The foundations of Suslin logic

1975 ◽  
Vol 40 (4) ◽  
pp. 567-575 ◽  
Author(s):  
Erik Ellentuck
Keyword(s):  
Finite Sequence ◽  
Order Logic ◽  
First Order Logic ◽  
Infinitary Logic ◽  
First Order ◽  
Finite Sequences ◽  
Image Position

Let L be a first order logic and the infinitary logic (as described in [K, p. 6] over L. Suslin logic is obtained from by adjoining new propositional operators and . Let f range over elements of ωω and n range over elements of ω. Seq is the set of all finite sequences of elements of ω. If θ: Seq → is a mapping into formulas of then and are formulas of LA. If is a structure in which we can interpret and h is an -assignment then we extend the notion of satisfaction from to by definingwhere f ∣ n is the finite sequence consisting of the first n values of f. We assume that has ω symbols for relations, functions, constants, and ω1 variables. θ is valid if θ ⊧ [h] for every h and is valid if -valid for every . We address ourselves to the problem of finding syntactical rules (or nearly so) which characterize validity .

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).

Strong extension axioms and Shelah's zero-one law for choiceless polynomial time

2003 ◽  
Vol 68 (1) ◽  
pp. 65-131 ◽  
Author(s):  
Andreas Blass ◽  
Yuri Gurevich
Keyword(s):  
Fixed Point ◽  
Polynomial Time ◽  
Order Logic ◽  
Complexity Class ◽  
First Order Logic ◽  
Infinitary Logic ◽  
Extensive Discussion ◽  
First Order ◽  
Detailed Proof ◽  
Strong Extension

AbstractThis paper developed from Shelah's proof of a zero-one law for the complexity class “choiceless polynomial time,” defined by Shelah and the authors. We present a detailed proof of Shelah's result for graphs, and describe the extent of its generalizability to other sorts of structures. The extension axioms, which form the basis for earlier zero-one laws (for first-order logic, fixed-point logic, and finite-variable infinitary logic) are inadequate in the case of choiceless polynomial time; they must be replaced by what we call the strong extension axioms. We present an extensive discussion of these axioms and their role both in the zero-one law and in general.

Infinitary logics

2018 ◽  
Author(s):  
Bernd Buldt
Keyword(s):  
Mathematical Logic ◽  
Order Logic ◽  
Logical Analysis ◽  
First Order Logic ◽  
Infinitary Logic ◽  
First Order ◽  
Essential Tool

An infinitary logic arises from ordinary first-order logic when one or more of its finitary properties is allowed to become infinite, for example, by admitting infinitely long formulas or infinitely long or infinitely branched proof figures. The need to extend first-order logic became pressing in the late 1950s when it was realized that many of the fundamental notions of mathematics cannot be expressed in first-order logic in a way that would allow for their logical analysis. Because infinitary logics often do not suffer the same limitation, they have become an essential tool in mathematical logic.

A probabilistic interpolation theorem

1985 ◽  
Vol 50 (3) ◽  
pp. 708-713 ◽  
Author(s):  
Douglas N. Hoover
Keyword(s):  
Interpolation Theorem ◽  
Order Logic ◽  
First Order Logic ◽  
Probability Logic ◽  
Admissible Set ◽  
First Order ◽  
Probability Spaces ◽  
Image Position ◽  
Dedekind Cuts ◽  
Exact Definition

The probability logic is a logic with a natural interpretation on probability spaces (thus, a logic whose model theory is part of probability theory rather than a system for putting probabilities on formulas of first order logic). Its exact definition and basic development are contained in the paper [3] of H. J. Keisler and the papers [1] and [2] of the author. Building on work in [2], we prove in this paper the following probabilistic interpolation theorem for .Let L be a countable relational language, and let A be a countable admissible set with ω ∈ A (in this paper some probabilistic notation will be used, but ω will always mean the least infinite ordinal). is the admissible fragment of corresponding to A. We will assume that L is a countable set in A, as is usual in practice, though all that is in fact needed for our proof is that L be a set in A which is wellordered in A.Theorem. Let ϕ(x) and ψ(x) be formulas of LAP such thatwhere ε ∈ [0, 1) is a real in A (reals may be defined in the usual way as Dedekind cuts in the rationals). Then for any real d > ε¼, there is a formula θ(x) of (L(ϕ) ∩ L(ψ))AP such thatand

scholarly journals THE FLUTED FRAGMENT REVISITED

2019 ◽  
Vol 84 (3) ◽  
pp. 1020-1048
Author(s):  
IAN PRATT-HARTMANN ◽  
WIESŁAW SZWAST ◽  
LIDIA TENDERA
Keyword(s):  
Order Logic ◽  
First Order Logic ◽  
Satisfiability Problem ◽  
Complexity Bounds ◽  
First Order ◽  
Complexity Results ◽  
Decidable Fragment ◽  
Image Position ◽  
Exponential Size

AbstractWe study the fluted fragment, a decidable fragment of first-order logic with an unbounded number of variables, motivated by the work of W. V. Quine. We show that the satisfiability problem for this fragment has nonelementary complexity, thus refuting an earlier published claim by W. C. Purdy that it is in NExpTime. More precisely, we consider ${\cal F}{{\cal L}^m}$, the intersection of the fluted fragment and the m-variable fragment of first-order logic, for all $m \ge 1$. We show that, for $m \ge 2$, this subfragment forces $\left\lfloor {m/2} \right\rfloor$-tuply exponentially large models, and that its satisfiability problem is $\left\lfloor {m/2} \right\rfloor$-NExpTime-hard. We further establish that, for $m \ge 3$, any satisfiable ${\cal F}{{\cal L}^m}$-formula has a model of at most ($m - 2$)-tuply exponential size, whence the satisfiability (= finite satisfiability) problem for this fragment is in ($m - 2$)-NExpTime. Together with other, known, complexity results, this provides tight complexity bounds for ${\cal F}{{\cal L}^m}$ for all $m \le 4$.

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

On the semantics of the Henkin quantifier

1979 ◽  
Vol 44 (2) ◽  
pp. 184-200 ◽  
Author(s):  
Michał Krynicki ◽  
Alistair H. Lachlan
Keyword(s):  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Original Definition ◽  
Definition Of ◽  
Image Position

In [5] Henkin defined a quantifier, which we shall denote by QH: linking four variables in one formula. This quantifier is related to the notion of formulas in which the usual universal and existential quantifiers occur but are not linearly ordered. The original definition of QH wasHere (QHx1x2y1y2)φ is true if for every x1 there exists y1 such that for every x2 there exists y2, whose choice depends only on x2 not on x1 and y1 such that φ(x14, x2, y1, y2). Another way of writing this isIn [5] it was observed that the logic L(QH) obtained by adjoining QH defined as in (1) is more powerful than first-order logic. In particular, it turned out that the quantifier “there exist infinitely many” denoted Q0 was definable from QH because

scholarly journals A Definability Theorem for First Order Logic

1997 ◽  
Vol 4 (3) ◽  
Author(s):  
Carsten Butz ◽  
Ieke Moerdijk
Keyword(s):  
Boolean Algebra ◽  
Topological Space ◽  
Model Theory ◽  
Order Logic ◽  
First Order Logic ◽  
Infinitary Logic ◽  
First Order ◽  
Definable Subset ◽  
Language L ◽  
The One

In this paper, we will present a definability theorem for first order logic.<br />This theorem is very easy to state, and its proof only uses elementary tools. To explain the theorem, let us first observe that if M is a model of a theory T in a language L, then, clearly, any definable subset S M (i.e., a subset S = {a | M |= phi(a)} defined by some formula phi) is invariant under all<br />automorphisms of M. The same is of course true for subsets of M" defined<br />by formulas with n free variables.<br /> Our theorem states that, if one allows Boolean valued models, the converse holds. More precisely, for any theory T we will construct a Boolean valued model M, in which precisely the T-provable formulas hold, and in which every (Boolean valued) subset which is invariant under all automorphisms of M is definable by a formula of L.<br />Our presentation is entirely selfcontained, and only requires familiarity<br />with the most elementary properties of model theory. In particular, we have added a first section in which we review the basic definitions concerning<br />Boolean valued models.<br />The Boolean algebra used in the construction of the model will be presented concretely as the algebra of closed and open subsets of a topological space X naturally associated with the theory T. The construction of this space is closely related to the one in [1]. In fact, one of the results in that paper could be interpreted as a definability theorem for infinitary logic, using topological rather than Boolean valued models.

scholarly journals The optimality of induction as an axiomatization of arithmetic

1983 ◽  
Vol 48 (1) ◽  
pp. 182-184 ◽  
Author(s):  
Daniel Leivant
Keyword(s):  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Primitive Recursive Arithmetic ◽  
Image Position ◽  
Primitive Recursive ◽  
Universal Closure

By induction for a formula φ we mean the schema(where the terms in brackets are implicitly substituted for some fixed variable, with the usual restrictions). Let be the schema IAφ for φ in Πn (i.e. ); similarly for . Each instance of is Δn+2, and each instance of is Σn+1 Thus the universal closure of an instance α is Πn+2 in either case. Charles Parsons [72] proved that and are equivalent over Z0, where Z0 is essentially Primitive Recursive Arithmetic augmented by classical First Order Logic [Parsons 70].Theorem. For each n > 0 there is a Πn formula π for whichis not derivable in Z0from(i) true Πn+1sentences; nor even(ii) Πn+1sentences consistent withZ0.

Independent Sets of Axioms In Lκα

1981 ◽  
Vol 24 (2) ◽  
pp. 219-223 ◽  
Author(s):  
Xavier Caicedo
Keyword(s):  
Independent Sets ◽  
Order Logic ◽  
First Order Logic ◽  
Countable Subset ◽  
Infinitary Logic ◽  
First Order

A set of sentences T is called independent if for every . It is countably independent if every countable subset is independent. In flnitary first order logic, Lωω, the two notions coincide because of compactness. This is not the case for infinitary logic.

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