Meta Theory of the Zeckendorf-Hilbert Universe

24.2 Intuitionistic Logic: Logical Duality of Quantum Superposition and Failure of Law of Excluded Middle

In Section 24.1, we established that mathematical model of physical reality is not static sets in set theory, but dynamic presheaves in topos. This structural change brings fundamental reconstruction of logical laws. In classical physics, we default to using Boolean Logic, whose core pillar is Law of Excluded Middle (LEM): any physical proposition $P$ (e.g., “electron is at A”), is either true or false ($P\vee \neg P=\text{True}$).

However, in QCA quantum universe, existence of superposition states challenges this binary logic. This section will prove that intrinsic logic of quantum mechanics is Intuitionistic Logic. We will show that superposition states are not logical paradoxes of “both A and B,” but natural results of failure of law of excluded middle. In topos logic, $\neg \neg P$ (not-not P) is not equivalent to $P$, and this logical gap is precisely logical origin of quantum uncertainty and measurement collapse.

24.2.1 Heyting Algebra of Physical Propositions

In classical phase space, propositions correspond to subsets (Borel sets), which form a Boolean algebra. But in physical topos ${\mathcal{T}}_{\mathcal{A}}$, propositions correspond to Subobjects, which form a Heyting Algebra.

Definition 24.2.1 (Truth Value of Physical Propositions)

Let $P$ be a proposition about physical system (e.g., “energy $E\in [E_1,E_2]$”). In topos ${\mathcal{T}}_{\mathcal{A}}$, truth value $[[P]]$ of $P$ is not simply $\{0,1\}$, but a Sieve on base category $\mathcal{V}(\mathcal{A})$.

Specifically, $[[P]](V)$ is a set containing all observation contexts (commutative subalgebras) $V$ where proposition $P$ is true in classical approximation.

Key difference between Heyting algebra and Boolean algebra lies in definition of Negation:

• Boolean Negation: $\neg P$ is complement of $P$.

• Heyting Negation: $\neg P$ is Pseudo-complement. It is the maximum element of set of all contexts incompatible with $P$.

  $$[[\neg P]]=\bigvee \{Q\mid Q\wedge P=\perp \}$$

24.2.2 Failure of Law of Excluded Middle and Superposition

Consider a spin $1/2$ particle. Let proposition $P_z$ be “spin $z$ component is $+1/2$.”

• In classical logic, $\neg P_z$ means “spin $z$ component is $-1/2$.” Therefore $P_z\vee \neg P_z$ covers all cases.

• In quantum logic (intuitionistic logic), if we are in $x$-direction eigenstate $|{\uparrow }_x⟩=\frac{1}{\sqrt{2}}(|{\uparrow }_z⟩+|{\downarrow }_z⟩)$, the situation is completely different.

Theorem 24.2.2 (Failure of Law of Excluded Middle)

In physical topos ${\mathcal{T}}_{\mathcal{A}}$, for non-trivial quantum systems, there exist propositions $P$ such that global truth value:

$$[[P\vee \neg P]]\ne 1(\text{absolute truth})$$

Proof:

In context $V_x$ (measuring $S_x$), proposition $P_z$ (statement about $S_z$) is undefined, because operator $S_z$ is not in algebra $V_x$.

Therefore, in $V_x$, we can neither assert $P_z$ is true, nor assert $\neg P_z$ is true.

$$[[P_z]](V_x)=\varnothing ,[[\neg P_z]](V_x)=\varnothing$$

So $[[P_z\vee \neg P_z]](V_x)=\varnothing \ne \text{True}$.

This means statement “electron spin is either up or down” is logically invalid in contexts where $z$-direction measurement is not performed.

Physical Corollary (Logical Essence of Superposition):

Superposition state $|\psi ⟩=\alpha |0⟩+\beta |1⟩$ is not “simultaneously in 0 and 1,” but in a logical state where $P_0\vee \neg P_0$ has not yet obtained truth value. Superposition is logical “pending zone.”

24.2.3 Double Negation Does Not Eliminate: $\neg \neg P\ne P$

Most famous feature of intuitionistic logic is failure of double negation law. In QCA universe, this has profound dynamical meaning.

Definition 24.2.3 (Physical Meaning of Double Negation)

• $P$: System is explicitly in state $P$ (e.g., measurement result confirms $P$).

• $\neg P$: System is in state orthogonal to $P$.

• $\neg \neg P$: System is in state where $P$ cannot be excluded.

In Hilbert space, for any non-zero state $|\psi ⟩$, as long as its projection onto subspace of $P$ is non-zero, it satisfies $\neg \neg P$.

For example, superposition state $|\psi ⟩=ϵ|0⟩+\sqrt{1-{ϵ}^2}|1⟩$.

• It is not $|0⟩$ (so $P_0$ is not fully true).

• It is not $|1⟩$ (so $\neg P_0$ is also not fully true).

• But it can never be classified as “not $P_0$,” because it contains component of $P_0$. Therefore $\neg \neg P_0$ is true.

Theorem 24.2.4 (Logical Approximation Theorem)

In physical topos, $P\subseteq \neg \neg P$ always holds.

$\neg \neg P$ represents Logical Closure or Densification of $P$.

Physical evolution (unitary dynamics) usually occurs at level of $\neg \neg P$ (possibility space), while measurement (topological fusion) is process of forcibly jumping from $\neg \neg P$ to $P$ (deterministic realization). This is precisely dual description of “topological fusion” in Section 21.4 at logical level.

24.2.4 Curry-Howard-Lambek Correspondence: Proofs as Programs

Why should physics follow this strange logic? Because Physics is Computation.

According to Curry-Howard-Lambek (CHL) correspondence:

• Proposition = Type / State Space

• Proof = Program / Physical Process

In constructive mathematics (mathematical foundation of intuitionistic logic), to prove $A\vee B$, you must provide an algorithm that explicitly outputs $A$ or outputs $B$.

In quantum mechanics, without performing measurement (running “collapse program”), system has no definite $A$ or $B$. Therefore, before measurement occurs, $A\vee B$ is unprovable in constructive sense.

Corollary 24.2.5 (Constructivity of Physical Reality)

QCA discrete ontology is a Constructive Theory. Universe does not contain physical quantities that “exist but are uncomputable.”

• Classical physics claims $x(t)$ has value at any moment (law of excluded middle), even if no one measures. This is non-constructive (God’s-eye view).

• QCA physics claims only information computed by $U$ operators and extracted by observers has truth value. This is intuitionistic (constructive perspective).

24.2.5 Conclusion: Geometrization of Logic

This section proved that quantum strangeness is not collapse of physics, but rigorization of logic.

1. Superposition is logical gap where law of excluded middle fails.

2. Uncertainty is manifestation that double negation does not eliminate ($\neg \neg P\ne P$).

3. Measurement is logical phase transition from intuitionistic logic ($\neg \neg P$) to Boolean logic ($P$).

By introducing topos and intuitionistic logic, we found the most fundamental logical foundation for physical theory of the entire book. In the next section 24.3, we will specifically demonstrate implementation of CHL correspondence in physics, proving that physical laws are type inference rules in type theory.