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Interrelation between weak fragments of double negation shift and related principles

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Journal of Symbolic Logic 83 (3):991-1012 (2018)

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  1. Prenex normal form theorems in semi-classical arithmetic.Makoto Fujiwara &Taishi Kurahashi -2021 -Journal of Symbolic Logic 86 (3):1124-1153.
    Akama et al. [1] systematically studied an arithmetical hierarchy of the law of excluded middle and related principles in the context of first-order arithmetic. In that paper, they first provide a prenex normal form theorem as a justification of their semi-classical principles restricted to prenex formulas. However, there are some errors in their proof. In this paper, we provide a simple counterexample of their prenex normal form theorem [1, Theorem 2.7], then modify it in an appropriate way which still serves (...) to largely justify the arithmetical hierarchy. In addition, we characterize a variety of prenex normal form theorems by logical principles in the arithmetical hierarchy. The characterization results reveal that our prenex normal form theorems are optimal. For the characterization results, we establish a new conservation theorem on semi-classical arithmetic. The theorem generalizes a well-known fact that classical arithmetic is $\Pi _2$ -conservative over intuitionistic arithmetic. (shrink)
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  • König's lemma, weak König's lemma, and the decidable fan theorem.Makoto Fujiwara -2021 -Mathematical Logic Quarterly 67 (2):241-257.
    We provide a fine‐grained analysis on the relation between König's lemma, weak König's lemma, and the decidable fan theorem in the context of constructive reverse mathematics. In particular, we show that double negated variants of König's lemma and weak König's lemma are equivalent to double negated variants of the general decidable fan theorem and the binary decidable fan theorem, respectively, over a nearly intuitionistic system containing a weak countable choice only. This implies that the general decidable fan theorem is not (...) equivalent to the binary decidable fan theorem in the absence of a strong countable choice. In addition, we introduce a principle which fills the gap between König's lemma and weak König's lemma, and show that König's lemma is equivalent to weak König's lemma plus this principle. We also solve several problems arising from the investigation. (shrink)
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  • Refining the arithmetical hierarchy of classical principles.Makoto Fujiwara &Taishi Kurahashi -2022 -Mathematical Logic Quarterly 68 (3):318-345.
    We refine the arithmetical hierarchy of various classical principles by finely investigating the derivability relations between these principles over Heyting arithmetic. We mainly investigate some restricted versions of the law of excluded middle, De Morgan's law, the double negation elimination, the collection principle and the constant domain axiom.
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  • $$\Delta ^0_1$$ variants of the law of excluded middle and related principles.Makoto Fujiwara -2022 -Archive for Mathematical Logic 61 (7):1113-1127.
    We systematically study the interrelations between all possible variations of \(\Delta ^0_1\) variants of the law of excluded middle and related principles in the context of intuitionistic arithmetic and analysis.
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