On the Conditions for the Validity of the Law of Excluded Middle—The Mutual Reinforcement of Weyl’s Mathematical Boundary and the Spatiotemporal-Observation Boundary

Author: Zhang Suhang

Luoyang School of Mathematics

Abstract

Since Aristotle founded formal logic, the law of excluded middle has long been regarded as an unconditionally universal foundational axiom, giving rise to the ingrained assumption that binary logic can be applied without qualification to all objects. In the twentieth century, Hermann Weyl, starting from constructive mathematics and infinite domains of discourse, was among the first to reflect on the scope of application of the law of excluded middle, pointing out that it cannot be directly applied to infinite continuum systems. Continuing Weyl’s line of thought, this paper introduces two classes of physical examples: the temporal indeterminacy corresponding to quantum superposition states, and the multi-scale coexistence characteristic of real fluid flow fields, thereby supplementing the constraints on the law of excluded middle from the dimension of physical observation. Weyl’s boundary belongs to the level of mathematical construction, whereas the boundary proposed in this paper belongs to the level of spatiotemporal observation; the two are mutually complementary and reinforcing. This paper derives the preconditions for the reasonable use of the law of excluded middle: a finite and constructible domain of discourse, a unified observational scale, and a determinate temporal state. When these conditions are absent, simple binary judgment is no longer appropriate. This paper does not negate the law of excluded middle; rather, it corrects its unconditional generalization and use beyond its proper domain, delineates its applicable boundary, and clarifies the limitations of unconditional binary thinking.

Keywords: law of excluded middle; binary logic; conditionality of logic; Weyl; quantum superposition; fluid scale; observational domain of discourse

1 Introduction

The law of excluded middle is one of the three basic laws of formal logic. Its classical formulation is: for any proposition, either the proposition or its negation must be true, and there is no intermediate state. For a long time, many have assumed that this rule is unrestricted, and that mathematical objects and natural phenomena alike can be judged directly in an “either-or” manner.

This understanding came under reflection in the twentieth century. Influenced by intuitionist thought, Hermann Weyl proposed that, within an infinite domain of discourse, an existence proposition lacking a constructed instance cannot simply invoke the law of excluded middle for judgment. Weyl’s discussion was rooted in mathematics itself, focusing on infinity and constructivity, and revealed one boundary of the law of excluded middle. However, he did not further extend this boundary from the observational phenomena of the physical world.

Many systems in nature inherently possess multi-layered structures and temporal indeterminacy, and are difficult to divide simply in binary terms. This paper selects two typical systems. The first is the quantum system: before measurement collapse, a particle is in a superposition state, embodying temporal indeterminacy. The second is the fluid system: according to the axioms of “tangential convergence and normal boundedness,” the same flow field exhibits directional flow globally while possessing turbulent disturbances locally.

Based on these two groups of phenomena, this paper supplements the constraints on the law of excluded middle in physical observation, and sets them against Weyl’s mathematical boundary in mutual contrast and mutual supplementation, jointly demonstrating that the law of excluded middle is a useful logical tool but has its conditions of use and cannot be generalized without qualification. What this paper seeks to correct is not the law of excluded middle itself, but its unconditional misuse and application beyond its proper domain.

2 Weyl’s Reflection: The Constructive Boundary of the Law of Excluded Middle in Infinite Domains

Weyl’s reflection centers on infinite sets and constructive proof, and can be summarized in several points:

First, finite objects can be examined one by one and judged true or false, and the law of excluded middle can be used normally in this setting; an infinite continuum cannot be traversed in finite steps, so we cannot assume that propositions about infinite objects naturally satisfy the principle that they are either true or false.

Second, a proposition that merely proves “something exists” without giving a concrete construction is empty and unsuitable for directly invoking the law of excluded middle.

Third, classical binary thinking derives from human experience with finite things; extending it directly to the infinite domain produces judgments lacking justification.

Weyl did not advocate abandoning the application of the law of excluded middle in finite settings; he merely reminded us that the infinite domain of discourse is one boundary of the law of excluded middle. More precisely, the law of excluded middle itself does not fail; what fails is its application beyond its proper domain in infinite, non-constructive domains. This is a reflection starting from mathematical proof itself; it lacks corroboration from objective physical phenomena and does not address the limitations imposed by observational scale and temporal sequence.

3 The Boundary of the Temporal Dimension: Quantum Superposition

Classical logic implicitly presupposes that the state of the object under discussion is determinate and unchanging. Quantum phenomena break this presupposition.

Before measurement occurs and the wavefunction collapses, a particle does not possess a determinate eigenstate and is in a multi-state superposition. If we then formulate the proposition “the particle is in state A,” more precisely, the proposition has not yet acquired a determinate truth value; the law of excluded middle requires that the proposition already have a determinate truth value, and this precondition is not satisfied here. This does not mean that quantum mechanics overturns logic, nor does it mean that the law of excluded middle fails. Rather, it shows that the use of the law of excluded middle requires that the system state already be determinate and that the temporal sequence has completed collapse. At the temporal stage when the state is not yet determinate, binary judgment lacks a basis for application. This is a constraint from the physical temporal dimension, independent of Weyl’s boundary of infinity.

4 The Boundary of the Scale Dimension: Local Turbulence and Global Directional Flow in Fluids

According to the two fluid axioms proposed by the author:

1. Axiom of tangential convergence:

\forall \, \mathbf{v} \in V(\Omega), \quad \lim_{t \to \infty} \mathbf{v}_\parallel(t) \rightarrow \mathbf{v}_\parallel^* \in \text{elliptic steady-state flow field}

2. Axiom of normal boundedness:

\exists \, C>0, \quad \text{s.t.} \quad \sup_{t} \|\nabla \mathbf{v}_\perp(t)\|_{L^\infty(\partial\Omega)} \le C

At the global statistical level, the flow tends over long times toward a stable directional steady state; in local space, finite-intensity turbulence and vortex disturbances persist.

If one departs from scale and simply asks, “Is this flow ordered or disordered?”, one encounters the appearance of contradiction: globally ordered, locally disordered. Yet the two conclusions do not conflict, because they are two propositions belonging to different observational scales and do not constitute a contradiction between P and \neg P; the error lies in compressing propositions from two scales into a single binary proposition. It follows that the law of excluded middle requires observational scale to remain unified. Mixing perspectives across scales and forcing binary division is precisely the misuse of the law of excluded middle beyond its proper domain, and is also a spatial-scale boundary that Weyl did not discuss.

5 Mutual Reinforcement: A Summary of the Conditions for the Validity of the Law of Excluded Middle

Weyl delineated the boundary on the mathematical side: the law of excluded middle should not be applied unconditionally to infinite, non-constructive domains of discourse.

This paper delineates the boundary on the side of physical observation: the law of excluded middle should not be applied unconditionally to systems with temporally indeterminate states or multi-scale nesting.

The two are mutually supplementary. It must be emphasized that the law of excluded middle itself is not negated; what is limited is its use beyond its proper domain. From this we summarize three preconditions for the reasonable use of the law of excluded middle:

1. The domain of discourse in which the reasoning takes place is finite and constructible;
2. Observation and judgment remain at the same scale, and the domain of discourse is not switched arbitrarily;
3. The state of the object of study is already determinate, and the temporal sequence has completed collapse.

These three preconditions correspond respectively to three dimensions: the nature of the domain of discourse, the mode of observation, and the state of the object. When all three conditions are satisfied simultaneously, the law of excluded middle can function reliably; if any one of them is not satisfied, we should use simple binary judgment with caution.

6 Discussion

For a long time, people have easily developed a habit of thinking in binary divisions. This habit itself is a convenience brought to us by the law of excluded middle under appropriate conditions, but once it is abused, it easily leads to cognitive bias. What this paper seeks to correct is precisely this unconditional misuse, not the law of excluded middle itself.

Recent research on blow-up solutions of the Navier–Stokes equations has found special initial data under unconstrained conditions. It holds within a single domain of discourse and a single perspective, but it does not mean that it can describe the vast majority of flows in nature. This is precisely the misunderstanding easily brought about by single-perspective binary thinking: taking a local special case directly as the global nature of the system.

Weyl, from within mathematics, initiated reflection on the unconditional use of the law of excluded middle; this paper, with the help of two classes of natural phenomena—quantum mechanics and fluids—extends this reflection from the observational level. The two paths do not conflict but corroborate each other, jointly reminding us that binary logic is a tool, not an ultimate standard naturally applicable to all scenarios; the law of excluded middle is not broken—what is broken is the forcing of propositions that should not be dichotomized into a binary division and the application of it beyond its proper domain.

In short: Let P stand for "the three prerequisites are satisfied simultaneously", and Q stand for "the classical binary judgment is valid and sound". When \neg P, superposed states and coexistent states may reasonably obtain; P is a necessary condition for Q, that is, Q holds only if P holds.

7 Conclusion

1. The law of excluded middle is not unconditionally applicable to all domains of discourse; its reliable use requires three preconditions: a finite and constructible domain of discourse, a unified observational scale, and a determinate state of the object. Applying binary judgment beyond its proper domain is precisely the problem this paper seeks to correct.

2. Hermann Weyl, starting from constructive mathematics, revealed one boundary: the infinite domain of discourse.

3. Based on the phenomenon of quantum superposition, this paper proposes that temporal indeterminacy is one boundary; based on the multi-scale characteristics of fluids, it proposes that non-unified observational scale is another boundary.

4. Weyl’s thought and the conclusions of this paper are mutually complementary and reinforcing, and together constitute a complete specification of the applicable conditions of the law of excluded middle.

5. Binary judgment is appropriate only within limited conditions; when faced with multi-scale systems, superposition states, and infinite objects, we should move beyond simple black-and-white thinking.

References

[1] Hermann Weyl. The Continuum and Intuitionistic Mathematics, 1918.

[2] Zhang Suhang. Fundamental Axioms of Fluid Mechanics: Normal Boundedness and Tangential Convergence—A Unified Proof of Turbulence Finiteness and Global Flow Direction, and a Differentiation from N-S Blow-up Special Cases, 2026.

[3] Zhang Suhang. Proof of Global Smooth Solutions of the Navier–Stokes Equations under the DOG/MOC/ECS/MIE Framework, 2026.

[4] OpenAI. On the Navier–Stokes Millennium Prize Problem [R/OL]. 2026.

[5] Dirac. The Principles of Quantum Mechanics.

[6] Aristotle. Organon.