Abstract
A non-classical subsystem of orthomodular quantum logic is proposed. This system employs two basic operations: the Sasaki hook as implication and the and-then operation as conjunction. These operations successfully satisfy modus ponens and the deduction theorem. In other words, they form an adjunction in terms of category theory. Two types of semantics are presented for this logic: one algebraic and one physical. The algebraic semantics deals with orthomodular lattices, as in traditional quantum logic. The physical semantics is given as a procedure for deriving a final segment of a series of yes-no experiments.
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Notes
Some symbols and names used here have been changed from Nishimura’s original ones.
For a sequent with an empty left-hand side, let \(\alpha \) be \(\bot \rightarrow \bot \), and for a sequent with an empty right-hand side, let \(\beta \) be \(\bot \).
The term counterfactual may possibly cause unnecessary confusion to the reader. As is well known, quantum theory is understood to be context-dependent due to the Kochen-Specker theorem, so it is problematic to make naive arguments based on counterfactual definiteness, such as “What would the result be if we measured momentum instead of position?” In the present paper, the counterfactual proposition Y/X is actually a simple rephrasing of the following situation, as can be seen from the deduction theorem: “Whenever X outputs yes, Y also outputs yes.”
The concept of distance between states is introduced by the norm determined by the inner product of \({\mathcal {H}}\).
Some technical conditions are not mentioned here.
The steps to introduce and eliminate double negation using \(\lnot \lnot \)-1, \(\lnot \lnot \) -2 and Cut rules are omitted here to avoid complexity.
References
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Appendix
Appendix
We provide a proof of Theorem 1.
Proof
Formulas of the proposed system which contain \(\bot \), \(\rightarrow \), or & will be identified with the formulas of GOM as follows:
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\(\bot \equiv \lnot \alpha \wedge \alpha \).
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\(\alpha \rightarrow \beta \equiv \lnot (\alpha \wedge \lnot (\alpha \wedge \beta ))\).
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\( \alpha \& \beta \equiv \lnot (\lnot \alpha \wedge \beta )\wedge \beta \).
It should be pointed out in advance that modus ponens in a single sequent form, i.e. \((\alpha \rightarrow \beta )\wedge \alpha \vdash \beta \), holds in GOM, as shown in the following diagram:Footnote 6
From this and the following identities, we can immediately see that modus ponens also holds for the combination of \(\rightarrow \) and & in GOM: \( (\alpha \rightarrow \beta ) \& \alpha = \lnot (\alpha \wedge \lnot (\alpha \wedge \beta )) \& \alpha =\lnot ((\alpha \wedge \lnot (\alpha \wedge \beta ))\wedge \alpha )\wedge \alpha =\lnot (\alpha \wedge \lnot (\alpha \wedge \beta ))\wedge \alpha =(\alpha \rightarrow \beta )\wedge \alpha \).
Axioms
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Axiom-1. For each formula \(\alpha \), \(\alpha \vdash \alpha \) is an axiom of GOM.
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Axiom-2.
Rules
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Weakening-1, Weakening-2 and Cut . We can simply apply Weakening and Cut rules of GOM.
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\(\rightarrow \) -1. Using modus ponens and&-2 rule, we have:
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\(\rightarrow \) -2. Using modus ponens and&-1 rule , we have:
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&-1. We can simply apply \(\wedge \) -1R rule of GOM.
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&-2. Using &-1 rule, we have:
Identities
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&-idempotency. Since we can prove \( \alpha \& \alpha \vdash \alpha \) and \( \alpha \vdash \alpha \& \alpha \) in GOM, we are justified in regarding \( \alpha \& \alpha \) and \(\alpha \) as identical.
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Double Negation. Since we can prove \(\alpha \rightarrow \bot \vdash \lnot \alpha \) and \(\lnot \alpha \vdash \alpha \rightarrow \bot \) in GOM, we are justified in regarding \(\alpha \rightarrow \bot \) and \(\lnot \alpha \) as identical. Then, by using \(\lnot \lnot \)-1 and \(\lnot \lnot \) -2 rules of GOM, we are justified in regarding \((\alpha \rightarrow \bot )\rightarrow \bot \) and \(\alpha \) as identical. \(\square \)
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Tokuo, K. Implicational quantum logic. Axiomathes 32 (Suppl 2), 473–483 (2022). https://doi.org/10.1007/s10516-021-09614-y
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DOI: https://doi.org/10.1007/s10516-021-09614-y