A generalization of the concept of ω-consistency

1954 ◽  
Vol 19 (3) ◽  
pp. 183-196 ◽  
Author(s):  
Leon Henkin
Keyword(s):  
Functional Calculus ◽  
Free Variable ◽  
Natural Numbers ◽  
Deductive Systems ◽  
Individual Variable ◽  
Special Cases ◽  
Definition Of ◽  
The Individual ◽  
Formal Properties ◽  
Use Of Models

In this paper we consider certain formal properties of deductive systems which, in special cases, reduce to the property of ω-consistency; and we then seek to understand the significance of these properties by relating them to the use of models in providing interpretations of the deductive systems.The notion of ω-consistency arises in connection with deductive systems of arithmetic. For definiteness, let us suppose that the system is a functional calculus whose domain of individuals is construed as the set of natural numbers, and that the system possesses individual constants ν0, ν1, ν2, … such that νi functions as a name for the number i. Such a system is called ω-consistent, if there is no well-formed formula A(x) (in which x is the only free variable) such that A(ν0), A(ν1), A(ν2), … and ∼(x)A(x) are all formal theorems of the system, where A(νi) is the formula resulting from A(x) by substituting the constant νi for each free occurrence of the individual variable x.Now consider an arbitrary applied functional calculus F, and let Γ be any non-empty set of its individual constants. In imitation of the definition of ω-consistency, we may say that the system F is Γ-consistent, if it contains no formula A(x) (in which x is the only free variable) such that ⊦ A (α) for every constant α in Γ, and also ⊦ ∼(x)A(x) (where an occurrence of “⊦” indicates that the formula which it precedes is a formal theorem). We easily see that the condition of Γ-consistency is equivalent to the condition that the system F contain no formula B(x) such that ⊦ ∼ B(α) for each α in Γ, and also ⊦ (∃x)B(x).

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.

scholarly journals Free and Forced Vibrations of Elastically Connected Structures

2010 ◽  
Vol 2010 ◽  
pp. 1-11 ◽  
Author(s):  
S. Graham Kelly
Keyword(s):  
Natural Frequencies ◽  
Iterative Solution ◽  
Forced Response ◽  
Inner Product ◽  
Mode Shapes ◽  
Square Roots ◽  
Parallel Structures ◽  
Special Cases ◽  
Definition Of ◽  
The Individual

A general theory for the free and forced responses of elastically connected parallel structures is developed. It is shown that if the stiffness operator for an individual structure is self-adjoint with respect to an inner product defined for , then the stiffness operator for the set of elastically connected structures is self-adjoint with respect to an inner product defined on . This leads to the definition of energy inner products defined on . When a normal mode solution is used to develop the free response, it is shown that the natural frequencies are the square roots of the eigenvalues of an operator that is self-adjoint with respect to the energy inner product. The completeness of the eigenvectors in is used to develop a forced response. Special cases are considered. When the individual stiffness operators are proportional, the problem for the natural frequencies and mode shapes reduces to a matrix eigenvalue problem, and it is shown that for each spatial mode there is a set of intramodal mode shapes. When the structures are identical, uniform, or nonuniform, the differential equations are uncoupled through diagonalization of a coupling stiffness matrix. The most general case requires an iterative solution.

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.

The Autobiographical Subject and the Death Penalty

2013 ◽  
Vol 35 (2) ◽  
pp. 165-187
Author(s):  
E. S. Burt
Keyword(s):  
Death Penalty ◽  
First Person ◽  
Philosophical Investigation ◽  
The Subject ◽  
Definition Of ◽  
The Individual

Why does writing of the death penalty demand the first-person treatment that it also excludes? The article investigates the role played by the autobiographical subject in Derrida's The Death Penalty, Volume I, where the confessing ‘I’ doubly supplements the philosophical investigation into what Derrida sees as a trend toward the worldwide abolition of the death penalty: first, to bring out the harmonies or discrepancies between the individual subject's beliefs, anxieties, desires and interests with respect to the death penalty and the state's exercise of its sovereignty in applying it; and second, to provide a new definition of the subject as haunted, as one that has been, but is no longer, subject to the death penalty, in the light of the worldwide abolition currently underway.

ГЕРМЕНЕВТИКА ЛИЧНОСТИ КАК МЕТОД СОВРЕМЕННОЙ ПЕРСОНОЛОГИИ

1970 ◽  
Vol 6 (1) ◽  
pp. 32-42
Author(s):  
Елена Старовойтенко
Keyword(s):  
Text Setting ◽  
Research Setting ◽  
Textual Representations ◽  
General Scientific ◽  
Definition Of ◽  
The Individual ◽  
The Continuum

Персонологическая интерпретация текстов предполагает реализацию общенаучных, а также специфических для персонологии, герменевтических установок, к которым относятся: установка на интерпретацию текста как исследование, установка на разнообразие герменевтических действий с текстом, установка на выявление неисследованных содержаний текста, установка на творческое постижение тайн текста, установка на целостное отношение к личности и "Я" автора текста, установка на выявление способности автора быть "практикующим феноменологом", установка на определение места изучаемого текста в континууме текстовых репрезентаций "личности", установка на соотнесение своего понимания текста с другими интерпретациями и их интеграцию, установка на раскрытие сущности авторской "идеи личности", возможное только в единстве интерпретаций, установка на построение и применение герменевтической модели, определяющей процедуру интерпретации как исследования и творчества, установка на определение места проделанного герменевтического поиска в культуре познания и жизни личности, установка на интерпретацию различных видов "текстов личности". Personological interpretation of texts suggests the implementation of the general scientific and also hermeneutical settings specific for Personology which include the setting of the interpretation of the text as a research, setting of a variety of hermeneutical actions with the text, setting to identify unexplored contents of the text, setting of the creative comprehension of the mysteries of the text, setting of the integrity of the attitude of the individual and the "I" of the author of the text, setting to reveal the author's ability to be "practicing phenomenologist", setting of the definition of the place in the text in the continuum of textual representations of the "personality", setting in the correlation of the understanding of the text with other interpretations and their integration, setting of the disclosure of the author's "ideas person" is possible only in the unity of interpretation, setting of the construction and usage of hermeneutical models defining the procedure for the interpretation of both studies and work, the setting to determine the place of hermeneutical research in culture and knowledge of a person's life, setting of the interpretation of various types of "texts of the individual."

¿Quién soy yo? Reflexiones teórico sistémicas acerca de la indentidad individual

2017 ◽  
Vol 1 (1) ◽  
pp. 15-31
Author(s):  
Francisco Xavier Morales
Keyword(s):  
Identity Construction ◽  
Social Systems ◽  
Social Contexts ◽  
Individual Identity ◽  
Contemporary Society ◽  
Self Identity ◽  
Moral Communication ◽  
Identity Based ◽  
Definition Of ◽  
The Individual

The problem of identity is an issue of contemporary society that is not only expressed in daily life concerns but also in discourses of politics and social movements. Nevertheless, the I and the needs of self-fulfillment usually are taken for granted. This paper offers thoughts regarding individual identity based on Niklas Luhmann’s systems theory. From this perspective, identity is not observed as a thing or as a subject, but rather as a “selfillusion” of a system of consciousness, which differentiates itself from the world, event after event, in a contingent way. As concerns the definition  of contents of self-identity, the structures of social systems define who is a person, how he or she should act, and how much esteem he or she should receive. These structures are adopted by consciousness as its own identity structures; however, some social contexts are more relevant for self-identity construction than others. Moral communication increases the probability that structure appropriation takes place, since the emotional element of identity is linked to the esteem/misesteem received by the individual from the interactions in which he or she participates.

Optimal Consumption and Insurance: A Continuous-time Markov Chain Approach

2008 ◽  
Vol 38 (01) ◽  
pp. 231-257 ◽  
Author(s):  
Holger Kraft ◽  
Mogens Steffensen
Keyword(s):  
Continuous Time ◽  
Optimal Solution ◽  
Decision Problems ◽  
Optimal Consumption ◽  
Continuous Time Markov Chain ◽  
Financial Decision Making ◽  
Markov Chain Approach ◽  
Special Cases ◽  
Financial Decision ◽  
The Individual

Personal financial decision making plays an important role in modern finance. Decision problems about consumption and insurance are in this article modelled in a continuous-time multi-state Markovian framework. The optimal solution is derived and studied. The model, the problem, and its solution are exemplified by two special cases: In one model the individual takes optimal positions against the risk of dying; in another model the individual takes optimal positions against the risk of losing income as a consequence of disability or unemployment.

Public Service Motivation in Buffy the Vampire Slayer

Public Voices ◽  
2016 ◽  
Vol 12 (1) ◽  
pp. 67 ◽  
Author(s):  
Sharon Mastracci
Keyword(s):  
Popular Culture ◽  
Public Service ◽  
Public Service Motivation ◽  
Public Servants ◽  
Buffy The Vampire Slayer ◽  
The Public ◽  
American Popular Culture ◽  
Public Servant ◽  
Definition Of ◽  
The Individual

In this paper, the author examines public service as depicted in the television series Buffy the Vampire Slayer (BtVS). First, she shows how slaying meets the economist’s definition of a public good, using the BtVS episode “Flooded” (6.04). Second, she discusses public service motivation (PSM) to determine whether or not Buffy, a public servant, operates from a public service ethic. Relying on established measures and evidence from shooting scripts and episode transcripts, the author concludes Buffy is a public servant motivated by a public service ethic. In this way, BtVS informs scholarship on public service by broadening the concept of PSM beyond the public sector; prompting one to wonder whether it is located in a sector, an occupation, or in the individual. These conclusions allow the author to situate Buffy alongside other idealized public servants in American popular culture.

The Concept of the Individual and Its Scope

2018 ◽  
Author(s):  
Ursula Renz
Keyword(s):  
Strict Sense ◽  
Physical Concept ◽  
Individual Group ◽  
Group Minds ◽  
Definition Of ◽  
The Individual

This chapter discusses the implications of Spinoza’s concept of individual bodies, as introduced in the definition of individuum in the physical digression. It begins by showing that this definition allows for an extremely wide application of the term; accordingly, very different sorts of physical entities can be described as Spinozistic individuals. Given the quite distinct use of the terms divisibilis and indivisibilis in his metaphysics, however, the chapter argues that the physical concept of individuality is not universally applied in the Ethics but reserved for physical or natural-philosophical considerations. The chapter concludes with a discussion of the problem of collective individuals. It is argued that, while societies or states are described as individual bodies, they do not constitute individual group minds in the strict sense of the term for Spinoza. This in turn indicates that minds are not individuated in the same way as bodies.

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