Remarks on identity and description in first-order axiom systems

1954 ◽  
Vol 19 (1) ◽  
pp. 14-20 ◽  
Author(s):  
Theodore Hailperin
Keyword(s):  
Subject Matter ◽  
Axiom System ◽  
Order System ◽  
Quantification Theory ◽  
Specific Subject ◽  
First Order ◽  
Finite Set ◽  
The One ◽  
Image Position ◽  
Axiom Systems

Hilbert and Ackermann ([1], p. 107) define a first order axiom system as one in which the axioms contain one or more predicate constants, but no predicate variables. Here “axiom” refers to the specific subject-matter axioms and not to the rules of the restricted predicate calculus (quantification theory), which rules are presupposed for each first-order system. It is pointed out by them that an exception could be made for the predicate of identity; for the axiom scheme for this predicate, namelywhich has in (b) the variable predicate F, could nevertheless be replaced, in any given first-order system, by a finite set of axioms without predicate variables. Thus, for example, if Φ[x, y) is the one constant predicate of such a system then PId(b) could be replaced byThus one postulates, in addition to the reflexivity, symmetry, and transitivity of identity, the substitutivity of identical entities in each of the possible “atomic” contexts of a variable (occurrences in the primitive predicates). In this method of introducing identity it has to be taken as an additional primitive predicate and further axioms are consequently needed. In such a system having PId(a), (b1)−(b3) as axioms, the scheme PId(b) can be derived as a meta-theorem of the system, F(x) then being any formula of the system.

scholarly journals THE VARIETY OF COSET RELATION ALGEBRAS

2018 ◽  
Vol 83 (04) ◽  
pp. 1595-1609 ◽  
Author(s):  
STEVEN GIVANT ◽  
HAJNAL ANDRÉKA
Keyword(s):  
Large Class ◽  
Identity Element ◽  
Relation Algebra ◽  
Order Theory ◽  
Relation Algebras ◽  
First Order ◽  
First Order Theory ◽  
Atomic Pair ◽  
Finite Set ◽  
Image Position

AbstractGivant [6] generalized the notion of an atomic pair-dense relation algebra from Maddux [13] by defining the notion of a measurable relation algebra, that is to say, a relation algebra in which the identity element is a sum of atoms that can be measured in the sense that the “size” of each such atom can be defined in an intuitive and reasonable way (within the framework of the first-order theory of relation algebras). In Andréka--Givant [2], a large class of examples of such algebras is constructed from systems of groups, coordinated systems of isomorphisms between quotients of the groups, and systems of cosets that are used to “shift” the operation of relative multiplication. In Givant--Andréka [8], it is shown that the class of these full coset relation algebras is adequate to the task of describing all measurable relation algebras in the sense that every atomic and complete measurable relation algebra is isomorphic to a full coset relation algebra.Call an algebra $\mathfrak{A}$ a coset relation algebra if $\mathfrak{A}$ is embeddable into some full coset relation algebra. In the present article, it is shown that the class of coset relation algebras is equationally axiomatizable (that is to say, it is a variety), but that no finite set of sentences suffices to axiomatize the class (that is to say, the class is not finitely axiomatizable).

Existence Theorems for Nonlinear Boundary Value Problems

1974 ◽  
Vol 26 (4) ◽  
pp. 884-892 ◽  
Author(s):  
W. L. McCandless
Keyword(s):  
Nonlinear Boundary ◽  
Existence Theorems ◽  
Boundary Value ◽  
Continuous Functions ◽  
Order System ◽  
Nonlinear Boundary Value Problems ◽  
First Order ◽  
Differentiable Functions ◽  
Image Position ◽  
General Boundary Value Problem

Let C(I) denote the linear space of continuous functions from the compact interval I = [a, b] into n-dimensional real arithmetic space Rn, and let C′(I) be the subspace of continuously differentiable functions on I. A general boundary value problem for a first-order system of n ordinary differential equations on I is given by

scholarly journals Predicate functors revisited

1981 ◽  
Vol 46 (3) ◽  
pp. 649-652 ◽  
Author(s):  
W. V. Quine
Keyword(s):  
Set Theory ◽  
Predicate Logic ◽  
Combinatory Logic ◽  
Homogeneous Case ◽  
Quantification Theory ◽  
Present Scheme ◽  
First Order ◽  
Image Position ◽  
First Order Predicate Logic

Quantification theory, or first-order predicate logic, can be formulated in terms purely of predicate letters and a few predicate functors which attach to predicates to form further predicates. Apart from the predicate letters, which are schematic, there are no variables. On this score the plan is reminiscent of the combinatory logic of Schönfinkel and Curry. Theirs, however, had the whole of higher set theory as its domain; the present scheme stays within the bounds of predicate logic.In 1960 I published an apparatus to this effect, and an improved version in 1971. In both versions I assumed two inversion functors, major and minor; also a cropping functor and the obvious complement functor. The effects of these functors, when applied to an n-place predicate, are as follows:The variables here are explanatory only and no part of the final notation. Ultimately the predicate letters need exponents showing the number of places, but I omit them in these pages.A further functor-to continue now with the 1971 version-was padding:Finally there was a zero-place predicate functor, which is to say simply a constant predicate, namely the predicate ‘I’ of identity, and there was a two-place predicate functor ‘∩’ of intersection. The intersection ‘F ∩ G’ received a generalized interpretation, allowing ‘F’ and ‘G’ to be predicates with unlike numbers of places. However, Steven T. Kuhn has lately shown me that the generalization is unnecessary and reducible to the homogeneous case.

On the first-order logic of terms

1973 ◽  
Vol 38 (2) ◽  
pp. 177-188
Author(s):  
Lars Svenonius
Keyword(s):  
Order Logic ◽  
First Order Logic ◽  
Identity Function ◽  
First Order ◽  
Elementary Condition ◽  
Combinatorial Function ◽  
The One ◽  
Definition Of ◽  
Image Position

By an elementary condition in the variablesx1, …, xn, we mean a conjunction of the form x1 ≤ i < j ≤ naij where each aij is one of the formulas xi = xj or xi ≠ xj. (We should add that the formula x1 = x1 should be regarded as an elementary condition in the one variable x1.)Clearly, according to this definition, some elementary conditions are inconsistent, some are consistent. For instance (in the variables x1, x2, x3) the conjunction x1 = x2 & x1 = x3 & x2 ≠ x3 is inconsistent.By an elementary combinatorial function (ex. function) we mean any function which can be given a definition of the formwhere E1(x1, …, xn), …, Ek(x1, …, xn) is an enumeration of all consistent elementary conditions in x1, …, xn, and all the numbers d1, …, dk are among 1, …, n.Examples. (1) The identity function is the only 1-ary e.c. function.(2) A useful 3-ary e.c. function will be called J. The definition is

Oscillation Criteria for Matrix Differential Equations

1967 ◽  
Vol 19 ◽  
pp. 184-199 ◽  
Author(s):  
H. C. Howard
Keyword(s):  
Existence Theorems ◽  
Symmetric Matrix ◽  
Order System ◽  
Matrix Differential Equation ◽  
First Order ◽  
The Matrix ◽  
Matrix Differential Equations ◽  
Matrix Differential ◽  
Image Position

We shall be concerned at first with some properties of the solutions of the matrix differential equation1.1whereis an n × n symmetric matrix whose elements are continuous real-valued functions for 0 < x < ∞, and Y(x) = (yij(x)), Y″(x) = (y″ ij(x)) are n × n matrices. It is clear such equations possess solutions for 0 < x < ∞, since one can reduce them to a first-order system and then apply known existence theorems (6, Chapter 1).

scholarly journals THE NEAT EMBEDDING PROBLEM FOR ALGEBRAS OTHER THAN CYLINDRIC ALGEBRAS AND FOR INFINITE DIMENSIONS

2014 ◽  
Vol 79 (01) ◽  
pp. 208-222 ◽  
Author(s):  
ROBIN HIRSCH ◽  
TAREK SAYED AHMED
Keyword(s):  
Embedding Problem ◽  
Infinite Dimensions ◽  
Cylindric Algebras ◽  
First Order ◽  
Polyadic Algebras ◽  
Finite Set ◽  
Image Position ◽  
Substitution Algebras

Abstract Hirsch and Hodkinson proved, for $3 \le m &lt; \omega $ and any $k &lt; \omega $ , that the class $SNr_m {\bf{CA}}_{m + k + 1} $ is strictly contained in $SNr_m {\bf{CA}}_{m + k} $ and if $k \ge 1$ then the former class cannot be defined by any finite set of first-order formulas, within the latter class. We generalize this result to the following algebras of m-ary relations for which the neat reduct operator $_m $ is meaningful: polyadic algebras with or without equality and substitution algebras. We also generalize this result to allow the case where m is an infinite ordinal, using quasipolyadic algebras in place of polyadic algebras (with or without equality).

scholarly journals Gradient Structures Associated with a Polynomial Differential Equation

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 535 ◽  
Author(s):  
Savin Treanţă
Keyword(s):  
Differential Equation ◽  
Order System ◽  
Mathematical Framework ◽  
Special Significance ◽  
First Order ◽  
System Method ◽  
Polynomial Differential Equation ◽  
The Cauchy Problem ◽  
Finite Set ◽  
Kernel Representation

In this paper, by using the characteristic system method, the kernel of a polynomial differential equation involving a derivation in R n is described by solving the Cauchy Problem for the corresponding first order system of PDEs. Moreover, the kernel representation has a special significance on the space of solutions to the corresponding system of PDEs. As very important applications, it has been established that the mathematical framework developed in this work can be used for the study of some second-order PDEs involving a finite set of derivations.

A system of implicit quantification

1968 ◽  
Vol 32 (4) ◽  
pp. 480-504 ◽  
Author(s):  
J. Jay Zeman
Keyword(s):  
Subject Matter ◽  
Traditional Method ◽  
Natural Deduction ◽  
Propositional Calculus ◽  
Axiom System ◽  
Symbolic Logic ◽  
Quantification Theory ◽  
The Subject

The “traditional” method of presenting the subject-matter of symbolic logic involves setting down, first of all, a basis for a propositional calculus—which basis might be a system of natural deduction, an axiom system, or a rule concerning tautologous formulas. The next step, ordinarily, consists of the introduction of quantifiers into the symbol-set of the system, and the stating of axioms or rules for quantification. In this paper I shall propose a system somewhat different from the ordinary; this system has rules for quantification and is, indeed, equivalent to classical quantification theory. It departs from the usual, however, in that it has no primitive quantifiers.

Axiom systems for first order logic with finitely many variables

1973 ◽  
Vol 38 (4) ◽  
pp. 576-578 ◽  
Author(s):  
James S. Johnson
Keyword(s):  
Finite Size ◽  
Order Logic ◽  
First Order Logic ◽  
Finite Sets ◽  
First Order ◽  
Finite Set ◽  
Axiom Systems

AbstractJ. D. Monk has shown that for first order languages with finitely many variables there is no finite set of schema which axiomatizes the universally valid formulas. There are such finite sets of schema which axiomatize the formulas valid in all structures of some fixed finite size.

On the derivative chain-rules in fractional calculus via fractional difference and their application to systems modelling

Open Physics ◽  
2013 ◽  
Vol 11 (6) ◽  
Author(s):  
Guy Jumarie
Keyword(s):  
Fractional Calculus ◽  
Short Note ◽  
Order System ◽  
Fractional Difference ◽  
First Order ◽  
Fractional Models ◽  
Derivative Chain ◽  
Systems Modelling ◽  
Fractional Differential ◽  
The One

AbstractIt has been pointed out that the derivative chains rules in fractional differential calculus via fractional calculus are not quite satisfactory as far as they can yield different results which depend upon how the formula is applied, that is to say depending upon where is the considered function and where is the function of function. The purpose of the present short note is to display some comments (which might be clarifying to some readers) on the matter. This feature is basically related to the non-commutativity of fractional derivative on the one hand, and furthermore, it is very close to the physical significance of the systems under consideration on the other hand, in such a manner that everything is right so. As an example, it is shown that the trivial first order system may have several fractional modelling depending upon the way by which it is observed. This suggests some rules to construct the fractional models of standard dynamical systems, in as meaningful a model as possible. It might happen that this pitfall comes from the feature that a function which is continuous everywhere, but is nowhere differentiable, exhibits random-like features.

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