THE ARITHMETIC OF THE EVEN AND THE ODD

2016 ◽  
Vol 9 (2) ◽  
pp. 359-369 ◽  
Author(s):  
VICTOR PAMBUCCIAN
Keyword(s):  
Formal Theory ◽  
Image Position ◽  
Formal Theories

AbstractWe present several formal theories for the arithmetic of the even and the odd, show that the irrationality of $\sqrt 2$ can be proved in one of them, that the proof must involve contradiction, and prove that the irrationality of $\sqrt {17}$ cannot be proved inside any formal theory of the even and the odd.

Toward a Formal Theory of Transactions and Transvections

1965 ◽  
Vol 2 (2) ◽  
pp. 117-127 ◽  
Author(s):  
Wroe Alderson ◽  
Miles W. Martin
Keyword(s):  
Formal Theory ◽  
Marketing Channels ◽  
System Behavior ◽  
Marketing System ◽  
Basic Concepts ◽  
Formal Theories

This article presents the initial steps in the formalization of a partial theory of marketing. The partial theory pertains to the movement of goods and information through marketing channels, and the theory utilizes two basic concepts of marketing system behavior, namely, transactions and transvections. Current approaches to the problem of constructing formal theories are compared and reasons are given for choosing the “molar” approach.

scholarly journals Modeling Psychopathology: From Data Models to Formal Theories

2019 ◽  
Author(s):  
Jonas M B Haslbeck ◽  
Oisín Ryan ◽  
Donald Robinaugh ◽  
Lourens Waldorp ◽  
Denny Borsboom
Keyword(s):  
Mental Disorders ◽  
Philosophy Of Science ◽  
Empirical Research ◽  
Formal Theory ◽  
Data Models ◽  
Science Literature ◽  
Prediction And Control ◽  
Using Data ◽  
And Control ◽  
Formal Theories

Over the past decade there has been a surge of empirical research investigating mental disorders as complex systems. In this paper, we investigate how to best make use of this growing body of empirical research and move the field toward its fundamental aims of explaining, predicting, and controlling psychopathology. We first review the contemporary philosophy of science literature on scientific theories and argue that fully achieving the aims of explanation, prediction, and control requires that we construct formal theories of mental disorders: theories expressed in the language of mathematics or a computational programming language. We then investigate three routes by which one can use empirical findings (i.e., data models) to construct formal theories: (a) using data models themselves as formal theories, (b) using data models to infer formal theories, and (c) comparing empirical data models to theory-implied data models in order to evaluate and refine an existing formal theory. We argue that the third approach is the most promising path forward. We conclude by introducing the Abductive Formal Theory Construction (AFTC) framework, informed by both our review of philosophy of science and our methodological investigation. We argue that this approach provides a clear and promising way forward for using empirical research to inform the generation, development, and testing of formal theories both in the domain of psychopathology and in the broader field of psychological science.

scholarly journals On Formal Theories

1968 ◽  
Vol 32 ◽  
pp. 361-371 ◽  
Author(s):  
Katuzi Ono
Keyword(s):  
Finite Number ◽  
Formal Theory ◽  
Axiom System ◽  
Axiomatic Theory ◽  
Traditional Pattern ◽  
Formal Theories

The main purpose of the present paper is to introduce a new understanding of formal theories.It has been a traditional pattern of formal theories to presuppose a logic and an axiom system for each formal theory. The axiom system of any formal theory consists of a finite number of axiom schemata in general, but occasionally it can be regarded as consisting of a finite number of axioms. I will call any formal theory of this kind an axiomatic theory or an axiom-schematic theory according as its axiom system is regarded as consisting of a finite number of axioms or as consisting of a finite number of axiom schemata.

scholarly journals Invisible Hands and Fine Calipers: A Call to Use Formal Theory as a Toolkit for Theory Construction

2021 ◽  
pp. 174569162097469
Author(s):  
Donald J. Robinaugh ◽  
Jonas M. B. Haslbeck ◽  
Oisín Ryan ◽  
Eiko I. Fried ◽  
Lourens J. Waldorp
Keyword(s):  
Theory Development ◽  
Formal Theory ◽  
Psychological Theory ◽  
Theory Testing ◽  
Theory Construction ◽  
Psychological Theories ◽  
Collaborative Construction ◽  
The Status ◽  
Formal Theories

In recent years, a growing chorus of researchers has argued that psychological theory is in a state of crisis: Theories are rarely developed in a way that indicates an accumulation of knowledge. Paul Meehl raised this very concern more than 40 years ago. Yet in the ensuing decades, little has improved. We aim to chart a better path forward for psychological theory by revisiting Meehl’s criticisms, his proposed solution, and the reasons his solution failed to meaningfully change the status of psychological theory. We argue that Meehl identified serious shortcomings in our evaluation of psychological theories and that his proposed solution would substantially strengthen theory testing. However, we also argue that Meehl failed to provide researchers with the tools necessary to construct the kinds of rigorous theories his approach required. To advance psychological theory, we must equip researchers with tools that allow them to better generate, evaluate, and develop their theories. We argue that formal theories provide this much-needed set of tools, equipping researchers with tools for thinking, evaluating explanation, enhancing measurement, informing theory development, and promoting the collaborative construction of psychological theories.

Linguistic Epiphenomenalism ‐ Davidson and Chomsky on the Status of Public Languages

2010 ◽  
Vol 4 (1) ◽  
pp. 1-22 ◽  
Author(s):  
Isaac Nevo
Keyword(s):  
Formal Theory ◽  
Linguistic Meaning ◽  
Historical Reality ◽  
Social Fact ◽  
The Status ◽  
Systematic Structure ◽  
Formal Theories

AbstractThe aim of this paper is to highlight an individualist streak in both Davidson’s conception of language and Chomsky’s. In the first part of the paper, I argue that in Davidson’s case this individualist streak is a consequence of an excessively strong conception of what the compositional nature of linguistic meaning requires, and I offer a weaker conception of that requirement that can do justice to both the publicity and the compositionality of language. In the second part of the paper, I offer a comparison between Davidson’s position on the unreality of public languages, and Chomsky’s position regarding the epiphenomenal status of “externalized” languages. In Chomsky’s case, as in Davidson’s, languages are individuated in terms of the formal theories that serve to account for their systematic structure, and this assumption rests upon a similarly strong and similarly questionable understanding of what it is to employ finite means in pursuit of an infinite task. The alternative, at which I can only hint, is a view of language as a social and historical reality, i.e., a realm of social fact that cannot be exhausted by any formal theory and cannot be reduced to properties of individual speakers.

Two questions from Dana Scott: Intuitionistic topologies and continuous functions

2009 ◽  
Vol 74 (2) ◽  
pp. 689-692
Author(s):  
Charles McCarty
Keyword(s):  
Set Theory ◽  
Continuous Functions ◽  
Unit Interval ◽  
Natural Numbers ◽  
Double Negation ◽  
Closed Set ◽  
Closed Sets ◽  
Image Position ◽  
Intuitionistic Set Theory ◽  
Formal Theories

Since intuitionistic sets are not generally stable – their membership relations are not always closed under double negation – the open sets of a topology cannot be recovered from the closed sets of that topology via complementation, at least without further ado. Dana Scott asked, first, whether it is possible intuitionistically for two distinct topologies, given as collections of open sets on the same carrier, to share their closed sets. Second, he asked whether there can be intuitionistic functions that are closed continuous in that the inverse of every closed set is closed without being continuous in the usual, open sense. Here, we prove that, as far as intuitionistic set theory is concerned, there can be infinitely-many distinct topologies on the same carrier sharing a single collection of closed sets. The proof employs Heyting-valued sets, and demonstrates that the intuitionistic set theory IZF [4, 624], as well as the theory IZF plus classical elementary arithmetic, are both consistent with the statement that infinitely many topologies on the set of natural numbers share the same closed sets. Without changing models, we show that these formal theories are also consistent with the statement that there are infinitely many endofunctions on the natural numbers that are closed continuous but not open continuous with respect to a single topology.For each prime k ∈ ω, let Ak be this ω-sequence of sets open in the standard topology on the closed unit interval: for each n ∈ ω,

scholarly journals On choice sequences determined by spreads

1984 ◽  
Vol 49 (3) ◽  
pp. 908-916 ◽  
Author(s):  
Gerrit van der Hoeven ◽  
Ieke Moerdijk
Keyword(s):  
Formal System ◽  
Formal Theory ◽  
Restricted Class ◽  
Continuous Choice ◽  
Choice Sequence ◽  
Finite Sequences ◽  
The Moment ◽  
Basic Axiom ◽  
Image Position ◽  
Do So

From the moment choice sequences appear in Brouwer's writings, they do so as elements of a spread. This led Kreisel to take the so-called axiom of spreaddata as the basic axiom in a formal theory of choice sequences (Kreisel [1965, pp. 133–136]). This axiom expresses the idea that to be given a choice sequence means to be given a spread to which the choice sequence belongs. Subsequently, however, it was discovered that there is a formal clash between this axiom and closure of the domain of choice sequences under arbitrary (lawlike) continuous operations (Troelstra [1968]). For this reason, the formal system CS was introduced (Kreisel and Troelstra [1970]), in which spreaddata is replaced by analytic data. In this system CS, the domain of choice sequences is closed under all continuous operations, and therefore it provides a workable basis for intuitionistic analysis. But the problem whether the axiom of spreaddata is compatible with closure of the domain of choice sequences under the continuous operations from a restricted class, which is still rich enough to validate the typical axioms of continuous choice, remained open. It is precisely this problem that we aim to discuss in this paper.Recall that a spread is a (lawlike, inhabited) decidable subtree S of the tree N< N of all finite sequences, having all branches infinite:

Ordinals connected with formal theories for transfinitely iterated inductive definitions

1978 ◽  
Vol 43 (2) ◽  
pp. 161-182 ◽  
Author(s):  
W. Pohlers
Keyword(s):  
Fixed Point ◽  
Number Theory ◽  
Formal Theory ◽  
Order Type ◽  
General Notion ◽  
Inductive Definitions ◽  
Primitive Recursive ◽  
Formal Theories

Let Th be a formal theory extending number theory. Call an ordinal ξ provable in Th if there is a primitive recursive ordering which can be proved in Th to be a wellordering and whose order type is > ξ. One may define the ordinal ∣ Th ∣ of Th to be the least ordinal which is not provable in Th. By [3] and [12] we get , where IDN is the formal theory for N-times iterated inductive definitions. We will generalize this result not only to the case of transfinite iterations but also to a more general notion of ‘the ordinal of a theory’.For an X-positive arithmetic formula [X,x] there is a natural norm ∣x∣: = μξ (x ∈ Iξ), where Iξ is defined as {x: [∪μ<ξIμ, x]} by recursion on ξ (cf. [7], [17]). By P we denote the least fixed point of [X,x]. Then P = ∪ξξ holds. If Th allows the formation of P, we get the canonical definitions ∥Th()∥ = sup{∣x∣ + 1: Th ⊢ x ∈ P} and ∥Th∥ = sup{∥Th()∥: P is definable in Th} (cf. [17]). If ≺ is any primitive recursive ordering define Q≺[X,x] as the formula ∀y(y ≺ x → y ∈ X) and ∣x∣≺:= ∣x∣O≺. Then ∣x∣≺ turns out to be the order type of the ≺ -predecessors of x and P is the accessible part of ≺. So Th ⊢ x ∈ P implies the provability of ∣x∣≺ in Th.

scholarly journals Provable wellorderings of formal theories for transfinitely iterated inductive definitions

1978 ◽  
Vol 43 (1) ◽  
pp. 118-125 ◽  
Author(s):  
W. Buchholz ◽  
W. Pohlers
Keyword(s):  
Transfinite Induction ◽  
Crucial Point ◽  
Inductive Definitions ◽  
Image Position ◽  
Intuitionistic Version ◽  
Formal Theories

By [12] we know that transfinite induction up to ΘεΩN+10 is not provable in IDN, the theory of N-times iterated inductive definitions. In this paper we will show that conversely transfinite induction up to any ordinal less than ΘεΩN+10 is provable in IDNi, the intuitionistic version of IDN, and extend this result to theories for transfinitely iterated inductive definitions.In [14] Schütte proves the wellordering of his notational systems using predicates is wellordered) with Mκ ≔ {x ∈ and 0 ≤ κ ≤ N. Obviously the predicates are definable in IDNi with the defining axioms:where Prog [Mκ, X] means that X is progressive with respect to Mκ, i.e.The crucial point in Schütte's wellordering proof is Lemma 19 [14, p. 130] which can be modified towhere TI[Mκ + 1, a] is the scheme of transfinite induction over Mκ + 1 up to a.

Simple Identification of Electron Diffraction Ring Patterns

1969 ◽  
Vol 27 ◽  
pp. 160-161
Author(s):  
Carolyn Nohr ◽  
Ann Ayres
Keyword(s):  
Electron Microscope ◽  
Electron Diffraction ◽  
Diffraction Ring ◽  
Image Position

Texts on electron diffraction recommend that the camera constant of the electron microscope be determine d by calibration with a standard crystalline specimen, using the equation

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