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Franco Parlamento [9]F. Parlamento [4]
  1.  2
    Absorbing the Structural Rules in the Sequent Calculus with Additional Atomic Rules.Franco Parlamento & Flavio Previale - forthcoming - Archive for Mathematical Logic:1-20.
    We show that if the structural rules are admissible over a set \ of atomic rules, then they are admissible in the sequent calculus obtained by adding the rules in \ to the multisuccedent minimal and intuitionistic \ calculi as well as to the classical one. Two applications to pure logic and to the sequent calculus with equality are presented.
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  2.  12
    Decidability of ∀*∀‐Sentences in Membership Theories.Eugenio G. Omodeo, Franco Parlamento & Alberto Policriti - 1996 - Mathematical Logic Quarterly 42 (1):41-58.
    The problem is addressed of establishing the satisfiability of prenex formulas involving a single universal quantifier, in diversified axiomatic set theories. A rather general decision method for solving this problem is illustrated through the treatment of membership theories of increasing strength, ending with a subtheory of Zermelo-Fraenkel which is already complete with respect to the ∀*∀ class of sentences. NP-hardness and NP-completeness results concerning the problems under study are achieved and a technique for restricting the universal quantifier is presented.
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  3.  3
    The Elimination of Atomic Cuts and the Semishortening Property for Gentzen’s Sequent Calculus with Equality.F. Parlamento & F. Previale - forthcoming - Review of Symbolic Logic:1-32.
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  4.  9
    Decidability of ∃*∀∀-Sentences in HF.D. Bellè & F. Parlamento - 2008 - Notre Dame Journal of Formal Logic 49 (1):55-64.
    Let HF be the collection of the hereditarily finite well-founded sets and let the primitive language of set theory be the first-order language which contains binary symbols for equality and membership only. As announced in a previous paper by the authors, "Truth in V for ∃*∀∀-sentences is decidable," truth in HF for ∃*∀∀-sentences of the primitive language is decidable. The paper provides the proof of that claim.
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  5.  19
    The Decision Problem for Restricted Universal Quantification in Set Theory and the Axiom of Foundation.Franco Parlamento & Alberto Policriti - 1992 - Zeitschrift fur mathematische Logik und Grundlagen der Mathematik 38 (1):143-156.
  6.  7
    The Decision Problem for Restricted Universal Quantification in Set Theory and the Axiom of Foundation.Franco Parlamento & Alberto Policriti - 1992 - Mathematical Logic Quarterly 38 (1):143-156.
    The still unsettled decision problem for the restricted purely universal formulae 0-formulae) of the first order set-theoretic language based over =, ∈ is discussed in relation with the adoption or rejection of the axiom of foundation. Assuming the axiom of foundation, the related finite set-satisfiability problem for the very significant subclass of the 0-formulae consisting of the formulae involving only nested variables of level 1 is proved to be semidecidable on the ground of a reflection property over the hereditarily finite (...)
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  7.  14
    Cut Elimination for Gentzen's Sequent Calculus with Equality and Logic of Partial Terms.Franco Parlamento & Flavio Previale - 2013 - In Kamal Lodaya (ed.), Logic and its Applications. Springer. pp. 161--172.
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  8.  18
    Decidability and Completeness for Open Formulas of Membership Theories.Dorella Bellè & Franco Parlamento - 1995 - Notre Dame Journal of Formal Logic 36 (2):304-318.
    We establish the decidability, with respect to open formulas in the first order language with equality =, the membership relation , the constant for the empty set, and a binary operation w which, applied to any two sets x and y, yields the results of adding y as an element to x, of the theory NW having the obvious axioms for and w. Furthermore we establish the completeness with respect to purely universal sentences of the theory , obtained from NW (...)
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  9.  18
    Truth in V for Ǝ ∀∀-Sentences Is Decidable.D. Bellé & F. Parlamento - 2006 - Journal of Symbolic Logic 71 (4):1200 - 1222.
    Let V be the cumulative set theoretic hierarchy, generated from the empty set by taking powers at successor stages and unions at limit stages and, following [2], let the primitive language of set theory be the first order language which contains binary symbols for equality and membership only. Despite the existence of ∀∀-formulae in the primitive language, with two free variables, which are satisfiable in V but not by finite sets ([5]), and therefore of ƎƎ∀∀ sentences of the same language, (...)
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  10.  22
    Expressing Infinity Without Foundation.Franco Parlamento & Alberto Policriti - 1991 - Journal of Symbolic Logic 56 (4):1230-1235.
    The axiom of infinity can be expressed by stating the existence of sets satisfying a formula which involves restricted universal quantifiers only, even if the axiom of foundation is not assumed.
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  11.  7
    The Decidability of the Class and the Axiom of Foundation.Dorella Bellè & Franco Parlamento - 2001 - Notre Dame Journal of Formal Logic 42 (1):41-53.
    We show that the Axiom of Foundation, as well as the Antifoundation Axiom AFA, plays a crucial role in determining the decidability of the following problem. Given a first-order theory T over the language , and a sentence F of the form with quantifier-free in the same language, are there models of T in which F is true? Furthermore we show that the Extensionality Axiom is quite irrelevant in that respect.
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  12.  7
    Truth-Value Semantics and Functional Extensions for Classical Logic of Partial Terms Based on Equality.F. Parlamento - 2014 - Notre Dame Journal of Formal Logic 55 (3):383-395.
    We develop a bottom-up approach to truth-value semantics for classical logic of partial terms based on equality and apply it to prove the conservativity of the addition of partial description and selection functions, independently of any strictness assumption.
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