Undecidability in diagonalizable algebras

1997 ◽  
Vol 62 (1) ◽  
pp. 79-116 ◽  
Author(s):  
V. Yu. Shavrukov
Keyword(s):  
Wide Class ◽  
Formal Theory ◽  
First Order ◽  
Unary Operator ◽  
Diagonalizable Algebra

AbstractIf a formal theory T is able to reason about its own syntax, then the diagonalizable algebra of T is defined as its Lindenbaum sentence algebra endowed with a unary operator □ which sends a sentence φ to the sentence □φ asserting the provability of φ in T. We prove that the first order theories of diagonalizable algebras of a wide class of theories are undecidable and establish some related results.

A formal theory of objects, space and time

1990 ◽  
Vol 55 (1) ◽  
pp. 74-89 ◽  
Author(s):  
Wayne D. Blizard
Keyword(s):  
Formal Theory ◽  
Order Theory ◽  
First Order ◽  
Space And Time ◽  
First Order Theory ◽  
Logical Nature

The two statements “Two different objects cannot occupy the same place at the same time” and “An object cannot be in two different places at the same time” are axioms of our everyday understanding of objects, space and time. We develop a first-order theory OST (Objects, Space and Time) in which formal equivalents of these two statements are taken as axioms. Using the theory OST, we uncover other fundamental principles of objects, space and time. We attempt to understand the logical nature of these principles, to investigate their formal consequences, and to identify logical alternatives to them. For easy reference, all of the nonlogical axioms of OST are listed together at the end of §2. In §3, we introduce two possible extensions of OST.

On the variance of the sample correlation between two independent lattice processes

1981 ◽  
Vol 18 (04) ◽  
pp. 943-948 ◽  
Author(s):  
Sylvia Richardson ◽  
Denis Hemon
Keyword(s):  
Wide Class ◽  
Asymptotic Expression ◽  
Asymptotic Variance ◽  
First Order ◽  
Sample Correlation ◽  
Lattice Processes ◽  
Independent Lattice

Consider two stochastically independent, stationary Gaussian lattice processes with zero means, {X(u), u (Z 2} and {Y(u), u (Z 2}. An asymptotic expression for the variance of the sample correlation between {X(u)} and {Y(u)} over a finite square is derived. This expression also holds for a wide class of domains in Z 2. As an illustration, the asymptotic variance of the correlation between two first-order autonormal schemes is evaluated.

On the notion of algebraic closedness for noncommutative groups and fields

1971 ◽  
Vol 36 (3) ◽  
pp. 441-444 ◽  
Author(s):  
Abraham Robinson
Keyword(s):  
Wide Class ◽  
First Order ◽  
Division Rings ◽  
Ordered Field ◽  
Ordered Fields ◽  
Skew Fields

The notion of algebraic closedness plays an important part in the theory of commutative fields. The corresponding notion in the theory of ordered fields is (not only intuitively but in a sense which can be made precise in a metamathematical framework, compare [4]) that of a real closed ordered field. Several suggestions have been made (see [2] and [8]) for the formulation of corresponding concepts in the theory of groups and in the theory of skew fields (division rings, noncommutative fields). Here we present a concept of this kind, which preserves the principal metamathematical properties of algebraically closed commutative fields and which applies to a wide class of first order theories K, including the theories of commutative and of skew fields and the theories of commutative and of general groups.

The degree of the set of sentences of predicate provability logic that are true under every interpretation

1987 ◽  
Vol 52 (1) ◽  
pp. 165-171 ◽  
Author(s):  
George Boolos ◽  
Vann McGee
Keyword(s):  
Predicate Symbol ◽  
Order Logic ◽  
First Order Logic ◽  
Provability Logic ◽  
First Order ◽  
Free Variables ◽  
Unary Operator

The formalism of P(redicate) P(rovability) L(ogic) is the result of adjoining the unary operator □ to first-order logic without identity, constants, or function symbols. The term “provability” indicates that □ is to be “read” as “it is provable in P(eano) A(rithmetic) that…” and that the formulae of predicate provability logic are to be interpreted via formulae of PA as follows.Pr(x), alias Bew(x), is the standard provability predicate of PA. For any formula F of PA, Pr[F] is the formula of PA that expresses the PA-provability of F “of” the values of the variables free in F, i.e., it is the formula of PA with the same free variables as F that expresses the PA-provability of the result of substituting for each variable free in F the numeral for the value of that variable. For the details of the construction of Pr[F], the reader may consult [B2, p. 42]. If F is a sentence of PA, then Pr[F] = Pr(‘F’), the sentence that expresses the PA-provability of F.Let υ1, υ2,… be an enumeration of the variables of PA. An interpretation * of a formula ϕ of PPL is a function which assigns to each predicate symbol P of ϕ a formula P* of the language of arithmetic whose free variables are the first n variables of PA, where n is the degree of P.

scholarly journals Much Ado About the Many

2021 ◽  
Vol 25 (1) ◽  
Author(s):  
Jonathan Mai
Keyword(s):  
Set Theory ◽  
Formal Theory ◽  
Order Logic ◽  
First Order Logic ◽  
Plural Quantification ◽  
First Order ◽  
Order Language ◽  
The Many ◽  
Second Order Logic ◽  
Alternative Theories

English distinguishes between singular quantifiers like "a donkey" and plural quantifiers like "some donkeys". Pluralists hold that plural quantifiers range in an unusual, irreducibly plural, way over common objects, namely individuals from first-order domains and not over set-like objects. The favoured framework of pluralism is plural first-order logic, PFO, an interpreted first-order language that is capable of expressing plural quantification. Pluralists argue for their position by claiming that the standard formal theory based on PFO is both ontologically neutral and really logic. These properties are supposed to yield many important applications concerning second-order logic and set theory that alternative theories supposedly cannot deliver. I will show that there are serious reasons for rejecting at least the claim of ontological innocence. Doubt about innocence arises on account of the fact that, when properly spelled out, the PFO-semantics for plural quantifiers is committed to set-like objects. The correctness of my worries presupposes the principle that for every plurality there is a coextensive set. Pluralists might reply that this principle leads straight to paradox. However, as I will argue, the true culprit of the paradox is the assumption that every definite condition determines a plurality.

scholarly journals DIFFERENTIATION IN P-MINIMAL STRUCTURES AND A p-ADIC LOCAL MONOTONICITY THEOREM

2014 ◽  
Vol 79 (4) ◽  
pp. 1133-1147 ◽  
Author(s):  
TRISTAN KUIJPERS ◽  
EVA LEENKNEGT
Keyword(s):  
Wide Class ◽  
Local Version ◽  
First Order ◽  
Almost Everywhere ◽  
Local Monotonicity ◽  
Definable Functions

AbstractWe prove a p-adic, local version of the Monotonicity Theorem for P-minimal structures. The existence of such a theorem was originally conjectured by Haskell and Macpherson. We approach the problem by considering the first order strict derivative. In particular, we show that, for a wide class of P-minimal structures, the definable functions f : K → K are almost everywhere strictly differentiable and satisfy the Local Jacobian Property.

scholarly journals Informal Provability, First-Order BAT Logic and First Steps Towards a Formal Theory of Informal Provability

2021 ◽  
pp. 1-27
Author(s):  
Pawel Pawlowski ◽  
Rafal Urbaniak
Keyword(s):  
Formal Theory ◽  
Order System ◽  
First Order ◽  
Do So

BAT is a logic built to capture the inferential behavior of informal provability. Ultimately, the logic is meant to be used in an arithmetical setting. To reach this stage it has to be extended to a first-order version. In this paper we provide such an extension. We do so by constructing non-deterministic three-valued models that interpret quantifiers as some sorts of infinite disjunctions and conjunctions. We also elaborate on the semantical properties of the first-order system and consider a couple of its strengthenings. It turns out that obtaining a sensible strengthening is not straightforward. We prove that most strategies commonly used for strengthening non-deterministic logics fail in our case. Nevertheless, we identify one method of extending the system which does not.

On the variance of the sample correlation between two independent lattice processes

1981 ◽  
Vol 18 (4) ◽  
pp. 943-948 ◽  
Author(s):  
Sylvia Richardson ◽  
Denis Hemon
Keyword(s):  
Wide Class ◽  
Asymptotic Expression ◽  
Asymptotic Variance ◽  
First Order ◽  
Sample Correlation ◽  
Lattice Processes ◽  
Independent Lattice

Consider two stochastically independent, stationary Gaussian lattice processes with zero means, {X(u), u (Z2} and {Y(u), u (Z2}. An asymptotic expression for the variance of the sample correlation between {X(u)} and {Y(u)} over a finite square is derived. This expression also holds for a wide class of domains in Z2. As an illustration, the asymptotic variance of the correlation between two first-order autonormal schemes is evaluated.

What is an Artefact Design?

2009 ◽  
Vol 13 (2) ◽  
pp. 137-149
Author(s):  
Pawel Garbacz ◽  
Keyword(s):  
Formal Theory ◽  
States Of Affairs ◽  
Intentional States ◽  
First Order ◽  
Design Specifications ◽  
Criterion Of Identity ◽  
Ontological Framework

The paper contains a first order formal theory pertaining to artefact designs, designs which are construed as the results of designing activities. The theory is based on a minimal ontology of states of affairs and it is inspired by the ideas of the Polish philosopher Roman Ingarden. After differentiating the philosophical notion of design from the engineering notion of design specifications, I then go on to argue that the philosophical category of artefact designs may be compared with Ingarden’s category of intentional states of affairs. At least some artefacts are found to be determined by more than one design. I also show how this ontological framework allows for the distinction between artefact tokens and artefact types. That leads to a proposal on how to define a criterion of identity for artefact types. The proposed theory serves as a basis both for a better understanding of what artefacts are and for the construction of computer-readable models of design specifications.

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