32 found
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  1.  24
    Propositional Superposition Logic.Athanassios Tzouvaras - 2018 - Logic Journal of the IGPL 26 (1):149-190.
  2.  3
    Algebraic Semantics for Propositional Superposition Logic.Athanassios Tzouvaras - forthcoming - Journal of Applied Non-Classical Logics:1-32.
    We provide a new semantics and a slightly different formalisation for the propositional logic with superposition introduced and studied in Tzouvaras [. Propositional superposition logic...
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  3.  35
    A Reduction of the NF Consistency Problem.Athanassios Tzouvaras - 2007 - Journal of Symbolic Logic 72 (1):285 - 304.
    We give a necessary and sufficient condition in order that a type-shifting automorphism be constructed on a model of the Theory of Simple Types (TST) by forcing. Namely it is proved that, if for every n ≥ 1 there is a model of TST in the ground model M of ZFC that contains an n-extendible coherent pair, then there is a generic extension M[G] of M that contains a model of TST with a type-shifting automorphism, and hence M[G] contains a (...)
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  4.  33
    Localizing the Axioms.Athanassios Tzouvaras - 2010 - Archive for Mathematical Logic 49 (5):571-601.
    We examine what happens if we replace ZFC with a localistic/relativistic system, LZFC, whose central new axiom, denoted by Loc(ZFC), says that every set belongs to a transitive model of ZFC. LZFC consists of Loc(ZFC) plus some elementary axioms forming Basic Set Theory (BST). Some theoretical reasons for this shift of view are given. All ${\Pi_2}$ consequences of ZFC are provable in LZFC. LZFC strongly extends Kripke-Platek (KP) set theory minus Δ0-Collection and minus ${\in}$ -induction scheme. ZFC+ “there is an (...)
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  5.  8
    Totally Non-Immune Sets.Athanassios Tzouvaras - 2015 - Mathematical Logic Quarterly 61 (1-2):103-116.
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  6.  10
    Worlds of Homogeneous Artifacts.Athanassios Tzouvaras - 1995 - Notre Dame Journal of Formal Logic 36 (3):454-474.
    We present a formal first-order theory of artificial objects, i.e., objects made out of a finite number of parts and subject to assembling and dismantling processes. These processes are absolutely reversible. The theory is an extension of the theory of finite sets with urelements. The notions of transformation and identity are defined and studied on the assumption that the objects are homogeneous, that is to say, all their atomic parts are of equal ontological importance. Particular emphasis is given to the (...)
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  7.  33
    Forcing and Antifoundation.Athanassios Tzouvaras - 2005 - Archive for Mathematical Logic 44 (5):645-661.
    It is proved that the forcing apparatus can be built and set to work in ZFCA (=ZFC minus foundation plus the antifoundation axiom AFA). The key tools for this construction are greatest fixed points of continuous operators (a method sometimes referred to as “corecursion”). As an application it is shown that the generic extensions of standard models of ZFCA are models of ZFCA again.
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  8.  23
    Cardinality Without Enumeration.Athanassios Tzouvaras - 2005 - Studia Logica 80 (1):121-141.
    We show that the notion of cardinality of a set is independent from that of wellordering, and that reasonable total notions of cardinality exist in every model of ZF where the axiom of choice fails. Such notions are either definable in a simple and natural way, or non-definable, produced by forcing. Analogous cardinality notions exist in nonstandard models of arithmetic admitting nontrivial automorphisms. Certain motivating phenomena from quantum mechanics are also discussed in the Appendix.
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  9.  97
    The Logic of Multisets Continued: The Case of Disjunction.Athanassios Tzouvaras - 2003 - Studia Logica 75 (3):287 - 304.
    We continue our work [5] on the logic of multisets (or on the multiset semantics of linear logic), by interpreting further the additive disjunction . To this purpose we employ a more general class of processes, called free, the axiomatization of which requires a new rule (not compatible with the full LL), the cancellation rule. Disjunctive multisets are modeled as finite sets of multisets. The -Horn fragment of linear logic, with the cut rule slightly restricted, is sound with respect to (...)
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  10.  62
    An Axiomatization of 'Very' Within Systiems of Set Theory.Athanassios Tzouvaras - 2003 - Studia Logica 73 (3):413 - 430.
    A structural (as opposed to Zadeh's quantitative) approach to fuzziness is given, based on the operator "very", which is added to the language of set theory together with some elementary axioms about it. Due to the axiom of foundation and to a lifting axiom, the operator is proved trivial on the cumulative hierarchy of ZF. So we have to drop either foundation or lifting. Since fuzziness concerns complemented predicates rather than sets, a class theory is needed for the very operator. (...)
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  11.  1
    Russell's Typicality as Another Randomness Notion.Athanassios Tzouvaras - 2020 - Mathematical Logic Quarterly 66 (3):355-365.
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  12. The Order Structure of Continua.Athanassios Tzouvaras - 1997 - Synthese 113 (3):381-421.
    A continuum is here a primitive notion intended to correspond precisely to a path-connected subset of the usual euclidean space. In contrast, however, to the traditional treatment, we treat here continua not as pointsets, but as irreducible entities equipped only with a partial ordering ≤ interpreted as parthood. Our aim is to examine what basic topological and geometric properties of continua can be expressed in the language of ≤, and what principles we need in order to prove elementary facts about (...)
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  13.  12
    Significant Parts and Identity of Artifacts.Athanassios Tzouvaras - 1993 - Notre Dame Journal of Formal Logic 34 (3):445-452.
  14.  20
    A Note on Real Subsets of A Recursively Saturated Model.Athanassios Tzouvaras - 1991 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 37 (13-16):207-216.
  15.  31
    Freiling's Axioms of Symmetry in a General Setting and Some Applications.Athanassios Tzouvaras - 2001 - Archive for Mathematical Logic 40 (2):131-145.
    We formulate C. Freiling's axioms of symmetry for general second-order structures with respect to a certain ideal of small sets contained in them and find several equivalent formulations of the principles. Then we focus on particular models, namely saturated and recursively saturated ones, and show that they are symmetric with respect to appropriate classes of small sets when their second-order part consists of definable sets. Some asymmetric models are also exhibited as well as partial asymmetric ones constructed by forcing.
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  16.  13
    Erratum: “Forcing and Antifoundation”. [REVIEW]Athanassios Tzouvaras - 2005 - Archive for Mathematical Logic 44 (5):663-663.
  17.  10
    A Note on Real Subsets of A Recursively Saturated Model.Athanassios Tzouvaras - 1991 - Mathematical Logic Quarterly 37 (13‐16):207-216.
  18.  22
    Aspects of Analytic Deduction.Athanassios Tzouvaras - 1996 - Journal of Philosophical Logic 25 (6):581-596.
    Let ⊢ be the ordinary deduction relation of classical first-order logic. We provide an "analytic" subrelation ⊢a of ⊢ which for propositional logic is defined by the usual "containment" criterion Γ ⊢a φ iff Γ⊢φ and Atom ⊆ Atom, whereas for predicate logic, ⊢a is defined by the extended criterion Γ⊢aφ iff Γ⊢aφ and Atom ⊆' Atom, where Atom ⊆' Atom means that every atomic formula occurring in φ "essentially occurs" also in Γ. If Γ, φ are quantifier-free, then the (...)
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  19.  34
    Logic of Knowledge and Utterance and the Liar.Athanassios Tzouvaras - 1998 - Journal of Philosophical Logic 27 (1):85-108.
    We extend the ordinary logic of knowledge based on the operator K and the system of axioms S₅ by adding a new operator Uφ, standing for "the agent utters φ", and certain axioms and a rule for U, forming thus a new system KU. The main advantage of KU is that we can express in it intentions of the speaker concerning the truth or falsehood of the claims he utters and analyze them logically. Specifically we can express in the new (...)
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  20.  39
    Classification of Non‐Well‐Founded Sets and an Application.Nitta Takashi, Okada Tomoko & Athanassios Tzouvaras - 2003 - Mathematical Logic Quarterly 49 (2):187-200.
    A complete list of Finsler, Scott and Boffa sets whose transitive closures contain 1, 2 and 3 elements is given. An algorithm for deciding the identity of hereditarily finite Scott sets is presented. Anti-well-founded sets, i. e., non-well-founded sets whose all maximal ∈-paths are circular, are studied. For example they form transitive inner models of ZFC minus foundation and empty set, and they include uncountably many hereditarily finite awf sets. A complete list of Finsler and Boffa awf sets with 2 (...)
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  21.  5
    An Axiomatization of 'Very' Within Systiems of Set Theory.Athanassios Tzouvaras - 2003 - Studia Logica 73 (3):413-430.
    A structural approach to fuzziness is given, based on the operator "very", which is added to the language of set theory together with some elementary axioms about it. Due to the axiom of foundation and to a lifting axiom, the operator is proved trivial on the cumulative hierarchy of ZF. So we have to drop either foundation or lifting. Since fuzziness concerns complemented predicates rather than sets, a class theory is needed for the very operator. And of them the Kelley-Morse (...)
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  22.  31
    On Expandability of Models of Peano Arithmetic to Models of the Alternative Set Theory.Athanassios Tzouvaras - 1992 - Journal of Symbolic Logic 57 (2):452-460.
    We give a sufficient condition for a countable model M of PA to be expandable to an ω-model of AST with absolute Ω-orderings. The condition is in terms of saturation schemes or, equivalently, in terms of the ability of the model to code sequences which have some kind of definition in (M, ω). We also show that a weaker scheme of saturation leads to the existence of wellorderings of the model with nice properties. Finally, we answer affirmatively the question of (...)
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  23.  23
    Omega‐ and Beta‐Models of Alternative Set Theory.Athanassios Tzouvaras - 1994 - Mathematical Logic Quarterly 40 (4):547-569.
    We present the axioms of Alternative Set Theory in the language of second-order arithmetic and study its ω- and β-models. These are expansions of the form , M ⊆ P, of nonstandard models M of Peano arithmetic such that ⊩ AST and ω ϵ M. Our main results are: A countable M ⊩ PA is β-expandable iff there is a regular well-ordering for M. Every countable β-model can be elementarily extended to an ω-model which is not a β-model. The Ω-orderings (...)
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  24.  39
    Periodicity of Negation.Athanassios Tzouvaras - 2001 - Notre Dame Journal of Formal Logic 42 (2):87-99.
    In the context of a distributive lattice we specify the sort of mappings that could be generally called ''negations'' and study their behavior under iteration. We show that there are periodic and nonperiodic ones. Natural periodic negations exist with periods 2, 3, and 4 and pace 2, as well as natural nonperiodic ones, arising from the interaction of interior and quasi interior mappings with the pseudocomplement. For any n and any even , negations of period n and pace s can (...)
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  25.  16
    A Combinatorial Result Related to the Consistency of New Foundations.Athanassios Tzouvaras - 2011 - Annals of Pure and Applied Logic 162 (5):373-383.
    We prove a combinatorial result for models of the 4-fragment of the Simple Theory of Types , TST4. The result says that if is a standard transitive and rich model of TST4, then satisfies the 0,0,n-property, for all n≥2. This property has arisen in the context of the consistency problem of the theory New Foundations . The result is a weak form of the combinatorial condition that was shown in Tzouvaras [5] to be equivalent to the consistency of NF. Such (...)
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  26.  17
    What is so Special with the Powerset Operation?Athanassios Tzouvaras - 2004 - Archive for Mathematical Logic 43 (6):723-737.
    The powerset operator, , is an operator which (1) sends sets to sets,(2) is defined by a positive formula and (3) raises the cardinality of its argument, i.e., | (x)|>|x|. As a consequence of (3), has a proper class as least fixed point (the universe itself). In this paper we address the questions: (a) How does contribute to the generation of the class of all positive operators? (b) Are there other operators with the above properties, “independent” of ? Concerning (a) (...)
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  27.  15
    How Effective Indeed is Present-Day Mathematics?Athanassios Tzouvaras - 2006 - Logic and Logical Philosophy 15 (2):131-153.
    We argue that E. Wigner’s well-known claim that mathematics is unreasonably effective in physics is only one side of the hill. The other side is the surprising insufficiency of present-day mathematics to capture the uniformities that arise in science outside physics. We describe roughly what the situation is in the areas of everyday reasoning, theory of meaning and vagueness. We make also the point that mathematics, as we know it today, founded on the concept of set, need not be a (...)
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  28.  11
    Large Transitive Models in Local ZFC.Athanassios Tzouvaras - 2014 - Archive for Mathematical Logic 53 (3-4):233-260.
    This paper is a sequel to Tzouvaras :571–601, 2010), where a local version of ZFC, LZFC, was introduced and examined and transitive models of ZFC with properties that resemble large cardinal properties, namely Mahlo and Π11\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Pi_1^1}$$\end{document}-indescribable models, were considered. By analogy we refer to such models as “large models”, and the properties in question as “large model properties”. Continuing here in the same spirit we consider further large model properties, that resemble (...)
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  29.  10
    Non‐Circular, Non‐Well‐Founded Set Universes.Athanassios Tzouvaras - 1993 - Mathematical Logic Quarterly 39 (1):454-460.
    We show that there are universes of sets which contain descending ϵ-sequences of length α for every ordinal α, though they do not contain any ϵ-cycle. It is also shown that there is no set universe containing a descending ϵ-sequence of length On. MSC: 03E30; 03E65.
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  30.  1
    Semantics for First-Order Superposition Logic.Athanassios Tzouvaras - 2019 - Logic Journal of the IGPL 27 (4):570-595.
    We investigate how the sentence choice semantics for propositional superposition logic developed in Tzouvaras could be extended so as to successfully apply to first-order superposition logic. There are two options for such an extension. The apparently more natural one is the formula choice semantics based on choice functions for pairs of arbitrary formulas of the basis language. It is proved however that the universal instantiation scheme of first-order logic, $\varphi \rightarrow \varphi $, is false, as a scheme of tautologies, with (...)
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  31.  5
    Erratum To: Localizing the Axioms.Athanassios Tzouvaras - 2011 - Archive for Mathematical Logic 50 (3-4):513-513.
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  32.  6
    Notions of Symmetry in Set Theory with Classes.Athanassios Tzouvaras - 2000 - Annals of Pure and Applied Logic 106 (1-3):275-296.
    We adapt C. Freiling's axioms of symmetry 190–200) to models of set theory with classes by identifying small classes with sets getting thus a sequence of principles An, for n2, of increasing strength. Several equivalents of A2 are given. A2 is incompatible both with the foundation axiom and the antifoundation axioms AFA considered in Aczel . A hierarchy of symmetry degrees of preorderings is introduced and compared with An. Models are presented in which this hierarchy is strict. The main result (...)
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