Quantum Epistemology and Constructivism

Abstract

Constructivist epistemology posits that all truths are knowable. One might ask to what extent constructivism is compatible with naturalized epistemology and knowledge obtained from inference-making using successful scientific theories. If quantum theory correctly describes the structure of the physical world, and if quantum theoretic inferences about which measurement outcomes will be observed with unit probability count as knowledge, we demonstrate that constructivism cannot be upheld. Our derivation is compatible with both intuitionistic and quantum propositional logic. This result is implied by the Frauchiger-Renner theorem, though it is of independent importance as well.

Appendix A Proofs

We do not assume that $\mathcal {L}$ satisfies the axioms of classical propositional logic, for we want it also to be compatible with an underlying quantum propositional logic while only allowing for intuitionistically valid deductions. However, we do stipulate that it satisfies the following axiom schemas (whose instances shall collectively be denoted $\Sigma$) for all $\mathcal {L}$-formulas $\phi$ and $\psi$:

• Double Negation Introduction (DNI): $\quad \vdash \phi \rightarrow \lnot \lnot \phi$

• Triple Negation Elimination (TNE): $\quad \vdash \lnot \lnot \lnot \phi \rightarrow \lnot \phi$

• Conjunction Negation Distribution (CND): $\quad \vdash (\lnot \lnot \phi \wedge \lnot \lnot \psi )\rightarrow \lnot \lnot (\phi \wedge \psi )$

• Conjunction Introduction (CI): $\quad \{\phi ,\psi \} \vdash \phi \wedge \psi$

• Conjunction Elimination (CE): $\quad \vdash \phi \wedge \psi \rightarrow \phi ,\quad \vdash \phi \wedge \psi \rightarrow \psi$

• Contraposition:$\quad \vdash (\phi \rightarrow \psi )\rightarrow (\lnot \psi \rightarrow \lnot \phi )$

• Propositional Identity: $\quad \vdash \lnot (\phi \wedge \lnot \psi )\rightarrow (\phi \rightarrow \lnot \lnot \psi )$

These axioms are quite minimal (indeed, we do not posit any modal axioms whatsoever) and insufficient on their own to prove completeness. This is a positive feature of the generality of our analysis. Note that all of these axioms are intuitionistically valid and are valid in quantum propositional logic as well (so they can be used in the proof of our main result while being consistent with a constructivist account of logic). The only rule of inference we shall assume is modus ponens (we do not assume the necessitation rule).

Before proving our main theorem, we prove Theorem 2 (which entails Fitch’s paradox) and several lemmas. Defining $\vdash$ with respect to $\Sigma$, Theorem 2 asserts that for any $\mathcal {L}$-formula $\phi$ and agent $\alpha \in A$, $\Gamma _\text {Const}\vdash \phi \rightarrow \lnot \lnot K_\alpha \phi$:

Proof

For any $\mathcal {L}$-formula $\phi$ we have:

• 1. $(\phi \wedge \lnot K_\alpha \phi )\rightarrow \Diamond K_\alpha (\phi \wedge \lnot K_\alpha \phi )\quad \textbf{CONST}.$

• 2. $\quad \lnot \Diamond K_\alpha (\phi \wedge \lnot K_\alpha \phi )\rightarrow \lnot (\phi \wedge \lnot K_\alpha \phi )\quad \text {Contraposition.}$

• 3. $\lnot \Diamond K_\alpha (\phi \wedge \lnot K_\alpha \phi ) \quad \textbf {KCONT}.$

• 4. $\quad \lnot (\phi \wedge \lnot K_\alpha \phi )\quad \text {2, 3, Modus Ponens.}$

• 5. $\quad \lnot (\phi \wedge \lnot K_\alpha \phi )\rightarrow (\phi \rightarrow \lnot \lnot K_\alpha \phi )\quad \text {Propositional Identity.}$

• 6. $\quad \phi \rightarrow \lnot \lnot K_\alpha \phi \quad \text {4, 5, Modus Ponens.}$

We shall denote by FITCH the set of all instances of $\phi \rightarrow \lnot \lnot K_\alpha \phi$ for all $\mathcal {L}$-formulas $\phi$ and agents $\alpha \in A$. We now prove several lemmas, taking $\phi$ and $\psi$ to range over all $\mathcal {L}$-formulas, and $\alpha$ and $\beta$ to range over all agents in A.

Lemma 1

  $\Gamma _{\text {Const}}\cup \{\lnot K_\alpha \phi \}\vdash \lnot \phi$.

Proof

 

• 1. $\quad \lnot K_\alpha \phi \quad \text {Assumption.}$

• 2. $\quad \phi \rightarrow \lnot \lnot K_\alpha \phi \quad {\textbf {FITCH}}~ \text {for }\alpha .$

• 3. $\quad \lnot \lnot \lnot K_\alpha \phi \rightarrow \lnot \phi \quad \text {2, Contraposition.}$

• 4. $\quad \lnot K_\alpha \phi \rightarrow \lnot \lnot \lnot K_\alpha \phi \quad {\textbf {DNI}}.$

• 5. $\quad \lnot \lnot \lnot K_\alpha \phi \quad \text {1, 4, Modus Ponens.}$

• 6. $\quad \lnot \phi \quad \text {3, 5, Modus Ponens.}$

Lemma 2

  $\Gamma _{\text {Const}}\cup \{\lnot \lnot K_\beta \phi \}\vdash \lnot \lnot K_\alpha K_\beta \phi$.

Proof

 

• 1. $\quad \lnot \lnot K_\beta \phi \quad \text {Assumption.}$

• 2. $\quad K_\beta \phi \rightarrow \lnot \lnot K_\alpha K_\beta \phi \quad {\textbf {FITCH}}~ \text {for }\alpha .$

• 3. $\quad (K_\beta \phi \rightarrow \lnot \lnot K_\alpha K_\beta \phi )\rightarrow (\lnot \lnot \lnot K_\alpha K_\beta \phi \rightarrow \lnot K_\beta \phi ) \quad \text {Contraposition.}$

• 4. $\quad \lnot \lnot \lnot K_\alpha K_\beta \phi \rightarrow \lnot K_\beta \phi \quad \text {2, 3, Modus Ponens.}$

• 5. $\quad (\lnot \lnot \lnot K_\alpha K_\beta \phi \rightarrow \lnot K_\beta \phi )\rightarrow (\lnot \lnot K_\beta \phi \rightarrow \lnot \lnot \lnot \lnot K_\alpha K_\beta \phi ) \quad \text {Contraposition.}$

• 6. $\quad \lnot \lnot K_\beta \phi \rightarrow \lnot \lnot \lnot \lnot K_\alpha K_\beta \phi \quad \text {4, 5, Modus Ponens.}$

• 7. $\quad \lnot \lnot \lnot \lnot K_\alpha K_\beta \phi \quad \text {1, 6, Modus Ponens.}$

• 8. $\quad \lnot \lnot \lnot \lnot K_\alpha K_\beta \phi \rightarrow \lnot \lnot K_\alpha K_\beta \phi \quad {\textbf {TNE}}.$

• 9. $\quad \lnot \lnot K_\alpha K_\beta \phi \quad \text {7, 8, Modus Ponens.}$

Lemma 3

  $\Gamma _{\text {Const}}\cup \{(\lnot \lnot K_\alpha \phi )\wedge (\lnot \lnot K_\alpha \psi )\}\vdash \lnot \lnot K_\alpha (\phi \wedge \psi )$.

Proof

 

• 1. $\quad (\lnot \lnot K_\alpha \phi )\wedge (\lnot \lnot K_\alpha \psi ) \quad \text {Assumption.}$

• 2. $\quad ((\lnot \lnot K_\alpha \phi )\wedge (\lnot \lnot K_\alpha \psi ))\rightarrow \lnot \lnot (K_\alpha \phi \wedge K_\alpha \psi ) \quad {\textbf {CND}}.$

• 3. $\quad \lnot \lnot (K_\alpha \phi \wedge K_\alpha \psi ) \quad \text {1, 2, Modus Ponens.}$

• 4. $\quad (K_\alpha \phi \wedge K_\alpha \psi )\rightarrow K_\alpha (\phi \wedge \psi ) \quad {\textbf {DIST}}.$

• 5. $\quad ((K_\alpha \phi \wedge K_\alpha \psi )\rightarrow K_\alpha (\phi \wedge \psi )) \quad \text {Contraposition.}$ $\qquad \qquad \rightarrow ( \lnot K_\alpha (\phi \wedge \psi )\rightarrow \lnot (K_\alpha \phi \wedge K_\alpha \psi ))$

• 6. $\quad \lnot K_\alpha (\phi \wedge \psi )\rightarrow \lnot (K_\alpha \phi \wedge K_\alpha \psi ) \quad \text {4, 5, Modus Ponens.}$

• 7. $\quad (\lnot K_\alpha (\phi \wedge \psi )\rightarrow \lnot (K_\alpha \phi \wedge K_\alpha \psi )) \quad \text {Contraposition.}$ $\qquad \qquad \rightarrow (\lnot \lnot (K_\alpha \phi \wedge K_\alpha \psi )\rightarrow \lnot \lnot K_\alpha (\phi \wedge \psi ))$

• 8. $\quad \lnot \lnot (K_\alpha \phi \wedge K_\alpha \psi )\rightarrow \lnot \lnot K_\alpha (\phi \wedge \psi ) \quad \text {6, 7, Modus Ponens.}$

• 9. $\quad \lnot \lnot K_\alpha (\phi \wedge \psi ) \quad \text {3, 8, Modus Ponens.}$

The proof of our main result, Theorem 1, is then as follows.

Proof

We must prove three separate results. First, a counter-instance of EQ is inconsistent with CONST. If we assume that EQ is violated, then there is some agent $\alpha \in A_{\text {QM}}$ and some quantum proposition $p\in P_{\text {QM}}$ such that $\lnot (p\rightarrow \Diamond K_\alpha p)$ (a formula which which we shall denote by $\lnot {\textbf {EQ}}$). Letting $\phi _p:=p\rightarrow \Diamond K_\alpha p$ we have:

• 1. $\quad p\rightarrow \Diamond K_\alpha p \quad {\textbf {CONST}}$.

• 2. $\quad \lnot (p\rightarrow \Diamond K_\alpha p) \quad \lnot {\textbf {EQ}}$.

• 3. $\quad \phi _{p}\wedge \lnot \phi _{p} \quad 1, 2, {\textbf {CI}}$.

Thus, ${\textbf {CONST}}\cup \{\lnot {\textbf {EQ}}\}\vdash \perp$. Next, a counter instance of ES is inconsistent with $\Gamma ^+_{\text {Const}}\setminus \Gamma _{\text {Const}}$. If we assume that ES is violated, then there is some agent $\alpha \in A_{\text {QM}}$ and some quantum proposition $p\in P_{\text {QM}}$ such that $\lnot \lnot (K_\alpha p\wedge K_\alpha (\lnot p))$ (a formula which we shall denote by $\lnot {\textbf {ES}}$). Letting $\psi _p:=\lnot K_\alpha (p\wedge \lnot p)$, we have:

• 1. $\quad \lnot K_\alpha (p\wedge \lnot p) \quad {\textbf {NCK}}.$

• 2. $\quad \lnot \lnot (K_\alpha p\wedge K_\alpha (\lnot p)) \quad \lnot {\textbf {ES}}.$

• 3. $\quad \lnot \lnot K_\alpha (p\wedge \lnot p) \quad 2, {\textbf {DIST}}.$

• 4. $\quad \psi _p\wedge \lnot \psi _p \quad 1, 3, {\textbf {CI}}.$

Thus, ${\textbf {NCK}}\cup {\textbf {DIST}}\cup \{\lnot {\textbf {ES}}\}\vdash \perp$ (where $\Gamma ^+_{\text {Const}}\setminus \Gamma _{\text {Const}}={\textbf {NCK}}\cup {\textbf {DIST}}$). Finally, a counter instance of EC is inconsistent with $\Gamma ^+_{\text {Const}}$. If we assume that EC is violated, then there are some agents $\alpha ,\beta \in A_{\text {QM}}$ and some quantum proposition $p\in P_{\text {QM}}$ such that $\lnot \lnot (K_\alpha K_\beta p\wedge \lnot K_\alpha p)$ (a formula we shall denote by $\lnot$EC). Letting $\xi _p:=\lnot K_\alpha ((K_\beta p\wedge K_\beta (\lnot p))\wedge \lnot (K_\beta p\wedge K_\beta (\lnot p)))$, we have:

• 1. $\quad \lnot \lnot (K_\alpha K_\beta p\wedge \lnot K_\alpha p) \quad \lnot {\textbf {EC}}.$

• 2. $\quad \lnot \lnot (K_\alpha K_\beta p\wedge \lnot K_\alpha p)\rightarrow (\lnot \lnot K_\alpha K_\beta p\wedge \lnot \lnot \lnot K_\alpha p) \quad {\textbf {CND}}.$

• 3. $\quad \lnot \lnot K_\alpha K_\beta p\wedge \lnot \lnot \lnot K_\alpha p \quad 1, 2$, Modus Ponens.

• 4. $\quad \lnot \lnot K_\alpha K_\beta p \quad 3, {\textbf {CE}}.$

• 5. $\quad \lnot \lnot \lnot K_\alpha p \quad 3, {\textbf {CE}}.$

• 6. $\quad \lnot \lnot \lnot K_\alpha p\rightarrow \lnot K_\alpha p \quad {\textbf {TNE}}.$

• 7. $\quad \lnot K_\alpha p \quad 5, 6$, Modus Ponens.

• 8. $\quad \lnot p$    7, Lemma 1.

• 9. $\quad \lnot p\rightarrow \lnot \lnot K_\beta (\lnot p) \quad {\textbf {FITCH}}~\text {for }\beta .$

• 10. $\quad \lnot \lnot K_\beta (\lnot p) \quad 8, 9$, Modus Ponens.

• 11. $\quad \lnot \lnot K_\alpha K_\beta (\lnot p)$    10, Lemma 2.

• 12. $\quad (\lnot \lnot K_\alpha K_\beta p)\wedge (\lnot \lnot K_\alpha K_\beta (\lnot p)) \quad 4, 11, {\textbf {CI}}.$

• 13. $\quad \lnot \lnot K_\alpha (K_\beta p\wedge K_\beta (\lnot p))$    12, Lemma 3.

• 14. $\quad (K_\beta p\wedge K_\beta (\lnot p))\rightarrow K_\beta (p\wedge \lnot p) \quad {\textbf {DIST}}.$

• 15. $\quad ((K_\beta p\wedge K_\beta (\lnot p))\rightarrow K_\beta (p\wedge \lnot p)) \quad$ Contraposition. $\quad \rightarrow (\lnot K_\beta (p\wedge \lnot p)\rightarrow \lnot (K_\beta p\wedge K_\beta (\lnot p)))$

• 16. $\quad \lnot K_\beta (p\wedge \lnot p)\rightarrow \lnot (K_\beta p\wedge K_\beta (\lnot p)) \quad 14, 15$, Modus Ponens.

• 17. $\quad \lnot K_\beta (p\wedge \lnot p) \quad {\textbf {NCK}}.$

• 18. $\quad \lnot (K_\beta p\wedge K_\beta (\lnot p)) \quad 16, 17$, Modus Ponens.

• 19. $\quad \lnot (K_\beta p\wedge K_\beta (\lnot p))\rightarrow \lnot \lnot K_\alpha (\lnot (K_\beta p\wedge K_\beta (\lnot p))) \quad {\textbf {FITCH}}~ \text {for }\alpha .$

• 20. $\quad \lnot \lnot K_\alpha (\lnot (K_\beta p\wedge K_\beta (\lnot p))) \quad 18, 19$, Modus Ponens.

• 21. $\quad (\lnot \lnot K_\alpha (K_\beta p\wedge K_\beta (\lnot p)))\wedge (\lnot \lnot K_\alpha (\lnot (K_\beta p\wedge K_\beta (\lnot p)))) \quad 13, 20, {\textbf {CI}}.$

• 22. $\quad \lnot \lnot K_\alpha ((K_\beta p\wedge K_\beta (\lnot p))\wedge \lnot (K_\beta p\wedge K_\beta (\lnot p)))$    21, Lemma 3.

• 23. $\quad \lnot K_\alpha ((K_\beta p\wedge K_\beta (\lnot p))\wedge \lnot (K_\beta p\wedge K_\beta (\lnot p))) \quad {\textbf {NCK}}.$

• 24. $\quad \xi _p\wedge \lnot \xi _p \quad 22, 23, {\textbf {CI}}.$

Thus, $\Gamma ^+_{\text {Const}}\cup \{\lnot {\textbf {EC}}\}\vdash \perp$.