L. P. Belluce and C. C. Chang. A weak completeness theorem for infinite valued first-order logic. The journal of symbolic logic, vol. 28 no. 1 (for 1963, pub. 1964), pp. 43–50.

1971 ◽  
Vol 36 (2) ◽  
pp. 332-332
Author(s):  
Bruno Scarpellini
Keyword(s):  
Completeness Theorem ◽  
Order Logic ◽  
First Order Logic ◽  
Symbolic Logic ◽  
First Order ◽  
Weak Completeness

A weak completeness theorem for infinite valued first-order logic

1963 ◽  
Vol 28 (1) ◽  
pp. 43-50 ◽  
Author(s):  
L. P. Belluce ◽  
C. C. Chang
Keyword(s):  
Completeness Theorem ◽  
Unit Interval ◽  
Order Logic ◽  
First Order Logic ◽  
Order System ◽  
First Order ◽  
Recursively Enumerable ◽  
Weak Completeness ◽  
Truth Value

This paper contains some results concerning the completeness of a first-order system of infinite valued logicThere are under consideration two distinct notions of completeness corresponding to the two notions of validity (see Definition 3) and strong validity (see Definition 4). Both notions of validity, whether based on the unit interval [0, 1] or based on linearly ordered MV-algebras, use the element 1 as the designated truth value. Originally, it was thought by many investigators in the field that one should be able to prove that the set of valid sentences is recursively enumerable. It was first proved by Rutledge in [9] that the set of valid sentences in the monadic first-order infinite valued logic is recursively enumerable.

A Weak Completeness Theorem for Infinite Valued First-Order Logic.

1971 ◽  
Vol 36 (2) ◽  
pp. 332
Author(s):  
Bruno Scarpellini ◽  
L. P. Belluce ◽  
C. C. Chang
Keyword(s):  
Completeness Theorem ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Weak Completeness

Logic in the early 20th century

2018 ◽  
Author(s):  
Gregory H. Moore
Keyword(s):  
Twentieth Century ◽  
Algebraic Logic ◽  
Order Logic ◽  
First Order Logic ◽  
Axiomatic Method ◽  
Symbolic Logic ◽  
Modern Logic ◽  
Genetic Method ◽  
First Order ◽  
Modern Development

The creation of modern logic is one of the most stunning achievements of mathematics and philosophy in the twentieth century. Modern logic – sometimes called logistic, symbolic logic or mathematical logic – makes essential use of artificial symbolic languages. Since Aristotle, logic has been a part of philosophy. Around 1850 the mathematician Boole began the modern development of symbolic logic. During the twentieth century, logic continued in philosophy departments, but it began to be seriously investigated and taught in mathematics departments as well. The most important examples of the latter were, from 1905 on, Hilbert at Göttingen and then, during the 1920s, Church at Princeton. As the twentieth century began, there were several distinct logical traditions. Besides Aristotelian logic, there was an active tradition in algebraic logic initiated by Boole in the UK and continued by C.S. Peirce in the USA and Schröder in Germany. In Italy, Peano began in the Boolean tradition, but soon aimed higher: to express all major mathematical theorems in his symbolic logic. Finally, from 1879 to 1903, Frege consciously deviated from the Boolean tradition by creating a logic strong enough to construct the natural and real numbers. The Boole–Schröder tradition culminated in the work of Löwenheim (1915) and Skolem (1920) on the existence of a countable model for any first-order axiom system having a model. Meanwhile, in 1900, Russell was strongly influenced by Peano’s logical symbolism. Russell used this as the basis for his own logic of relations, which led to his logicism: pure mathematics is a part of logic. But his discovery of Russell’s paradox in 1901 required him to build a new basis for logic. This culminated in his masterwork, Principia Mathematica, written with Whitehead, which offered the theory of types as a solution. Hilbert came to logic from geometry, where models were used to prove consistency and independence results. He brought a strong concern with the axiomatic method and a rejection of the metaphysical goal of determining what numbers ‘really’ are. In his view, any objects that satisfied the axioms for numbers were numbers. He rejected the genetic method, favoured by Frege and Russell, which emphasized constructing numbers rather than giving axioms for them. In his 1917 lectures Hilbert was the first to introduce first-order logic as an explicit subsystem of all of logic (which, for him, was the theory of types) without the infinitely long formulas found in Löwenheim. In 1923 Skolem, directly influenced by Löwenheim, also abandoned those formulas, and argued that first-order logic is all of logic. Influenced by Hilbert and Ackermann (1928), Gödel proved the completeness theorem for first-order logic (1929) as well as incompleteness theorems for arithmetic in first-order and higher-order logics (1931). These results were the true beginning of modern logic.

Proof theory

2004 ◽  
Author(s):  
Shawn Hedman
Keyword(s):  
Propositional Logic ◽  
Completeness Theorem ◽  
Formal Proof ◽  
Order Logic ◽  
Truth Table ◽  
First Order Logic ◽  
Formal Proofs ◽  
Systematic Method ◽  
First Order

As with any logic, the semantics of first-order logic yield rules for deducing the truth of one sentence from that of another. In this chapter, we develop both formal proofs and resolution for first-order logic. As in propositional logic, each of these provides a systematic method for proving that one sentence is a consequence of another. Recall the Consequence problem for propositional logic. Given formulas F and G, the problemis to decide whether or not G is a consequence of F. From Chapter 1, we have three approaches to this problem: • We could compute the truth table for the formula F → G. If the truth values are all 1s then we conclude that F → G is a tautology and G is a consequence of F. Otherwise, G is not a consequence of F. • Using Tables 1.5 and 1.6, we could try to formally derive G from {F}. By the Completeness Theorem for propositional logic, G is a consequence of F if and only if {F} ├ G. • We could use resolution. By Theorem1.76, G is a consequence of F if and only if ∅ ∈ Res(H) where H is a formula in CNF equivalent to (F ∧￢G). Using these methods not only can we determine whether one formula is a consequence of another, but also we can determine whether a given formula is a tautology or a contradiction. A formula F is a tautology if and only if F is a consequence of (A∨￢A) if and only if ￢F is a contradiction. In this chapter, we consider the analogous problems for first-order logic. Given formulas φ and ψ, how can we determine whether ψ is a consequence of φ? Equivalently, how can we determine whether a given formula is a tautology or a contradiction? We present three methods for answering these questions. • In Section 3.1, we define a notion of formal proof for first-order logic by extending Table 1.5. • In Section 3.3, we “reduce” formulas of first-order logic to sets of formulas of propositional logic where we use resolution as defined in Chapter 1.

Prerequisites

1993 ◽  
Author(s):  
Raymond M. Smullyan
Keyword(s):  
Completeness Theorem ◽  
Order Logic ◽  
First Order Logic ◽  
Natural Numbers ◽  
Logical Validity ◽  
First Order ◽  
Gödel’S Completeness Theorem ◽  
Logical Connectives ◽  
Incompleteness Theorems ◽  
Individual Variables

As we remarked in the preface, although this volume is a sequel to our earlier volume G.I.T. (Gödel’s Incompleteness Theorems), it can be read independently by those readers familiar with at least one proof of Gödel’s first incompleteness theorem. In this chapter we give the notation, terminology and main results of G.I.T. that are needed for this volume. Readers familiar with G.I.T. can skip this chapter or perhaps glance through it briefly as a refresher. §0. Preliminaries. we assume the reader to be familiar with the basic notions of first-order logic—the logical connectives, quantifiers, terms, formulas, free and bound occurrences of variables, the notion of interpretations (or models), truth under an interpretation, logical validity (truth under all interpretations), provability (in some complete system of first-order logic with identity) and its equivalence to logical validity (Gödel’s completeness theorem). we let S be a system (theory) couched in the language of first-order logic with identity and with predicate and/or function symbols and with names for the natural numbers. A system S is usually presented by taking some standard axiomatization of first-order logic with identity and adding other axioms called the non-logical axioms of S.we associate with each natural number n an expression n̅ of S called the numeral designating n (or the name of n).we could, for example, take 0̅,1̅,2̅, . . . ,to be the expressions 0,0', 0",..., as we did in G.I.T. we have our individual variables arranged in some fixed infinite sequence v1, v2,..., vn , . . . . By F(v1, ..., vn) we mean any formula whose free variables are all among v1,... ,vn, and for any (natural) numbers k1,...,kn by F(к̅1 ,... к̅n), we mean the result of substituting the numerals к̅1 ,... к̅n, for all free occurrences of v1,... ,vn in F respectively.

Ranked partial structures

2003 ◽  
Vol 68 (4) ◽  
pp. 1109-1144
Author(s):  
Timothy J. Carlson
Keyword(s):  
Model Theory ◽  
Completeness Theorem ◽  
Order Logic ◽  
Basic Theory ◽  
First Order Logic ◽  
Theoretic Method ◽  
First Order ◽  
Partial Structures ◽  
Skolem Theorem

AbstractThe theory of ranked partial structures allows a reinterpretation of several of the standard results of model theory and first-order logic and is intended to provide a proof-theoretic method which allows for the intuitions of model theory. A version of the downward Löwenheim-Skolem theorem is central to our development. In this paper we will present the basic theory of ranked partial structures and their logic including an appropriate version of the completeness theorem.

scholarly journals A proof of completeness for continuous first-order logic

2010 ◽  
Vol 75 (1) ◽  
pp. 168-190 ◽  
Author(s):  
Itaï Ben Yaacov ◽  
Arthur Paul Pedersen
Keyword(s):  
Model Theory ◽  
Completeness Theorem ◽  
Order Logic ◽  
First Order Logic ◽  
Complete Theory ◽  
Strong Completeness ◽  
First Order ◽  
Completeness Result ◽  
Metric Structures ◽  
Natural Classes

AbstractContinuous first-order logic has found interest among model theorists who wish to extend the classical analysis of “algebraic” structures (such as fields, group, and graphs) to various natural classes of complete metric structures (such as probability algebras, Hilbert spaces, and Banach spaces). With research in continuous first-order logic preoccupied with studying the model theory of this framework, we find a natural question calls for attention. Is there an interesting set of axioms yielding a completeness result?The primary purpose of this article is to show that a certain, interesting set of axioms does indeed yield a completeness result for continuous first-order logic. In particular, we show that in continuous first-order logic a set of formulae is (completely) satisfiable if (and only if) it is consistent. From this result it follows that continuous first-order logic also satisfies anapproximatedform of strong completeness, whereby Σ⊧φ(if and) only if Σ⊢φ∸2−nfor alln < ω. This approximated form of strong completeness asserts that if Σ⊧φ, then proofs from Σ, being finite, can provide arbitrarily better approximations of the truth ofφ.Additionally, we consider a different kind of question traditionally arising in model theory—that of decidability. When is the set of all consequences of a theory (in a countable, recursive language) recursive? Say that a complete theoryTisdecidableif for every sentenceφ, the valueφTis a recursive real, and moreover, uniformly computable fromφ. IfTis incomplete, we say it is decidable if for every sentenceφthe real numberφTois uniformly recursive fromφ, whereφTois the maximal value ofφconsistent withT. As in classical first-order logic, it follows from the completeness theorem of continuous first-order logic that if a complete theory admits a recursive (or even recursively enumerable) axiomatization then it is decidable.

scholarly journals Completud débil y Post completud en la escuela de Hilbert

2019 ◽  
pp. 449
Author(s):  
Víctor Aranda
Keyword(s):  
Open Problem ◽  
Propositional Logic ◽  
Normal Forms ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Weak Completeness ◽  
Arithmetical Interpretation

The aim of this paper is to clarify why propositional logic is Post complete and its weak completeness was almost unnoticed by Hilbert and Bernays, while first-order logic is Post incomplete and its weak completeness was seen as an open problem by Hilbert and Ackermman. Thus, I will compare propositional and first-order logic in the Prinzipien der Mathematik, Bernays’s second Habilitationsschrift and the Grundzüge der Theoretischen Logik. The so called “arithmetical interpretation”, the conjunctive and disjunctive normal forms and the soundness of the propositional rules of inference deserve special emphasis.

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