U$ ). Left: $L_{B_{3}}(\Box\,\text{safe})$ . Right: $U_{K_{\text{swarm}}}(\Diamond\,\text{collision})$ . White dashed contours show detection thresholds.

Figure 10: Sub-Exp B: 2-rover sustained escort. The rescuer (blue) shields Rover 3 (orange) past the true chasm while R3 homes toward the goal (star). Final actual safety:

0.82

Figure 11: Escort training metrics. Left: actual safety margin improves. Centre: believed safety remains high (R3 never perceives the true hazard). Right: Wasserstein trust gap (

W=1.53

Figure 12: Sub-Exp C: 5-agent swarm defense (VIP protection). Two scouts defend a VIP against two enemies while the VIP explores toward the goal.

Figure 13: Defense formula satisfaction. Five temporal logic objectives jointly optimised.

Figure 14: Training dynamics. (a) MSE pre-training. (b) Logic loss and

\lambda

ramp. (c) Formula satisfaction converges.

Panel of Figure 15 — Figure 15: (a) $\Diamond\phi-\Box\phi$ over forecast horizon: SDE+LINN tracks the ground truth while deterministic models remain at zero (quantifier collapse). (b) Lobe visitation: SDE+LINN matches the ground truth ${\sim}0.5/0.5$ balance.

Figure 16: Soundness verification: predicted

\Box\leq

\Box

(left) and predicted

\Diamond\geq

\Diamond

(right). SDE+LINN satisfies both.

Figure 17: Phase portraits (

x

–

z

projection) for all eight models plus ground truth. SDE+LINN produces the richest two-lobe attractor coverage.

Figure 18: Overview dashboard. Top-left: temporal (red) vs deontic (blue) trajectory bundles. Top-right: modal truth values and quantifier gap. Bottom-left: 3D toroidal embedding. Bottom-right: learned drift fields; the deontic drift curves inward at the wall.

Figure 19: Per-path safety

\phi_{\text{safe}}(z_{t})

. Left: temporal dynamics, where safety degrades over time. Right: deontic dynamics, where safety is maintained throughout the horizon.

Panel of Figure 20 — Figure 20: (a) Exit fraction: temporal rises steadily while deontic remains at zero. (b) Radial distributions: deontic paths are concentrated inside the vessel.

Figure 21: Deontic training curves (multi-seed). (a)

L_{\Box}

and

U_{\Diamond}

increase. (b) Exit fraction drops to zero. (c) Loss converges.

Figure 22: Drift magnitude heatmaps. Left: temporal drift (unconstrained physics). Right: deontic drift, showing a learned strong restoring force (bright ring) near the vessel boundary.

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Lawrence Berkeley National Laboratory, Berkeley, CA, USA · The University of Queensland, Brisbane, QLD, Australia

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Neural Networks are indispensable to natural sciences and society. Their impact extends from applications in public health to workforce productivity. Here, we introduce Modal Logic Neural Networks (MLNNs) -- an end-to-end differentiable logical neural network realisation of modal logic which evaluates a learnable truth function across possible-world semantics. This neural architecture handles para-consistency and inconsistency via a learnable world accessibility relation and valuation function. Because the modality is fixed by which frame axioms the relation satisfies rather than by the operator, one differentiable engine covers the epistemic, doxastic, deontic and temporal readings, with applications from verification of reactive and distributed systems to legal discourse and microeconomic utility models. In this paper, we introduce a model of differentiable Kripke semantics, and establish their soundness, convergence, and structural guarantees. We show four applications, in which the learned relation reads as a trust matrix, an operating-regime embedding with safety bounds, a temporal precedence order, and a recovered constraint graph.

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