Logic Realism Theory

The Instantiation Barrier Explained

February 2026 | Introductory

You can think of a round square. You can describe one. You can even draw rules for what would count as one: four equal sides, ninety-degree corners, all points equidistant from the center. The description is perfectly clear.

Yet no round square has ever existed. Not because we haven’t looked hard enough, or because nature is hiding one somewhere. A round square cannot exist. The description is coherent as description; it’s impossible as thing.

This gap (between what we can represent and what can exist) is the foundation of Logic Realism Theory.

Two Domains

LRT distinguishes two domains with carefully chosen names:

$I_\infty$ (I-infinity) is the space of all representable configurations. Everything that can be described, specified, or conceived lives here. Round squares. Married bachelors. Contradictions of every kind. The space is closed under all the operations of thought: you can negate anything, conjoin anything, quantify over anything. If you can specify it, it’s in $I_\infty$.

$A_\Omega$ (A-omega) is the space of instantiable configurations. These are the configurations that can actually exist: physical things, stable records, concrete realities. This space is much smaller.

The claim: $A_\Omega$ is exactly the subset of $I_\infty$ that satisfies the three classical logical laws.

What the Barrier Is

The “instantiation barrier” is the boundary between $I_\infty$ and $A_\Omega$. It’s the threshold configurations must cross to move from representable to real.

Some configurations make it through. “Electron with spin-up” passes. “Red apple on table” passes. “Energy conserved in closed system” passes. These configurations satisfy Identity, Non-Contradiction, and Excluded Middle. They’re coherent not just as descriptions but as things.

Other configurations don’t make it. They hit the barrier. They exist as thoughts, as descriptions, as formal constructions. They cannot exist as things.

Examples that hit the barrier:

The barrier isn’t a physical wall. It’s a constraint on what it means to be instantiated.

Why This Matters

Here’s what makes this philosophically significant: the barrier is asymmetric.

We can represent $L_3$ violations easily. Philosophers spend careers exploring contradictions, impossibilities, and logical violations. Paraconsistent logics formalize them. Modal logic models them. The human mind navigates impossible scenarios fluently.

But nature never instantiates them.

Across a century of precision physics (quantum mechanics, particle accelerators, gravitational wave detectors), no stable experimental record has ever displayed a genuine $L_3$ violation. No detector has simultaneously fired and not-fired. No measurement log has recorded both P and not-P for the same property at the same time.

The asymmetry is stark: representation permits what instantiation forbids.

This isn’t obvious. Logic could have constrained thought while reality did whatever it wants. We find the reverse: thought unconstrained, reality rule-following.

LRT takes this asymmetry seriously.

The Barrier in Quantum Mechanics

Quantum mechanics makes this distinction vivid.

Consider a qubit in superposition:

\[|\psi\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)\]

This state is a perfectly well-defined mathematical object. It lives in the theory’s formalism. It’s as precisely specified as any classical state. But what does it describe?

Not a configuration that is “both 0 and 1.” That would hit the barrier. It would violate Non-Contradiction if instantiated as a record.

The superposition describes a physical situation before outcome-determination. The state encodes how the situation is disposed toward definite outcomes. When measurement occurs, one of those outcomes is instantiated (“0” or “1”) and that outcome is $L_3$ admissible.

The state vector lives in $I_\infty$ as a representational tool. The measurement outcome lives in $A_\Omega$ as an instantiated record. The measurement process is the interface where the barrier operates.

Common Misunderstandings

“So you’re saying contradictions are impossible?”

Contradictions are impossible to instantiate. They’re perfectly possible to represent, to analyze, to work with formally. Paraconsistent logics are legitimate mathematical frameworks. Dialetheia (true contradictions) are defensible philosophical positions. The claim is about what can exist, not what can be thought.

“Isn’t this just defining your way to victory?”

The definition ($A_\Omega$ = $L_3$-admissible configurations) is a starting point. The substantive claim is that this boundary is load-bearing: it generates mathematical structure. Vehicle-invariance forces the Born rule. Local tomography requires complex Hilbert space. The $L_3$ constraint has consequences.

“How do you know there aren’t $L_3$ violations we just haven’t detected?”

We don’t. That’s what makes LRT falsifiable. If a laboratory ever produced a stable record displaying a genuine contradiction (detector both triggered and not-triggered in the same respect), LRT would be refuted.

The Vehicle/Content Distinction

The barrier is why the “vehicle/content” distinction matters.

When you think about a round square, your thought is perfectly coherent. The neural states, the mental representations, the cognitive process: all $L_3$ admissible. The content of your thought (what you’re representing) violates $L_3$.

The vehicle (your thought-process) is instantiated. The content (round square) is not.

Quantum mechanics works the same way. The state vector is a representational vehicle. It encodes outcome-possibilities in a determinate, mathematically precise form. The vehicle satisfies $L_3$. The content (“situation disposed toward outcome 0 with probability $ \alpha ^2$”) describes something not yet instantiated as a definite record.

When measurement occurs, a definite outcome crosses the barrier. The record “0” or “1” is instantiated. The vehicle has done its job: encoding how the physical situation was disposed toward that outcome.

What Crosses the Barrier

What crosses always has the same character: definite, non-contradictory, and complete with respect to the relevant properties.

Definite: There’s a fact of the matter about what it is.

Non-contradictory: It doesn’t simultaneously possess and lack properties.

Complete: For any applicable property, it either has or lacks that property.

These are the three laws (Identity, Non-Contradiction, Excluded Middle) stated in terms of what they permit.

The barrier isn’t arbitrary. It’s the enforcement of determinacy. To be instantiated is to be something rather than nothing and not another thing. That’s Identity. To be instantiated is to avoid possessing contradictory properties. That’s Non-Contradiction. To be instantiated is to have definite status regarding applicable properties. That’s Excluded Middle.

The Deeper Question

Why does the barrier exist?

LRT treats the observation as foundational: this is what instantiation requires. The research program then asks what follows if the constraint is real.

The answer: a lot follows. The Born rule. Complex Hilbert space. Correlation bounds. Information preservation. These are consequences of taking the instantiation barrier seriously.

Maybe someday we’ll understand why the barrier exists. For now, LRT works with the observation that it does.

Next in the Series

This article introduced the instantiation barrier (the boundary between $I_\infty$ and $A_\Omega$). The next article, “Why Distinguishability Requires Quantum Mechanics,” examines what happens when we take Determinate Identity seriously for physical systems. The answer involves Hilbert space.

For the full technical treatment, see the Position Paper. For concept-by-concept breakdowns, explore Topics.

Logic Realism Theory was developed by James (JD) Longmire. This article is part of the LRT documentation at jdlongmire.github.io/logic-realism-theory.