19 found
Order:
Disambiguations
Joost J. Joosten [16]Joost Joosten [3]
  1.  12
    On Provability Logics with Linearly Ordered Modalities.Lev D. Beklemishev, David Fernández-Duque & Joost J. Joosten - 2014 - Studia Logica 102 (3):541-566.
    We introduce the logics GLP Λ, a generalization of Japaridze’s polymodal provability logic GLP ω where Λ is any linearly ordered set representing a hierarchy of provability operators of increasing strength. We shall provide a reduction of these logics to GLP ω yielding among other things a finitary proof of the normal form theorem for the variable-free fragment of GLP Λ and the decidability of GLP Λ for recursive orderings Λ. Further, we give a restricted axiomatization of the variable-free fragment (...)
    Direct download (6 more)  
     
    Export citation  
     
    Bookmark   7 citations  
  2.  15
    The Omega-Rule Interpretation of Transfinite Provability Logic.David Fernández-Duque & Joost J. Joosten - 2018 - Annals of Pure and Applied Logic 169 (4):333-371.
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark   2 citations  
  3.  2
    Models of Transfinite Provability Logic.David Fernández-Duque & Joost J. Joosten - 2013 - Journal of Symbolic Logic 78 (2):543-561.
    For any ordinal $\Lambda$, we can define a polymodal logic $\mathsf{GLP}_\Lambda$, with a modality $[\xi]$ for each $\xi < \Lambda$. These represent provability predicates of increasing strength. Although $\mathsf{GLP}_\Lambda$ has no Kripke models, Ignatiev showed that indeed one can construct a Kripke model of the variable-free fragment with natural number modalities, denoted $\mathsf{GLP}^0_\omega$. Later, Icard defined a topological model for $\mathsf{GLP}^0_\omega$ which is very closely related to Ignatiev's. In this paper we show how to extend these constructions for arbitrary $\Lambda$. (...)
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark   6 citations  
  4.  22
    Hyperations, Veblen Progressions and Transfinite Iteration of Ordinal Functions.David Fernández-Duque & Joost J. Joosten - 2013 - Annals of Pure and Applied Logic 164 (7-8):785-801.
    Ordinal functions may be iterated transfinitely in a natural way by taking pointwise limits at limit stages. However, this has disadvantages, especially when working in the class of normal functions, as pointwise limits do not preserve normality. To this end we present an alternative method to assign to each normal function f a family of normal functions Hyp[f]=〈fξ〉ξ∈OnHyp[f]=〈fξ〉ξ∈On, called its hyperation, in such a way that f0=idf0=id, f1=ff1=f and fα+β=fα∘fβfα+β=fα∘fβ for all α, β.Hyperations are a refinement of the Veblen hierarchy (...)
    Direct download (5 more)  
     
    Export citation  
     
    Bookmark   5 citations  
  5.  8
    Predicativity Through Transfinite Reflection.Andrés Cordón-Franco, David Fernández-Duque, Joost J. Joosten & Francisco Félix Lara-martín - 2017 - Journal of Symbolic Logic 82 (3):787-808.
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark   1 citation  
  6.  5
    A New Principle In The Interpretability Logic Of All Reasonable Arithmetical Theories.Evan Goris & Joost Joosten - 2011 - Logic Journal of the IGPL 19 (1):1-17.
    The interpretability logic of a mathematical theory describes the structural behavior of interpretations over that theory. Different theories have different logics. This paper revolves around the question what logic describes the behavior that is present in all theories with a minimum amount of arithmetic; the intersection over all such theories so to say. We denote this target logic by IL.In this paper we present a new principle R in IL. We show that R does not follow from the logic ILP0W* (...)
    Direct download  
     
    Export citation  
     
    Bookmark   3 citations  
  7.  9
    Modal Matters for Interpretability Logics.Evan Goris & Joost Joosten - 2008 - Logic Journal of the IGPL 16 (4):371-412.
    This paper is the first in a series of three related papers on modal methods in interpretability logics and applications. In this first paper the fundaments are laid for later results. These fundaments consist of a thorough treatment of a construction method to obtain modal models. This construction method is used to reprove some known results in the area of interpretability like the modal completeness of the logic IL. Next, the method is applied to obtain new results: the modal completeness (...)
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark   4 citations  
  8.  10
    Turing–Taylor Expansions for Arithmetic Theories.Joost J. Joosten - 2016 - Studia Logica 104 (6):1225-1243.
    Turing progressions have been often used to measure the proof-theoretic strength of mathematical theories: iterate adding consistency of some weak base theory until you “hit” the target theory. Turing progressions based on n-consistency give rise to a \ proof-theoretic ordinal \ also denoted \. As such, to each theory U we can assign the sequence of corresponding \ ordinals \. We call this sequence a Turing-Taylor expansion or spectrum of a theory. In this paper, we relate Turing-Taylor expansions of sub-theories (...)
    Direct download (6 more)  
     
    Export citation  
     
    Bookmark   1 citation  
  9.  15
    Empirical Encounters with Computational Irreducibility and Unpredictability.Hector Zenil, Fernando Soler-Toscano & Joost J. Joosten - 2012 - Minds and Machines 22 (3):149-165.
    The paper presents an exploration of conceptual issues that have arisen in the course of investigating speed-up and slowdown phenomena in small Turing machines, in particular results of a test that may spur experimental approaches to the notion of computational irreducibility. The test involves a systematic attempt to outrun the computation of a large number of small Turing machines (3 and 4 state, 2 symbol) by means of integer sequence prediction using a specialized function for that purpose. The experiment prompts (...)
    Direct download (7 more)  
     
    Export citation  
     
    Bookmark   2 citations  
  10.  44
    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.
    Direct download (8 more)  
     
    Export citation  
     
    Bookmark   4 citations  
  11.  31
    Provability and Interpretability Logics with Restricted Realizations.Thomas F. Icard & Joost J. Joosten - 2012 - Notre Dame Journal of Formal Logic 53 (2):133-154.
    The provability logic of a theory $T$ is the set of modal formulas, which under any arithmetical realization are provable in $T$. We slightly modify this notion by requiring the arithmetical realizations to come from a specified set $\Gamma$. We make an analogous modification for interpretability logics. We first study provability logics with restricted realizations and show that for various natural candidates of $T$ and restriction set $\Gamma$, the result is the logic of linear frames. However, for the theory Primitive (...)
    Direct download (5 more)  
     
    Export citation  
     
    Bookmark   1 citation  
  12.  9
    The Closed Fragment of the Interpretability Logic of PRA with a Constant for $\Mathrm{I}\Sigma_1$.Joost J. Joosten - 2005 - Notre Dame Journal of Formal Logic 46 (2):127-146.
    In this paper we carry out a comparative study of $\mathrm{I}\Sigma_1$ and PRA. We will in a sense fully determine what these theories have to say about each other in terms of provability and interpretability. Our study will result in two arithmetically complete modal logics with simple universal models.
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark   2 citations  
  13.  5
    Self Provers and Σ1 Sentences.Evan Goris & Joost Joosten - 2012 - Logic Journal of the IGPL 20 (1):1-21.
    This paper is the second in a series of three papers. All three papers deal with interpretability logics and related matters. In the first paper a construction method was exposed to obtain models of these logics. Using this method, we obtained some completeness results, some already known, and some new. In this paper, we will set the construction method to work to obtain more results. First, the modal completeness of the logic ILM is proved using the construction method. This is (...)
    Direct download (2 more)  
     
    Export citation  
     
    Bookmark   1 citation  
  14.  11
    The Closed Fragment of the Interpretability Logic of PRA with a Constant For.Joost J. Joosten - 2005 - Notre Dame Journal of Formal Logic 46 (2):127-146.
    In this paper we carry out a comparative study of and PRA. We will in a sense fully determine what these theories have to say about each other in terms of provability and interpretability. Our study will result in two arithmetically complete modal logics with simple universal models.
    Direct download (5 more)  
     
    Export citation  
     
    Bookmark   1 citation  
  15.  32
    Consistency Statements and Iterations of Computable Functions in IΣ1 and PRA.Joost J. Joosten - 2010 - Archive for Mathematical Logic 49 (7-8):773-798.
    In this paper we will state and prove some comparative theorems concerning PRA and IΣ1. We shall provide a characterization of IΣ1 in terms of PRA and iterations of a class of functions. In particular, we prove that for this class of functions the difference between IΣ1 and PRA is exactly that, where PRA is closed under iterations of these functions, IΣ1 is moreover provably closed under iteration. We will formulate a sufficient condition for a model of PRA to be (...)
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark  
  16.  7
    Interpretability in PRA.Marta Bílková, Dick de Jongh & Joost J. Joosten - 2009 - Annals of Pure and Applied Logic 161 (2):128-138.
    Direct download (3 more)  
     
    Export citation  
     
    Bookmark  
  17.  24
    Propositional Proof Systems and Fast Consistency Provers.Joost J. Joosten - 2007 - Notre Dame Journal of Formal Logic 48 (3):381-398.
    A fast consistency prover is a consistent polytime axiomatized theory that has short proofs of the finite consistency statements of any other polytime axiomatized theory. Krajíček and Pudlák have proved that the existence of an optimal propositional proof system is equivalent to the existence of a fast consistency prover. It is an easy observation that NP = coNP implies the existence of a fast consistency prover. The reverse implication is an open question. In this paper we define the notion of (...)
    Direct download (7 more)  
     
    Export citation  
     
    Bookmark  
  18.  3
    Kripke Models of Transfinite Provability Logic.David Fernández-Duque & Joost J. Joosten - 2012 - In Thomas Bolander, Torben Braüner, Silvio Ghilardi & Lawrence Moss (eds.), Advances in Modal Logic, Volume 9. CSLI Publications. pp. 185-199.
    No categories
    Direct download  
     
    Export citation  
     
    Bookmark  
  19.  6
    Interpretability In.Marta Bílková, Dick de Jongh & Joost J. Joosten - 2009 - Annals of Pure and Applied Logic 161 (2):128-138.
    In this paper, we study IL(), the interpretability logic of . As is neither an essentially reflexive theory nor finitely axiomatizable, the two known arithmetical completeness results do not apply to : IL() is not or . IL() does, of course, contain all the principles known to be part of IL, the interpretability logic of the principles common to all reasonable arithmetical theories. In this paper, we take two arithmetical properties of and see what their consequences in the modal logic (...)
    Direct download (4 more)  
     
    Export citation  
     
    Bookmark