A THREE-VALUED QUANTIFIED ARGUMENT CALCULUS: DOMAIN-FREE MODEL-THEORY, COMPLETENESS, AND EMBEDDING OF FOL

2017 ◽  
Vol 10 (3) ◽  
pp. 549-582 ◽  
Author(s):  
RAN LANZET
Keyword(s):  
Natural Language ◽  
Model Theory ◽  
Predicate Calculus ◽  
Extended Version ◽  
First Order ◽  
Completeness Result ◽  
Free Model ◽  
Simplifying Assumptions ◽  
Syntactic Restrictions ◽  
Order Predicate Calculus

AbstractThis paper presents an extended version of the Quantified Argument Calculus (Quarc). Quarc is a logic comparable to the first-order predicate calculus. It employs several nonstandard syntactic and semantic devices, which bring it closer to natural language in several respects. Most notably, quantifiers in this logic are attached to one-place predicates; the resulting quantified constructions are then allowed to occupy the argument places of predicates. The version presented here is capable of straightforwardly translating natural-language sentences involving defining clauses. A three-valued, model-theoretic semantics for Quarc is presented. Interpretations in this semantics are not equipped with domains of quantification: they are just interpretation functions. This reflects the analysis of natural-language quantification on which Quarc is based. A proof system is presented, and a completeness result is obtained. The logic presented here is capable of straightforward translation of the classical first-order predicate calculus, the translation preserving truth values as well as entailment. The first-order predicate calculus and its devices of quantification can be seen as resulting from Quarc on certain semantic and syntactic restrictions, akin to simplifying assumptions. An analogous, straightforward translation of Quarc into the first-order predicate calculus is impossible.

LePUS

2011 ◽  
pp. 357-372 ◽  
Author(s):  
Epameinondas Gasparis
Keyword(s):  
Mathematical Logic ◽  
Conceptual Framework ◽  
Model Theory ◽  
Design Patterns ◽  
Frame Of Reference ◽  
Predicate Calculus ◽  
Tool Support ◽  
First Order ◽  
Finite Structures ◽  
Order Predicate Calculus

We present LePUS, a formal language for modeling object oriented (O-O) Design patterns. We demonstrate the language’s unique efficacy in producing precise, concise, generic, and appropriately abstract specifications that effectively model the Gang of Four’s Design patterns. Mathematical logic is used as a main frame of reference: LePUS is defined as a subset of first-order predicate calculus and implementations (programs) are modeled as finite structures in model theory. We also demonstrate the conceptual framework in which the verification of implementations against pattern specifications is possible and our ongoing endeavour to develop effective tool support for LePUS.

A reduction class containing formulas with one monadic predicate and one binary function symbol

1976 ◽  
Vol 41 (1) ◽  
pp. 45-49
Author(s):  
Charles E. Hughes
Keyword(s):  
Normal Form ◽  
Predicate Symbol ◽  
Conjunctive Normal Form ◽  
Function Symbol ◽  
Predicate Calculus ◽  
Binary Function ◽  
First Order ◽  
The Matrix ◽  
Reduction Class ◽  
Order Predicate Calculus

AbstractA new reduction class is presented for the satisfiability problem for well-formed formulas of the first-order predicate calculus. The members of this class are closed prenex formulas of the form ∀x∀yC. The matrix C is in conjunctive normal form and has no disjuncts with more than three literals, in fact all but one conjunct is unary. Furthermore C contains but one predicate symbol, that being unary, and one function symbol which symbol is binary.

On the strong semantical completeness of the intuitionistic predicate calculus

1968 ◽  
Vol 33 (1) ◽  
pp. 1-7 ◽  
Author(s):  
Richmond H. Thomason
Keyword(s):  
Model Theory ◽  
Completeness Theorem ◽  
Predicate Calculus ◽  
First Order ◽  
Semantic Tableaux ◽  
Weak Completeness ◽  
Skolem Theorem ◽  
Completeness Theorems

In Kripke [8] the first-order intuitionjstic predicate calculus (without identity) is proved semantically complete with respect to a certain model theory, in the sense that every formula of this calculus is shown to be provable if and only if it is valid. Metatheorems of this sort are frequently called weak completeness theorems—the object of the present paper is to extend Kripke's result to obtain a strong completeness theorem for the intuitionistic predicate calculus of first order; i.e., we will show that a formula A of this calculus can be deduced from a set Γ of formulas if and only if Γ implies A. In notes 3 and 5, below, we will indicate how to account for identity, as well. Our proof of the completeness theorem employs techniques adapted from Henkin [6], and makes no use of semantic tableaux; this proof will also yield a Löwenheim-Skolem theorem for the modeling.

scholarly journals On a Propositional Calculus Whose Decision Problem is Recursively Unsolvable

1970 ◽  
Vol 38 ◽  
pp. 145-152
Author(s):  
Akira Nakamura
Keyword(s):  
Decision Problem ◽  
Propositional Calculus ◽  
Predicate Calculus ◽  
Reduction Theorem ◽  
List Type ◽  
First Order ◽  
Order Predicate Calculus

The purpose of this paper is to present a propositional calculus whose decision problem is recursively unsolvable. The paper is based on the following ideas: (1) Using Löwenheim-Skolem’s Theorem and Surányi’s Reduction Theorem, we will construct an infinitely many-valued propositional calculus corresponding to the first-order predicate calculus.(2) It is well known that the decision problem of the first-order predicate calculus is recursively unsolvable.(3) Thus it will be shown that the decision problem of the infinitely many-valued propositional calculus is recursively unsolvable.

Translating Advanced Integrity Checking Technology to SQL

2002 ◽  
pp. 203-249 ◽  
Author(s):  
Hendrik Decker
Keyword(s):  
Natural Language ◽  
Order Logic ◽  
Predicate Calculus ◽  
Integrity Constraints ◽  
First Order Logic ◽  
First Order ◽  
Integrity Checking ◽  
Natural Language Interface ◽  
User Friendly ◽  
Declarative Specifications

The main goal of this chapter is to arrive at a coherent technology for deriving efficient SQL triggers from declarative specifications of arbitrary integrity constraints. The user may specify integrity constraints declaratively as closed queries in predicate calculus syntax (i.e., sentences in the language of first-order logic, abbr. FOL), as datalog denials, as query conditions in SQL WHERE clauses, or in some other, possibly more user-friendly manner (e.g., via a dialog-driven graphical or natural language interface which internally translates to equivalent WHERE clause conditions). As we are going to see, the triggers derived from such specifications behave such that whenever some update event would violate any of the integrity constraints, one or several of the triggers derived from that constraint are activated in order to enforce the constraint. That is, the violation is either prevented by rolling back the update or repaired instantly by subsequent further updates.

Infinitary logic and admissible sets

1969 ◽  
Vol 34 (2) ◽  
pp. 226-252 ◽  
Author(s):  
Jon Barwise
Keyword(s):  
Second Order ◽  
Order Logic ◽  
Predicate Calculus ◽  
Generalized Quantifiers ◽  
Infinitary Logic ◽  
First Order ◽  
Admissible Sets ◽  
Classical Language ◽  
Second Order Logic ◽  
Order Predicate Calculus

In recent years much effort has gone into the study of languages which strengthen the classical first-order predicate calculus in various ways. This effort has been motivated by the desire to find a language which is(I) strong enough to express interesting properties not expressible by the classical language, but(II) still simple enough to yield interesting general results. Languages investigated include second-order logic, weak second-order logic, ω-logic, languages with generalized quantifiers, and infinitary logic.

Solution of a problem of Leon Henkin

1955 ◽  
Vol 20 (2) ◽  
pp. 115-118 ◽  
Author(s):  
M. H. Löb
Keyword(s):  
Number Theory ◽  
Recursive Function ◽  
Formal System ◽  
Free Variable ◽  
Formal Proof ◽  
Predicate Calculus ◽  
First Order ◽  
Free Variables ◽  
Order Predicate Calculus

If Σ is any standard formal system adequate for recursive number theory, a formula (having a certain integer q as its Gödel number) can be constructed which expresses the proposition that the formula with Gödel number q is provable in Σ. Is this formula provable or independent in Σ? [2].One approach to this problem is discussed by Kreisel in [4]. However, he still leaves open the question whether the formula (Ex)(x, a), with Gödel-number a, is provable or not. Here (x, y) is the number-theoretic predicate which expresses the proposition that x is the number of a formal proof of the formula with Gödel-number y.In this note we present a solution of the previous problem with respect to the system Zμ [3] pp. 289–294, and, more generally, with respect to any system whose set of theorems is closed under the rules of inference of the first order predicate calculus, and satisfies the subsequent five conditions, and in which the function (k, l) used below is definable.The notation and terminology is in the main that of [3] pp. 306–326, viz. if is a formula of Zμ containing no free variables, whose Gödel number is a, then ({}) stands for (Ex)(x, a) (read: the formula with Gödel number a is provable in Zμ); if is a formula of Zμ containing a free variable, y say, ({}) stands for (Ex)(x, g(y)}, where g(y) is a recursive function such that for an arbitrary numeral the value of g() is the Gödel number of the formula obtained from by substituting for y in throughout. We shall, however, depart trivially from [3] in writing (), where is an arbitrary numeral, for (Ex){x, ).

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