A note on direct products

1958 ◽  
Vol 23 (1) ◽  
pp. 1-6 ◽  
Author(s):  
L. Novak Gál
Keyword(s):  
Binary Relation ◽  
Direct Product ◽  
Functional Calculus ◽  
Ordered Set ◽  
Infinite Sequence ◽  
Direct Products ◽  
First Order ◽  
Individual Variables ◽  
The Individual ◽  
Infinite Sequences

By an algebra is meant an ordered set Γ = 〈V,R1, …, Rn, O1, …, Om〉, where V is a class, Ri (1 ≤ i ≤, n) is a relation on nj elements of V (i.e. Ri ⊆ Vni), and Oj (1 ≤ i ≤ n) is an operation on elements of V such that Oj(x1, … xmj) ∈ V) for all x1, …, xmj ∈ V). A sentence S of the first-order functional calculus is valid in Γ, if it contains just individual variables x1, x2, …, relation variables ϱ1, …,ϱn, where ϱi,- is nj-ary (1 ≤ i ≤ n), and operation variables σ1, …, σm, where σj is mj-ary (1 ≤, j ≤ m), and S holds if the individual variables are interpreted as ranging over V, ϱi is interpreted as Ri, and σi as Oj. If {Γi}i≤α is a (finite or infinite) sequence of algebras Γi, where Γi = 〈Vi, Ri〉 and Ri, is a binary relation, then by the direct productΓ = Πi<αΓi is meant the algebra Γ = 〈V, R〉, where V consists of all (finite or infinite) sequences x = 〈x1, x2, …, xi, …〉 with Xi ∈ Vi and where R is a binary relation such that two elements x and y of V are in the relation R if and only if xi and yi- are in the relation Ri for each i < α.

Topologies of Lattice Products

1966 ◽  
Vol 18 ◽  
pp. 1004-1014 ◽  
Author(s):  
Richard A. Alo ◽  
Orrin Frink
Keyword(s):  
Direct Product ◽  
Partially Ordered Set ◽  
Ordered Set ◽  
Order Relation ◽  
Direct Products ◽  
Ordered Sets ◽  
Partially Ordered

A number of different ways of defining topologies in a lattice or partially ordered set in terms of the order relation are known. Three of these methods have proved to be useful and convenient for lattices of special types, namely the ideal topology, the interval topology, and the new interval topology of Garrett Birkhoff. In another paper (2) we have shown that these three topologies are equivalent for chains (totally ordered sets), where they reduce to the usual intrinsic topology of the chain.Since many important lattices are either direct products of chains or sublattices of such products, it is natural to ask what relationships exist between the various order topologies of a direct product of lattices and those of the lattices themselves.

Completeness in the theory of types

1950 ◽  
Vol 15 (2) ◽  
pp. 81-91 ◽  
Author(s):  
Leon Henkin
Keyword(s):  
Functional Calculus ◽  
Formal System ◽  
Axiom System ◽  
Second Order ◽  
True Proposition ◽  
Arbitrary Domain ◽  
Individual Variables ◽  
Definition Of ◽  
The Individual ◽  
Positive Integers

The first order functional calculus was proved complete by Gödel in 1930. Roughly speaking, this proof demonstrates that each formula of the calculus is a formal theorem which becomes a true sentence under every one of a certain intended class of interpretations of the formal system.For the functional calculus of second order, in which predicate variables may be bound, a very different kind of result is known: no matter what (recursive) set of axioms are chosen, the system will contain a formula which is valid but not a formal theorem. This follows from results of Gödel concerning systems containing a theory of natural numbers, because a finite categorical set of axioms for the positive integers can be formulated within a second order calculus to which a functional constant has been added.By a valid formula of the second order calculus is meant one which expresses a true proposition whenever the individual variables are interpreted as ranging over an (arbitrary) domain of elements while the functional variables of degree n range over all sets of ordered n-tuples of individuals. Under this definition of validity, we must conclude from Gödel's results that the calculus is essentially incomplete.It happens, however, that there is a wider class of models which furnish an interpretation for the symbolism of the calculus consistent with the usual axioms and formal rules of inference. Roughly, these models consist of an arbitrary domain of individuals, as before, but now an arbitrary class of sets of ordered n-tuples of individuals as the range for functional variables of degree n. If we redefine the notion of valid formula to mean one which expresses a true proposition with respect to every one of these models, we can then prove that the usual axiom system for the second order calculus is complete: a formula is valid if and only if it is a formal theorem.

On the definition of ‘formal deduction’

1956 ◽  
Vol 21 (2) ◽  
pp. 129-136 ◽  
Author(s):  
Richard Montague ◽  
Leon Henkin
Keyword(s):  
Functional Calculus ◽  
Modus Ponens ◽  
Formal Proof ◽  
Axiom Schema ◽  
First Order ◽  
Individual Variable ◽  
Individual Variables ◽  
Definition Of ◽  
Formal Rules

The following remarks apply to many functional calculi, each of which can be variously axiomatized, but for clarity of exposition we shall confine our attention to one particular system Σ. This system is to have the usual primitive symbols and formation rules of the pure first-order functional calculus, and the following formal axiom schemata and formal rules of inference.Axiom schema 1. Any tautologous wff (well-formed formula).Axiom schema 2. (a) A ⊃ B, where A is any wff, a and b are any individual variables, and B arises from A by replacing all free occurrences of a by free occurrences of b.Axiom schema 3. (a)(A ⊃ B)⊃(A⊃ (a)B). where A and B are any wffs, and a is any individual variable not free in A.Rule of Modus Ponens: applies to wffs A and A ⊃ B, and yields B.Rule of Generalization: applies to a wff A and yields (a)A, where a is any individual variable.A formal proof in Σ is a finite column of wffs each of whose lines is a formal axiom or arises from two preceding lines by the Rule of Modus Ponens or arises from a single preceding line by the Rule of Generalization. A formal theorem of Σ is a wff which occurs as the last line of some formal proof.

Products of two-sorted structures

1972 ◽  
Vol 37 (1) ◽  
pp. 75-80 ◽  
Author(s):  
Philip Olin
Keyword(s):  
Finite Number ◽  
Direct Product ◽  
Direct Products ◽  
Direct Sums ◽  
First Order ◽  
Order Language ◽  
Universal Equivalence ◽  
Relational Systems ◽  
Multiple Operation ◽  
Definition Of

First order properties of direct products and direct sums (weak direct products) of relational systems have been studied extensively. For example, in Feferman and Vaught [3] an effective procedure is given for reducing such properties of the product to properties of the factors, and thus in particular elementary equivalence is preserved. We consider here two-sorted relational systems, with the direct product and sum operations keeping one of the sorts stationary. (See Feferman [4] for a similar definition of extensions.)These considerations are motivated by the example of direct products and sums of modules [8], [9]. In [9] examples are given to show that the direct product of two modules (even having only a finite number of module elements) does not preserve two-sorted (even universal) equivalence for any finite or infinitary language Lκ, λ. So we restrict attention here to direct powers and multiples (many copies of one structure). Also in [9] it is shown (for modules, but the proofs generalize immediately to two-sorted structures with a finite number of relations) that the direct multiple operation preserves first order ∀E-equivalence and the direct power operation preserves first order ∀-equivalence. We show here that these results for general two-sorted structures in a finite first order language are, in a sense, best-possible. Examples are given to show that does not imply , and that does not imply .

A model theoretic approach to Malcev conditions

1977 ◽  
Vol 42 (2) ◽  
pp. 277-288 ◽  
Author(s):  
John T. Baldwin ◽  
Joel Berman
Keyword(s):  
Extensive Study ◽  
Second Order ◽  
Theoretic Approach ◽  
Easy Consequence ◽  
First Order ◽  
Congruence Relations ◽  
Individual Variables ◽  
Definition Of ◽  
The Individual ◽  
Necessary And Sufficient

A varietyV(equational class of algebras) satisfies a strong Malcev condition ∃f1,…, ∃fnθ(f1, …,fn,x1, …,xm) where θ is a conjunction of equations in the function variablesf1, …,fnand the individual variablesx1, …,xm, if there are polynomial symbolsp1, …,pnin the language ofVsuch that ∀x1, …,xmθ(p1…,pn,x1, …,xm) is a law ofV. Thus a strong Malcev condition involves restricted second order quantification of a strange sort. The quantification is restricted to functions which are “polynomially definable”. This notion was introduced by Malcev [6] who used it to describe those varieties all of whose members have permutable congruence relations. The general formal definition of Malcev conditions is due to Grätzer [1]. Since then and especially since Jónsson's [3] characterization of varieties with distributive congruences there has been extensive study of strong Malcev conditions and the related concepts: Malcev conditions and weak Malcev conditions.In [9], Taylor gives necessary and sufficient semantic conditions for a class of varieties to be defined by a (strong) Malcev condition. A key to the proof is the translation of the restricted second order concepts into first order concepts in a certain many sorted language. In this paper we show that, given this translation, Taylor's theorem is an easy consequence of a result of Tarski [8] and the standard preservation theorems of first order logic.

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).

On the rules of proof in the pure functional calculus of the first order

1951 ◽  
Vol 16 (2) ◽  
pp. 107-111 ◽  
Author(s):  
Andrzej Mostowski
Keyword(s):  
Functional Calculus ◽  
Free Variable ◽  
Propositional Function ◽  
Cartesian Power ◽  
First Order ◽  
A Value ◽  
Free Variables ◽  
Individual Variables ◽  
Definition Of

We consider here the pure functional calculus of first order as formulated by Church.Church, l.c., p. 79, gives the definition of the validity of a formula in a given set I of individuals and shows that a formula is provable in if and only if it is valid in every non-empty set I. The definition of validity is preceded by the definition of a value of a formula; the notion of a value is the basic “semantical” notion in terms of which all other semantical notions are definable.The notion of a value of a formula retains its meaning also in the case when the set I is empty. We have only to remember that if I is empty, then an m-ary propositional function (i.e. a function from the m-th cartesian power Im to the set {f, t}) is the empty set. It then follows easily that the value of each well-formed formula with free individual variables is the empty set. The values of wffs without free variables are on the contrary either f or t. Indeed, if B has the unique free variable c and ϕ is the value of B, then the value of (c)B according to the definition given by Church is the propositional constant f or t according as ϕ(j) is f for at least one j in I or not. Since, however, there is no j in I, the condition ϕ(j) = t for all j in I is vacuously satisfied and hence the value of (c)B is t.

Some theorems on definability and decidability

1952 ◽  
Vol 17 (3) ◽  
pp. 179-187 ◽  
Author(s):  
Alonzo Church ◽  
W. V. Quine
Keyword(s):  
Number Theory ◽  
Functional Calculus ◽  
Natural Numbers ◽  
Quantification Theory ◽  
First Order ◽  
Dyadic Relation ◽  
Truth Value ◽  
Individual Variables

In this paper a theorem about numerical relations will be established and shown to have certain consequences concerning decidability in quantification theory, as well as concerning the foundation of number theory. The theorem is that relations of natural numbers are reducible in elementary fashion to symmetric ones; i.e.:Theorem I. For every dyadic relation R of natural numbers there is a symmetric dyadic relation H of natural numbers such that R is definable in terms of H plus just truth-functions and quantification over natural numbers.To state the matter more fully, there is a (well-formed) formula ϕ of pure quantification theory, or first-order functional calculus, which meets these conditions:(a) ϕ has ‘x’ and ‘y’ as sole free individual variables;(b) ϕ contains just one predicate letter, and it is dyadic;(c) for every dyadic relation R of natural numbers there is a symmetric dyadic relation H of natural numbers such that, when the predicate letter in ϕ is interpreted as expressing H, ϕ comes to agree in truth-value with ‘x bears R to y’ for all values of ‘x’ and ‘y’.

Some forms of completeness

1962 ◽  
Vol 27 (3) ◽  
pp. 344-352 ◽  
Author(s):  
P. C. Gilmore
Keyword(s):  
Predicate Symbol ◽  
Constant Term ◽  
Predicate Calculus ◽  
First Order ◽  
Individual Variables ◽  
The Individual ◽  
And Function ◽  
Order Predicate Calculus

By a theory is meant an applied first-order predicate calculus with at least one predicate symbol and perhaps some individual constants and function symbols and a specified set of axioms. In addition to the terms defined by means of the individual variables, constants, and function symbols a theory may also include among its terms those constructed by means of operators such as the epsilon or iota operators; that is, expressions like (εχΡ) or (οχΡ), where P is a well formed formula (wff) of the theory, may also be terms. A constant term of a theory F is then a term in which no variable occurs free. We are interested only in theories which have at least one constant term so that if a theory doesn't have any individual constants it must necessarily admit as terms expressions constructed by means of operators. A sentence of a theory F is a closed wff.

Development of a German-Language Questionnaire to Measure Collective Orientation as an Individual Attitude

2017 ◽  
Vol 76 (3) ◽  
pp. 91-105 ◽  
Author(s):  
Vera Hagemann
Keyword(s):  
Team Performance ◽  
Reliability And Validity ◽  
Future Research ◽  
German Language ◽  
Team Members ◽  
Collective Orientation ◽  
Team Research ◽  
German Speaking ◽  
Individual Variables ◽  
The Individual

Abstract. The individual attitudes of every single team member are important for team performance. Studies show that each team member’s collective orientation – that is, propensity to work in a collective manner in team settings – enhances the team’s interdependent teamwork. In the German-speaking countries, there was previously no instrument to measure collective orientation. So, I developed and validated a German-language instrument to measure collective orientation. In three studies (N = 1028), I tested the validity of the instrument in terms of its internal structure and relationships with other variables. The results confirm the reliability and validity of the instrument. The instrument also predicts team performance in terms of interdependent teamwork. I discuss differences in established individual variables in team research and the role of collective orientation in teams. In future research, the instrument can be applied to diagnose teamwork deficiencies and evaluate interventions for developing team members’ collective orientation.

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