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  1.  1
    Embeddings between Partial Combinatory Algebras.Anton Golov & Sebastiaan A. Terwijn - 2023 - Notre Dame Journal of Formal Logic 64 (1):129-158.
    Partial combinatory algebras (pcas) are algebraic structures that serve as generalized models of computation. In this article, we study embeddings of pcas. In particular, we systematize the embeddings between relativizations of Kleene’s models, of van Oosten’s sequential computation model, and of Scott’s graph model, showing that an embedding between two relativized models exists if and only if there exists a particular reduction between the oracles. We obtain a similar result for the lambda calculus, showing in particular that it cannot be (...)
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  2.  11
    Bicontextualism.Lorenzo Rossi - 2023 - Notre Dame Journal of Formal Logic 64 (1):95-127.
    Can one quantify over absolutely everything? Absolutists answer positively, while relativists answer negatively. Here, I focus on the absolutism versus relativism debate in the framework of theories of truth, where relativism becomes a form of contextualism about truth predications. Contextualist theories of truth provide elegant and uniform solutions to the semantic paradoxes while preserving classical logic. However, they interpret harmless generalizations (such as “everything is self-identical”) in less than absolutely comprehensive domains, thus systematically misconstruing them. In this article, I show (...)
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  3.  10
    Core Gödel.Neil Tennant - 2023 - Notre Dame Journal of Formal Logic 64 (1):15-59.
    This study examines how the Gödel phenomena are to be treated in core logic. We show in formal detail how one can use core logic in the metalanguage to prove Gödel’s incompleteness theorems for arithmetic even when classical logic is used for logical closure in the object language.
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  4. A Completeness Proof for a Regular Predicate Logic with Undefined Truth Value.Antti Valmari & Lauri Hella - 2023 - Notre Dame Journal of Formal Logic 64 (1):61-93.
    We provide a sound and complete proof system for an extension of Kleene’s ternary logic to predicates. The concept of theory is extended with, for each function symbol, a formula that specifies when the function is defined. The notion of “is defined” is extended to terms and formulas via a straightforward recursive algorithm. The “is defined” formulas are constructed so that they themselves are always defined. The completeness proof relies on the Henkin construction. For each formula, precisely one of the (...)
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  5. The Complexity of Decomposability of Computable Rings.Huishan Wu - 2023 - Notre Dame Journal of Formal Logic 64 (1):1-14.
    This article studies the complexity of decomposability of rings from the perspective of computability. Based on the equivalence between the decomposition of rings and that of the identity of rings, we propose four kinds of rings, namely, weakly decomposable rings, decomposable rings, weakly block decomposable rings, and block decomposable rings. Let R be the index set of computable rings. We study the complexity of subclasses of computable rings, showing that the index set of computable weakly decomposable rings is m-complete Σ10 (...)
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