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.

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

scholarly journals Formalization Of Theories of Design Using: First Order Logic Augmented With A Causal Calculus; And Entropy of a Design Defined In Terms Of The Function, Behavior and Structure Ontology.

2021 ◽  
Author(s):  
KARTHIK GURUMURTHI
Keyword(s):  
High Probability ◽  
Formal Proof ◽  
Order Logic ◽  
First Order Logic ◽  
Logical Framework ◽  
Seamless Integration ◽  
Good Design ◽  
First Order ◽  
Individual Variables ◽  
And Behavior

A symbolic logical framework (L) consisting of first order logic augmented with a causal calculus has been provided to formalize, axiomatize and integrate theories of design. L is used to represent designs in the Function-Behavior-Structure (FBS) ontology in a single, widely applicable language that enables the following: seamless integration of representations of function, behavior and structure; and generality in the formalization of theories of design. FRs, constraints, structure and behavior are represented as sentences in L. FRs are represented (as abstractions of behavior) in the form of existentially quantified sentences, the instantiation of whose individual variables yields the representation of behavior. This enables the logical implication of FRs by behavior, without recourse to apriori criteria for satisfaction of FRs by behavior. Functional decomposition is represented to enable lower level FRs to logically imply the satisfaction of higher level FRs. The theory of whether and how structure and behavior satisfy FRs and constraints is represented as a formal proof in L. Important general attributes of designs such as solution-neutrality of FRs, probability of satisfaction of requirements and constraints (calculated in a Bayesian framework using Monte Carlo simulation), extent and nature of coupling, etc. have been defined in terms of the representation of a design in L. The entropy of a design is defined in terms of the above attributes of a design, based on which a general theory of what constitutes a good design has been formalized to include the desirability of solution-neutrality of (especially higher level) FRs, high probability of satisfaction of requirements and constraints, wide specifications, low variability and bias, use of fewer attributes to specify the design, less coupling (especially circular coupling at higher levels of FRs), parametrization, standardization, etc..

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.

Structures elementarily closed relative to a model for arithmetic

1968 ◽  
Vol 33 (1) ◽  
pp. 101-104
Author(s):  
Eugene W. Madison
Keyword(s):  
General Theory ◽  
Characteristic Zero ◽  
Order Logic ◽  
First Order Logic ◽  
Natural Numbers ◽  
Algebraic Numbers ◽  
First Order

The present paper is a sequel to [1]. It is our purpose to formulate a general theory derived from the methods used to obtain three results for the field of real algebraic numbers in [1]. As there, we shall concern ourselves almost exclusively with fields of characteristic zero; thus we assume a convenient formulation of first order logic with extralogical constants E(x, y), S(x, y, z), F(x, y, z), F(x, y), N(x) and 0, whose intended interpretations are equality, sum, product, y is the successor of x, x ∈ (where is a substructure satisfying all first order truths of the natural numbers) and zero, respectively. In addition, we shall use Q(x, y) for x ≤ y in those cases where our field is ordered, e.g. the field of real algebraic numbers.

Stability theory for topological logic, with applications to topological modules

1986 ◽  
Vol 51 (3) ◽  
pp. 755-769 ◽  
Author(s):  
T. G. Kucera
Keyword(s):  
Stability Theory ◽  
Quantifier Elimination ◽  
Order Logic ◽  
First Order Logic ◽  
Elementary Substructure ◽  
First Order ◽  
Topological Modules ◽  
Individual Variables ◽  
The Stability ◽  
First Main Theorem

In this paper I show how to develop stability theory within the context of the topological logic first introduced by McKee [Mc 76], Garavaglia [G 78] and Ziegler [Z 76]. I then study some specific applications to topological modules; in particular I prove two quantifier élimination theorems, one a generalization of a result of Garavaglia.In the first section I present a summary of basic results on topological model theory, mostly taken from the book of Flum and Ziegler [FZ 80]. This is done primarily to fix notation, but I also introduce the notion of an Lt-elementary substructure. The important point with this concept, as with many others, appears to be to allow only individuals to appear as parameters, not open sets.In the second section I begin the study of stability theory for Lt. I first develop a translation of the topological language Lt into an ordinary first-order language L*. The first main theorem is (2.3), which shows that the translation is faithful to the model-theoretic content of Lt, and provides the necessary tools for studying Lt theories in the context of ordinary first-order logic. The translation allows me to consider individual stability theory for Lt: the stability-theoretic study of those types of Lt in which only individual variables occur freely and in which only individuals occur as parameters. I originally developed this stability theory entirely within Lt; the fact that the theorems and their proofs were virtually identical to those in ordinary first order logic suggested the reduction from Lt to L*.

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.

On an algebra of sets of finite sequences

1970 ◽  
Vol 35 (1) ◽  
pp. 19-28 ◽  
Author(s):  
J. Donald Monk
Keyword(s):  
Infinite Sequence ◽  
Finite Sequence ◽  
Order Logic ◽  
First Order Logic ◽  
Cylindric Algebras ◽  
First Order ◽  
Basic Model ◽  
Finite Sequences ◽  
Individual Variables ◽  
Algebra Of Sets

The algebras studied in this paper were suggested to the author by William Craig as a possible substitute for cylindric algebras. Both kinds of algebras may be considered as algebraic versions of first-order logic. Cylindric algebras can be introduced as follows. Let ℒ be a first-order language, and let be an ℒ-structure. We assume that ℒ has a simple infinite sequence ν0, ν1, … of individual variables, and we take as known what it means for a sequence ν0, ν1, … of individual variables, and we take as known what it means for a sequence x = 〈x0, x1, …〉 of elements of to satisfy a formula ϕ of ℒ in . Let ϕ be the collection of all sequences x which satisfy ϕ in . We can perform certain natural operations on the sets ϕ, of basic model-theoretic significance: Boolean operations = cylindrifications diagonal elements (0-ary operations) . In this way we make the class of all sets ϕ into an algebra; a natural abstraction gives the class of all cylindric set algebras (of dimension ω). Thus this method of constructing an algebraic counterpart of first-order logic is based upon the notion of satisfaction of a formula by an infinite sequence of elements. Since, however, a formula has only finitely many variables occurring in it, it may seem more natural to consider satisfaction by a finite sequence of elements; then ϕ becomes a collection of finite sequences of varying ranks (cf. Tarski [10]). In forming an algebra of sets of finite sequences it turns out to be possible to get by with only finitely many operations instead of the infinitely many ci's and dij's of cylindric algebras. Let be the class of all algebras of sets of finite sequences (an exact definition is given in §1).

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