ETA-RULES IN MARTIN-LÖF TYPE THEORY

2019 ◽  
Vol 25 (03) ◽  
pp. 333-359
Author(s):  
ANSTEN KLEV
Keyword(s):  
Lower Order ◽  
Category Theory ◽  
Type Theory ◽  
Proof Theory ◽  
Arbitrary Element ◽  
Higher Order

AbstractThe eta rule for a set A says that an arbitrary element of A is judgementally identical to an element of constructor form. Eta rules are not part of what may be called canonical Martin-Löf type theory. They are, however, justified by the meaning explanations, and a higher order eta rule is part of that type theory. The main aim of this article is to clarify this somewhat puzzling situation. It will be argued that lower order eta rules do not, whereas the higher order eta rule does, accord with the understanding of judgemental identity as definitional identity. A subsidiary aim is to clarify precisely what an eta rule is. This will involve showing how such rules relate to various other notions of type theory, proof theory, and category theory.

An intensional type theory: motivation and cut-elimination

2001 ◽  
Vol 66 (1) ◽  
pp. 383-400 ◽  
Author(s):  
Paul C Gilmore
Keyword(s):  
Type Theory ◽  
Proof Theory ◽  
Predicate Logic ◽  
Order Term ◽  
Higher Order ◽  
Cut Elimination ◽  
Consistency Proof ◽  
Completeness Proof ◽  
Higher Order Term

AbstractBy the theory TT is meant the higher order predicate logic with the following recursively defined types:(1) 1 is the type of individuals and [] is the type of the truth values:(2) [τ1…..τn] is the type of the predicates with arguments of the types τ1…..τn.The theory ITT described in this paper is an intensional version of TT. The types of ITT are the same as the types of TT, but the membership of the type 1 of individuals in ITT is an extension of the membership in TT. The extension consists of allowing any higher order term, in which only variables of type 1 have a free occurrence, to be a term of type 1. This feature of ITT is motivated by a nominalist interpretation of higher order predication.In ITT both well-founded and non-well-founded recursive predicates can be defined as abstraction terms from which all the properties of the predicates can be derived without the use of non-logical axioms.The elementary syntax, semantics, and proof theory for ITT are defined. A semantic consistency proof for ITT is provided and the completeness proof of Takahashi and Prawitz for a version of TT without cut is adapted for ITT: a consequence is the redundancy of cut.

A logical framework combining model and proof theory

2013 ◽  
Vol 23 (5) ◽  
pp. 945-1001 ◽  
Author(s):  
FLORIAN RABE
Keyword(s):  
Model Theory ◽  
Category Theory ◽  
Type Theory ◽  
Proof Theory ◽  
Software Verification ◽  
Logical Framework ◽  
Dependent Type Theory ◽  
Logical Frameworks ◽  
Balanced Approach ◽  
Dependent Type

Mathematical logic and computer science have driven the design of a growing number of logics and related formalisms such as set theories and type theories. In response to this population explosion, logical frameworks have been developed as formal meta-languages in which to represent, structure, relate and reason about logics.Research on logical frameworks has diverged into separate communities, often with conflicting backgrounds and philosophies. In particular, two of the most important logical frameworks are the framework of institutions, from the area of model theory based on category theory, and the Edinburgh Logical Framework LF, from the area of proof theory based on dependent type theory. Even though their ultimate motivations overlap – for example in applications to software verification – they have fundamentally different perspectives on logic.In the current paper, we design a logical framework that integrates the frameworks of institutions and LF in a way that combines their complementary advantages while retaining the elegance of each of them. In particular, our framework takes a balanced approach between model theory and proof theory, and permits the representation of logics in a way that comprises all major ingredients of a logic: syntax, models, satisfaction, judgments and proofs. This provides a theoretical basis for the systematic study of logics in a comprehensive logical framework. Our framework has been applied to obtain a large library of structured and machine-verified encodings of logics and logic translations.

Computational Models for Multilayered Composite Shells with Application to Tires

1996 ◽  
Vol 24 (1) ◽  
pp. 11-38 ◽  
Author(s):  
G. M. Kulikov
Keyword(s):  
Type Theory ◽  
Computational Models ◽  
Geometrical Nonlinearity ◽  
Higher Order ◽  
Stress Strain ◽  
Strain Fields ◽  
First Order ◽  
Timoshenko Type ◽  
Joint Influence ◽  
Discrete Layer

Abstract This paper focuses on four tire computational models based on two-dimensional shear deformation theories, namely, the first-order Timoshenko-type theory, the higher-order Timoshenko-type theory, the first-order discrete-layer theory, and the higher-order discrete-layer theory. The joint influence of anisotropy, geometrical nonlinearity, and laminated material response on the tire stress-strain fields is examined. The comparative analysis of stresses and strains of the cord-rubber tire on the basis of these four shell computational models is given. Results show that neglecting the effect of anisotropy leads to an incorrect description of the stress-strain fields even in bias-ply tires.

READING EXERCISES FOR VIII GRADE STUDENTS BASED ON BLOOM’S TAXONOMY

2017 ◽  
Vol 6 (3) ◽  
Author(s):  
Intan Permata Sari And Indra Hartoyo
Keyword(s):  
Data Analysis ◽  
Lower Order ◽  
Bloom's Taxonomy ◽  
Higher Order Thinking ◽  
Thinking Skills ◽  
Higher Order ◽  
Critical Thinking Skills ◽  
Thinking Skill ◽  
Level Of Education ◽  
Bloom’S Taxonomy

This study is aimed at (1) analyzing reading exercises based Bloom’s taxonomy for VIII grade in English on Sky textbook. (2) Found the distribution of the lower and higher order thinking skill in reading exercises. (3) To reason for level reading exercises. After analyzed the data, the result of the data analysis also infers that the six levels of Bloom’s taxonomy in reading exercises weren’t applied totally. The creating skill doesn’t have distribution in reading exercise, and the understanding – remembering level more dominant than another levels. The distribution of the higher order thinking level was lower than the lower order thinking level and the six levels are not appropriate with the proportion for each level of education based Bloom’s taxonomy, such as the distribution of the creating level in the reading exercise must be a concern because no question that belong to the creating level. It was concluded that reading exercises in English on Sky textbook cannot improve students' critical thinking skills for VIII grade.

Factors associated with complications in higher order compared to lower order repeat

2016 ◽  
Vol 1 (1) ◽  
pp. 32-39
Author(s):  
Tahira Akhtar ◽  
◽  
Aahsann Kazemi ◽  
Keyword(s):  
Lower Order ◽  
Higher Order ◽  
Factors Associated

Internal categoricity and truth

2018 ◽  
Author(s):  
Tim Button ◽  
Sean Walsh
Keyword(s):  
Model Theory ◽  
Lower Order ◽  
Higher Order ◽  
Mathematical Truth ◽  
Object Language ◽  
Famous Result ◽  
Famous Paper ◽  
Truth Value ◽  
Truth Operator

This chapter considers whether internal categoricity can be used to leverage any claims about mathematical truth. We begin by noting that internal categoricity allows us to introduce a truth-operator which gives an object-language expression to the supervaluationist semantics. In this way, the univocity discussed in previous chapters might seem to secure an object-language expression of determinacy of truth-value; but this hope falls short, because such truth-operators must be carefully distinguished from truth-predicates. To introduce these truth-predicates, we outline an internalist attitude towards model theory itself. We then use this to illuminate the cryptic conclusions of Putnam's justly-famous paper ‘Models and Reality’. We close this chapter by presenting Tarski’s famous result that truth for lower-order languages can be defined in higher-order languages.

scholarly journals Alzheimer’s Disease Progressively Reduces Visual Functional Network Connectivity

2021 ◽  
pp. 1-14
Author(s):  
Jie Huang ◽  
Paul Beach ◽  
Andrea Bozoki ◽  
David C. Zhu
Keyword(s):  
Visual Cortex ◽  
Lower Order ◽  
Resting State ◽  
Visual Processing ◽  
Network Connectivity ◽  
Higher Order ◽  
Functional Network ◽  
Functional Networks ◽  
The Face ◽  
Processing Network

Background: Postmortem studies of brains with Alzheimer’s disease (AD) not only find amyloid-beta (Aβ) and neurofibrillary tangles (NFT) in the visual cortex, but also reveal temporally sequential changes in AD pathology from higher-order association areas to lower-order areas and then primary visual area (V1) with disease progression. Objective: This study investigated the effect of AD severity on visual functional network. Methods: Eight severe AD (SAD) patients, 11 mild/moderate AD (MAD), and 26 healthy senior (HS) controls undertook a resting-state fMRI (rs-fMRI) and a task fMRI of viewing face photos. A resting-state visual functional connectivity (FC) network and a face-evoked visual-processing network were identified for each group. Results: For the HS, the identified group-mean face-evoked visual-processing network in the ventral pathway started from V1 and ended within the fusiform gyrus. In contrast, the resting-state visual FC network was mainly confined within the visual cortex. AD disrupted these two functional networks in a similar severity dependent manner: the more severe the cognitive impairment, the greater reduction in network connectivity. For the face-evoked visual-processing network, MAD disrupted and reduced activation mainly in the higher-order visual association areas, with SAD further disrupting and reducing activation in the lower-order areas. Conclusion: These findings provide a functional corollary to the canonical view of the temporally sequential advancement of AD pathology through visual cortical areas. The association of the disruption of functional networks, especially the face-evoked visual-processing network, with AD severity suggests a potential predictor or biomarker of AD progression.

scholarly journals Examining Bloom’s Taxonomy in Multiple Choice Questions: Students’ Approach to Questions

2021 ◽  
Author(s):  
J. K. Stringer ◽  
Sally A. Santen ◽  
Eun Lee ◽  
Meagan Rawls ◽  
Jean Bailey ◽  
...  
Keyword(s):  
Lower Order ◽  
Multiple Choice ◽  
Bloom's Taxonomy ◽  
Thinking Skills ◽  
Higher Order ◽  
Multiple Choice Questions ◽  
Analytic Thinking ◽  
Bloom’S Taxonomy ◽  
Degree Of Confidence ◽  
Clinical Reasoning Skills

Abstract Background Analytic thinking skills are important to the development of physicians. Therefore, educators and licensing boards utilize multiple-choice questions (MCQs) to assess these knowledge and skills. MCQs are written under two assumptions: that they can be written as higher or lower order according to Bloom’s taxonomy, and students will perceive questions to be the same taxonomical level as intended. This study seeks to understand the students’ approach to questions by analyzing differences in students’ perception of the Bloom’s level of MCQs in relation to their knowledge and confidence. Methods A total of 137 students responded to practice endocrine MCQs. Participants indicated the answer to the question, their interpretation of it as higher or lower order, and the degree of confidence in their response to the question. Results Although there was no significant association between students’ average performance on the content and their question classification (higher or lower), individual students who were less confident in their answer were more than five times as likely (OR = 5.49) to identify a question as higher order than their more confident peers. Students who responded incorrectly to the MCQ were 4 times as likely to identify a question as higher order than their peers who responded correctly. Conclusions The results suggest that higher performing, more confident students rely on identifying patterns (even if the question was intended to be higher order). In contrast, less confident students engage in higher-order, analytic thinking even if the question is intended to be lower order. Better understanding of the processes through which students interpret MCQs will help us to better understand the development of clinical reasoning skills.

REACTION OF ALDOSES AND KETOSES WITH LEAD TETRAACETATE

1956 ◽  
Vol 34 (4) ◽  
pp. 541-553 ◽  
Author(s):  
A. S. Perlin ◽  
Carol Brice
Keyword(s):  
Lower Order ◽  
Higher Order ◽  
Lead Tetraacetate ◽  
Pyranose Form ◽  
Reaction Proceeds ◽  
Furanose Form ◽  
Rare Sugars

Lead tetraacetate is highly selective for oxidation of α-hydroxy-hemiacetal groups and hence most readily attacks cyclic forms of the sugars. The reaction proceeds stepwise; the hemiacetal α-glycol being cleaved and the monoester of a correspondingly shorter-chained sugar formed. After cyclization the new sugar in turn is oxidized at the hemiacetal α-glycol group to yield a diester of a still-lower-order member of the series. In this manner D-glucose first yields mono-O-formyl-D-arabinose and then di-O-formyl-D-erythrose. Similarly, D-fructose is degraded to a glycolate ester of D-erythrose and finally to a formate–glycolate diester of D-glyceraldehyde. Some relatively rare sugars thus may conveniently be prepared directly from abundant higher-order members of the series. The reactions appear to involve preferential attack of the furanose form of a sugar rather than of the normally-predominant pyranose form, or possibly migration of ester groups towards the reducing end of the sugars.

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