84 found
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  1. Semantics and the liar paradox.Albert Visser - 1989 - Handbook of Philosophical Logic 4 (1):617--706.
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  2. A propositional logic with explicit fixed points.Albert Visser - 1981 - Studia Logica 40 (2):155 - 175.
    This paper studies a propositional logic which is obtained by interpreting implication as formal provability. It is also the logic of finite irreflexive Kripke Models.A Kripke Model completeness theorem is given and several completeness theorems for interpretations into Provability Logic and Peano Arithmetic.
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  3. Self-reference in arithmetic I.Volker Halbach & Albert Visser - 2014 - Review of Symbolic Logic 7 (4):671-691.
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  4.  53
    Self-Reference Upfront: A Study of Self-Referential Gödel Numberings.Balthasar Grabmayr & Albert Visser - 2023 - Review of Symbolic Logic 16 (2):385-424.
    In this paper we examine various requirements on the formalisation choices under which self-reference can be adequately formalised in arithmetic. In particular, we study self-referential numberings, which immediately provide a strong notion of self-reference even for expressively weak languages. The results of this paper suggest that the question whether truly self-referential reasoning can be formalised in arithmetic is more sensitive to the underlying coding apparatus than usually believed. As a case study, we show how this sensitivity affects the formal study (...)
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  5. Four valued semantics and the liar.Albert Visser - 1984 - Journal of Philosophical Logic 13 (2):181 - 212.
  6. New Constructions of Satisfaction Classes.Albert Visser & Ali Enayat - 2015 - In T. Achourioti, H. Galinon, J. Martínez Fernández & K. Fujimoto (eds.), Unifying the Philosophy of Truth. Dordrecht: Imprint: Springer.
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  7. Dynamics.Reinhard Muskens, Johan Van Benthem & Albert Visser - 1997 - In Johan Van Benthem & Alice Ter Meulen (eds.), Handbook of Logic and Language. Amsterdam: Elsevier. pp. 587-648.
  8.  61
    Self-reference in arithmetic II.Volker Halbach & Albert Visser - 2014 - Review of Symbolic Logic 7 (4):692-712.
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  9. On the termination of russell’s description elimination algorithm.Clemens Grabmayer, Joop Leo, Vincent van Oostrom & Albert Visser - 2011 - Review of Symbolic Logic 4 (3):367-393.
    In this paper we study the termination behavior of Russell’s description elimination rewrite system. We discuss certain claims made by Kripke (2005) in his paper concerning the possible nontermination of elimination of descriptions.
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  10.  60
    Another look at the second incompleteness theorem.Albert Visser - 2020 - Review of Symbolic Logic 13 (2):269-295.
    In this paper we study proofs of some general forms of the Second Incompleteness Theorem. These forms conform to the Feferman format, where the proof predicate is fixed and the representation of the set of axioms varies. We extend the Feferman framework in one important point: we allow the interpretation of number theory to vary.
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  11. The formalization of interpretability.Albert Visser - 1991 - Studia Logica 50 (1):81 - 105.
    This paper contains a careful derivation of principles of Interpretability Logic valid in extensions of I0+1.
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  12.  83
    Growing Commas. A Study of Sequentiality and Concatenation.Albert Visser - 2009 - Notre Dame Journal of Formal Logic 50 (1):61-85.
    In his paper "Undecidability without arithmetization," Andrzej Grzegorczyk introduces a theory of concatenation $\mathsf{TC}$. We show that pairing is not definable in $\mathsf{TC}$. We determine a reasonable extension of $\mathsf{TC}$ that is sequential, that is, has a good sequence coding.
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  13.  33
    An Inside View of Exp; or, The Closed Fragment of the Provability Logic of IΔ0+ Ω1 with a Propositional Constant for.Albert Visser - 1992 - Journal of Symbolic Logic 57 (1):131-165.
  14.  34
    Provability logic and the completeness principle.Albert Visser & Jetze Zoethout - 2019 - Annals of Pure and Applied Logic 170 (6):718-753.
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  15.  51
    The predicative Frege hierarchy.Albert Visser - 2009 - Annals of Pure and Applied Logic 160 (2):129-153.
    In this paper, we characterize the strength of the predicative Frege hierarchy, , introduced by John Burgess in his book [J. Burgess, Fixing frege, in: Princeton Monographs in Philosophy, Princeton University Press, Princeton, 2005]. We show that and are mutually interpretable. It follows that is mutually interpretable with Q. This fact was proved earlier by Mihai Ganea in [M. Ganea, Burgess’ PV is Robinson’s Q, The Journal of Symbolic Logic 72 619–624] using a different proof. Another consequence of the our (...)
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  16.  51
    Faith & falsity.Albert Visser - 2004 - Annals of Pure and Applied Logic 131 (1-3):103-131.
    A theory T is trustworthy iff, whenever a theory U is interpretable in T, then it is faithfully interpretable. In this paper we give a characterization of trustworthiness. We provide a simple proof of Friedman’s Theorem that finitely axiomatized, sequential, consistent theories are trustworthy. We provide an example of a theory whose schematic predicate logic is complete Π20.
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  17.  33
    The unprovability of small inconsistency.Albert Visser - 1993 - Archive for Mathematical Logic 32 (4):275-298.
    We show that a consistent, finitely axiomatized, sequential theory cannot prove its own inconsistency on every definable cut. A corollary is that there are at least three degrees of global interpretability of theories equivalent modulo local interpretability to a consistent, finitely axiomatized, sequential theory U.
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  18.  70
    Peano's smart children: a provability logical study of systems with built-in consistency.Albert Visser - 1989 - Notre Dame Journal of Formal Logic 30 (2):161-196.
  19.  64
    Rules and Arithmetics.Albert Visser - 1999 - Notre Dame Journal of Formal Logic 40 (1):116-140.
    This paper is concerned with the logical structure of arithmetical theories. We survey results concerning logics and admissible rules of constructive arithmetical theories. We prove a new theorem: the admissible propositional rules of Heyting Arithmetic are the same as the admissible propositional rules of Intuitionistic Propositional Logic. We provide some further insights concerning predicate logical admissible rules for arithmetical theories.
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  20.  45
    The provability logics of recursively enumerable theories extending peano arithmetic at arbitrary theories extending peano arithmetic.Albert Visser - 1984 - Journal of Philosophical Logic 13 (1):97 - 113.
  21.  73
    Pairs, sets and sequences in first-order theories.Albert Visser - 2008 - Archive for Mathematical Logic 47 (4):299-326.
    In this paper we study the idea of theories with containers, like sets, pairs, sequences. We provide a modest framework to study such theories. We prove two concrete results. First, we show that first-order theories of finite signature that have functional non-surjective ordered pairing are definitionally equivalent to extensions in the same language of the basic theory of non-surjective ordered pairing. Second, we show that a first-order theory of finite signature is sequential (is a theory of sequences) iff it is (...)
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  22. An Overview of Interpretability Logic.Albert Visser - 1998 - In Marcus Kracht, Maarten de Rijke, Heinrich Wansing & Michael Zakharyaschev (eds.), Advances in Modal Logic. CSLI Publications. pp. 307-359.
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  23. An Overview of Interpretability Logic.Albert Visser - 1998 - In Marcus Kracht, Maarten de Rijke, Heinrich Wansing & Michael Zakharyaschev (eds.), Advances in Modal Logic. CSLI Publications. pp. 307-359.
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  24.  37
    Substitutions of Σ10-sentences: explorations between intuitionistic propositional logic and intuitionistic arithmetic.Albert Visser - 2002 - Annals of Pure and Applied Logic 114 (1-3):227-271.
    This paper is concerned with notions of consequence. On the one hand, we study admissible consequence, specifically for substitutions of Σ 1 0 -sentences over Heyting arithmetic . On the other hand, we study preservativity relations. The notion of preservativity of sentences over a given theory is a dual of the notion of conservativity of formulas over a given theory. We show that admissible consequence for Σ 1 0 -substitutions over HA coincides with NNIL -preservativity over intuitionistic propositional logic . (...)
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  25.  27
    Modal Logic and Self-Reference.Albert Visser & Craig Smorynski - 1989 - Journal of Symbolic Logic 54 (4):1479.
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  26.  35
    On the completenes principle: A study of provability in heyting's arithmetic and extensions.Albert Visser - 1982 - Annals of Mathematical Logic 22 (3):263-295.
  27.  42
    Peano Corto and Peano Basso: A Study of Local Induction in the Context of Weak Theories.Albert Visser - 2014 - Mathematical Logic Quarterly 60 (1-2):92-117.
    In this paper we study local induction w.r.t. Σ1‐formulas over the weak arithmetic. The local induction scheme, which was introduced in, says roughly this: for any virtual class that is progressive, i.e., is closed under zero and successor, and for any non‐empty virtual class that is definable by a Σ1‐formula without parameters, the intersection of and is non‐empty. In other words, we have, for all Σ1‐sentences S, that S implies, whenever is progressive. Since, in the weak context, we have (at (...)
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  28. Cardinal arithmetic in the style of Baron Von münchhausen.Albert Visser - 2009 - Review of Symbolic Logic 2 (3):570-589.
    In this paper we show how to interpret Robinson’s arithmetic Q and the theory R of Tarski, Mostowski, and Robinson as theories of cardinals in very weak theories of relations over a domain.
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  29. Intermediate Logics and the de Jongh property.Dick de Jongh, Rineke Verbrugge & Albert Visser - 2011 - Archive for Mathematical Logic 50 (1-2):197-213.
    We prove that all extensions of Heyting Arithmetic with a logic that has the finite frame property possess the de Jongh property.
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  30.  30
    On a Question of Krajewski's.Fedor Pakhomov & Albert Visser - 2019 - Journal of Symbolic Logic 84 (1):343-358.
    In this paper, we study finitely axiomatizable conservative extensions of a theoryUin the case whereUis recursively enumerable and not finitely axiomatizable. Stanisław Krajewski posed the question whether there are minimal conservative extensions of this sort. We answer this question negatively.Consider a finite expansion of the signature ofUthat contains at least one predicate symbol of arity ≥ 2. We show that, for any finite extensionαofUin the expanded language that is conservative overU, there is a conservative extensionβofUin the expanded language, such that$\alpha (...)
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  31.  24
    A course on bimodal provability logic.Albert Visser - 1995 - Annals of Pure and Applied Logic 73 (1):109-142.
    In this paper we study 1. the frame-theory of certain bimodal provability logics involving the reflection principle and we study2. certain specific bimodal logics with a provability predicate for a subtheory of Peano arithmetic axiomatized by a non-standardly finite number of axioms.
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  32.  30
    Submodels of Kripke models.Albert Visser - 2001 - Archive for Mathematical Logic 40 (4):277-295.
    A Kripke model ? is a submodel of another Kripke model ℳ if ? is obtained by restricting the set of nodes of ℳ. In this paper we show that the class of formulas of Intuitionistic Predicate Logic that is preserved under taking submodels of Kripke models is precisely the class of semipositive formulas. This result is an analogue of the Łoś-Tarski theorem for the Classical Predicate Calculus.In Appendix A we prove that for theories with decidable identity we can take (...)
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  33.  16
    The small‐is‐very‐small principle.Albert Visser - 2019 - Mathematical Logic Quarterly 65 (4):453-478.
    The central result of this paper is the small‐is‐very‐small principle for restricted sequential theories. The principle says roughly that whenever the given theory shows that a definable property has a small witness, i.e., a witness in a sufficiently small definable cut, then it shows that the property has a very small witness: i.e., a witness below a given standard number. Which cuts are sufficiently small will depend on the complexity of the formula defining the property. We draw various consequences from (...)
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  34.  69
    Hume’s principle, beginnings.Albert Visser - 2011 - Review of Symbolic Logic 4 (1):114-129.
    In this note we derive Robinson???s Arithmetic from Hume???s Principle in the context of very weak theories of classes and relations.
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  35.  24
    Friedman-reflexivity.Albert Visser - 2022 - Annals of Pure and Applied Logic 173 (9):103160.
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  36.  21
    Explicit Fixed Points in Interpretability Logic.Dick de Jongh & Albert Visser - 1991 - Studia Logica 50 (1):39-49.
    The problem of Uniqueness and Explicit Definability of Fixed Points for Interpretability Logic is considered. It turns out that Uniqueness is an immediate corollary of a theorem of Smoryński.
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  37.  67
    The interpretability logic of all reasonable arithmetical theories.Joost J. Joosten & Albert Visser - 2000 - Erkenntnis 53 (1-2):3-26.
    This paper is a presentation of astatus quæstionis, to wit of the problemof the interpretability logic of all reasonablearithmetical theories.We present both the arithmetical side and themodal side of the question.Dedicated to Dick de Jongh on the occasion of his 60th birthday.
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  38.  17
    Uniform Density in Lindenbaum Algebras.V. Yu Shavrukov & Albert Visser - 2014 - Notre Dame Journal of Formal Logic 55 (4):569-582.
    In this paper we prove that the preordering $\lesssim $ of provable implication over any recursively enumerable theory $T$ containing a modicum of arithmetic is uniformly dense. This means that we can find a recursive extensional density function $F$ for $\lesssim $. A recursive function $F$ is a density function if it computes, for $A$ and $B$ with $A\lnsim B$, an element $C$ such that $A\lnsim C\lnsim B$. The function is extensional if it preserves $T$-provable equivalence. Secondly, we prove a (...)
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  39.  12
    On the completeness principle: A study of provability in Heyting's Arithmetic and extensions.Albert Visser - 1982 - Annals of Mathematical Logic 22 (3):263.
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  40.  59
    Explicit fixed points in interpretability logic.Dick Jongh & Albert Visser - 1991 - Studia Logica 50 (1):39 - 49.
    The problem of Uniqueness and Explicit Definability of Fixed Points for Interpretability Logic is considered. It turns out that Uniqueness is an immediate corollary of a theorem of Smoryski.
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  41.  26
    The Arithmetics of a Theory.Albert Visser - 2015 - Notre Dame Journal of Formal Logic 56 (1):81-119.
    In this paper we study the interpretations of a weak arithmetic, like Buss’s theory $\mathsf{S}^{1}_{2}$, in a given theory $U$. We call these interpretations the arithmetics of $U$. We develop the basics of the structure of the arithmetics of $U$. We study the provability logic of $U$ from the standpoint of the framework of the arithmetics of $U$. Finally, we provide a deeper study of the arithmetics of a finitely axiomatized sequential theory.
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  42.  69
    The Second Incompleteness Theorem and Bounded Interpretations.Albert Visser - 2012 - Studia Logica 100 (1-2):399-418.
    In this paper we formulate a version of Second Incompleteness Theorem. The idea is that a sequential sentence has ‘consistency power’ over a theory if it enables us to construct a bounded interpretation of that theory. An interpretation of V in U is bounded if, for some n , all translations of V -sentences are U -provably equivalent to sentences of complexity less than n . We call a sequential sentence with consistency power over T a pro-consistency statement for T (...)
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  43.  57
    Vaught's theorem on axiomatizability by a scheme.Albert Visser - 2012 - Bulletin of Symbolic Logic 18 (3):382-402.
    In his 1967 paper Vaught used an ingenious argument to show that every recursively enumerable first order theory that directly interprets the weak system VS of set theory is axiomatizable by a scheme. In this paper we establish a strengthening of Vaught's theorem by weakening the hypothesis of direct interpretability of VS to direct interpretability of the finitely axiomatized fragment VS2 of VS. This improvement significantly increases the scope of the original result, since VS is essentially undecidable, but VS2 has (...)
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  44. Intermediate Logics and the de Jongh property.Dick Jongh, Rineke Verbrugge & Albert Visser - 2011 - Archive for Mathematical Logic 50 (1-2):197-213.
    We prove that all extensions of Heyting Arithmetic with a logic that has the finite frame property possess the de Jongh property.
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  45.  40
    Transductions in arithmetic.Albert Visser - 2016 - Annals of Pure and Applied Logic 167 (3):211-234.
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  46.  23
    On the limit existence principles in elementary arithmetic and Σ n 0 -consequences of theories.Lev D. Beklemishev & Albert Visser - 2005 - Annals of Pure and Applied Logic 136 (1-2):56-74.
    We study the arithmetical schema asserting that every eventually decreasing elementary recursive function has a limit. Some other related principles are also formulated. We establish their relationship with restricted parameter-free induction schemata. We also prove that the same principle, formulated as an inference rule, provides an axiomatization of the Σ2-consequences of IΣ1.Using these results we show that ILM is the logic of Π1-conservativity of any reasonable extension of parameter-free Π1-induction schema. This result, however, cannot be much improved: by adapting a (...)
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  47.  66
    Dynamic Bracketing and Discourse Representation.Albert Visser & Kees Vermeulen - 1996 - Notre Dame Journal of Formal Logic 37 (2):321-365.
    In this paper we describe a framework for the construction of entities that can serve as interpretations of arbitrary contiguous chunks of text. An important part of the paper is devoted to describing stacking cells, or the proposed meanings for bracket-structures.
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  48.  78
    Contexts in dynamic predicate logic.Albert Visser - 1998 - Journal of Logic, Language and Information 7 (1):21-52.
    In this paper we introduce a notion of context for Groenendijk & Stokhof's Dynamic Predicate Logic DPL. We use these contexts to give a characterization of the relations on assignments that can be generated by composition from tests and random resettings in the case that we are working over an infinite domain. These relations are precisely the ones expressible in DPL if we allow ourselves arbitrary tests as a starting point. We discuss some possible extensions of DPL and the way (...)
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  49.  19
    Extension and Interpretability.Albert Visser - 2021 - In Mojtaba Mojtahedi, Shahid Rahman & MohammadSaleh Zarepour (eds.), Mathematics, Logic, and their Philosophies: Essays in Honour of Mohammad Ardeshir. Springer. pp. 53-92.
    In this paper we study the combined structure of the relations of theory-extension and interpretability between theories for the case of finitely axiomatised theories. We focus on two main questions. The first is definability of salient notions in terms of the structure. We show, for example, that local tolerance, locally faithful interpretability and the finite model property are definable over the structure. The second question is how to think about ‘good’ properties of theories that are independent of implementation details and (...)
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  50.  39
    No Escape from Vardanyan's theorem.Albert Visser & Maartje de Jonge - 2006 - Archive for Mathematical Logic 45 (5):539-554.
    Vardanyan's theorem states that the set of PA-valid principles of Quantified Modal Logic, QML, is complete Π0 2. We generalize this result to a wide class of theories. The crucial step in the generalization is avoiding the use of Tennenbaum's Theorem.
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