76 found
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  1.  50
    Self-Reference in Arithmetic I.Volker Halbach & Albert Visser - 2014 - Review of Symbolic Logic 7 (4):671-691.
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  2.  68
    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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  3. New Constructions of Satisfaction Classes.Albert Visser & Ali Enayat - 2015 - In Kentaro Fujimoto, José Martínez Fernández, Henri Galinon & Theodora Achourioti (eds.), Unifying the Philosophy of Truth. Springer Verlag.
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  4.  63
    Semantics and the Liar Paradox.Albert Visser - 1989 - Handbook of Philosophical Logic 4 (1):617--706.
  5.  25
    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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  6.  32
    Self-Reference in Arithmetic II.Volker Halbach & Albert Visser - 2014 - Review of Symbolic Logic 7 (4):692-712.
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  7.  43
    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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  8. 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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  9. 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.
  10.  82
    Four Valued Semantics and the Liar.Albert Visser - 1984 - Journal of Philosophical Logic 13 (2):181 - 212.
  11.  18
    Faith & Falsity.Albert Visser - 2004 - Annals of Pure and Applied Logic 131 (1):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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  12.  16
    On a Question of Krajewski's.Fedor Pakhomov & Albert Visser - 2019 - Journal of Symbolic Logic 84 (1):343-358.
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  13.  83
    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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  14.  42
    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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  15.  30
    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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  16.  65
    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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  17.  24
    An Inside View of Exp; or, The Closed Fragment of the Provability Logic of $I\Delta0 + \Omega1$ with a Propositional Constant for $\Operatorname{Exp}$. [REVIEW]Albert Visser - 1992 - Journal of Symbolic Logic 57 (1):131-165.
    In this paper I give a characterization of the closed fragment of the provability logic of $I \triangle_0 + \mathrm{EXP}$ with a propositional constant for $\mathrm{EXP}$. In three appendices many details on arithmetization are provided.
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  18.  15
    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.
  19.  13
    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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  20.  12
    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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  21.  19
    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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  22.  27
    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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  23.  47
    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.
  24. Stanford Encyclopedia of Philosophy.Jan van Eijck & Albert Visser - unknown
    Notice: This PDF version was distributed by request to members of the Friends of the SEP Society and by courtesy to SEP content contributors. It is solely for their fair use. Unauthorized distribution is prohibited. To learn how to join the Friends of the..
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  25.  9
    Substitutions of< I> Σ_< Sub> 1< Sup> 0-Sentences: Explorations Between Intuitionistic Propositional Logic and Intuitionistic Arithmetic. [REVIEW]Albert Visser - 2002 - Annals of Pure and Applied Logic 114 (1):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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  26.  41
    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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  27.  13
    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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  28.  23
    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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  29. An Overview of Interpretability Logic.Albert Visser - 1998 - In Marcus Kracht, Maarten de Rijke, Heinrich Wansing & Michael Zakharyaschev (eds.), Advances in Modal Logic, Volume 1. CSLI Publications. pp. 307-359.
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  30.  29
    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.
  31.  13
    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.
  32.  3
    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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  33.  17
    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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  34.  25
    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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  35. 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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  36.  38
    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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  37.  8
    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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  38.  16
    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.
  39.  45
    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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  40.  8
    Modal Logic and Self-Reference.Albert Visser & Craig Smorynski - 1989 - Journal of Symbolic Logic 54 (4):1479.
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  41.  36
    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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  42.  15
    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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  43.  60
    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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  44.  17
    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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  45.  41
    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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  46.  18
    Interpretability Degrees of Finitely Axiomatized Sequential Theories.Albert Visser - 2014 - Archive for Mathematical Logic 53 (1-2):23-42.
    In this paper we show that the degrees of interpretability of finitely axiomatized extensions-in-the-same-language of a finitely axiomatized sequential theory—like Elementary Arithmetic EA, IΣ1, or the Gödel–Bernays theory of sets and classes GB—have suprema. This partially answers a question posed by Švejdar in his paper (Commentationes Mathematicae Universitatis Carolinae 19:789–813, 1978). The partial solution of Švejdar’s problem follows from a stronger fact: the convexity of the degree structure of finitely axiomatized extensions-in-the-same-language of a finitely axiomatized sequential theory in the degree (...)
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  47.  4
    The Small‐is‐Very‐Small Principle.Albert Visser - 2019 - Mathematical Logic Quarterly 65 (4):453-478.
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  48. Logic and Information Flow.J. van Eijck & Albert Visser - 1994
     
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  49.  5
    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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  50.  55
    A Small Reflection Principle for Bounded Arithmetic.Rineke Verbrugge & Albert Visser - 1994 - Journal of Symbolic Logic 59 (3):785-812.
    We investigate the theory IΔ 0 + Ω 1 and strengthen [Bu86. Theorem 8.6] to the following: if NP ≠ co-NP. then Σ-completeness for witness comparison formulas is not provable in bounded arithmetic. i.e. $I\delta_0 + \Omega_1 + \nvdash \forall b \forall c (\exists a(\operatorname{Prf}(a.c) \wedge \forall = \leq a \neg \operatorname{Prf} (z.b))\\ \rightarrow \operatorname{Prov} (\ulcorner \exists a(\operatorname{Prf}(a. \bar{c}) \wedge \forall z \leq a \neg \operatorname{Prf}(z.\bar{b})) \urcorner)).$ Next we study a "small reflection principle" in bounded arithmetic. We prove that for (...)
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