LINDSTRÖM QUANTIFIERS AND LEAF LANGUAGE DEFINABILITY

1998 ◽  
Vol 09 (03) ◽  
pp. 277-294 ◽  
Author(s):  
HANS-JÖRG BURTSCHICK ◽  
HERIBERT VOLLMER
Keyword(s):  
Polynomial Time ◽  
Second Order ◽  
New Model ◽  
First Order ◽  
Theoretic Characterization ◽  
Polynomial Time Hierarchy ◽  
The Way

We introduce second-order Lindström quantifiers and examine analogies to the concept of leaf language definability. The quantifier structure in a second-order sentence defining a language and the quantifier structure in a first-order sentence characterizing the appropriate leaf language correspond to one another. Under some assumptions, leaf language definability and definability with second-order Lindström quantifiers may be seen as equivalent. Along the way we tighten the best up to now known leaf language characterization of the classes of the polynomial time hierarchy and give a new model-theoretic characterization of PSPACE.

Amphibole and mica 40Ar/39Ar ages from the Kaipokok and Aillik domains, Makkovik Province, Labrador: towards a characterization of back-arc processes in the Paleoproterozoic

2002 ◽  
Vol 39 (5) ◽  
pp. 749-764 ◽  
Author(s):  
Nicholas Culshaw ◽  
Peter Reynolds ◽  
Gavin Sinclair ◽  
Sandra Barr
Keyword(s):  
Thermal Effect ◽  
Shear Zone ◽  
Second Order ◽  
Greenschist Facies ◽  
Metamorphic Rocks ◽  
Arc Magmatism ◽  
First Order ◽  
Back Arc ◽  
Order Pattern

We report amphibole and mica 40Ar/39Ar ages from the Makkovik Province. Amphibole ages from metamorphic rocks decrease towards the interior of the province, indicating a first-order pattern of monotonic cooling with progressive migration of the province into a more distal back-arc location. The amphibole data, in combination with muscovite ages, reveal a second-order pattern consisting of four stages corresponding to changing spatial and temporal configurations of plutonism and deformation. (1) The western Kaipokok domain cooled through muscovite closure by 1810 Ma, long after the cessation of arc magmatism. (2) The Kaipokok Bay shear zone, bounding the Kaipokok and Aillik domains, cooled through amphibole closure during 1805–1780 Ma, synchronous with emplacement of syn-tectonic granitoid plutons. (3) Between 1740 and 1700 Ma, greenschist-facies shearing occurred along the boundary between the Kaipokok domain and Nain Province synchronous with A-type plutonism and localized shearing in the western Kaipokok domain, cooling to muscovite closure temperatures in the Kaipokok Bay shear zone, and A-type plutonism and amphibole closure or resetting in the Aillik domain. (4) In the period 1650–1640 Ma, muscovite ages, an amphibole age from a shear zone, and resetting of plutonic amphibole indicate a thermal effect coinciding in part with Labradorian plutonism in the Aillik domain. Amphibole ages from dioritic sheets in the juvenile Aillik domain suggest emplacement between 1715 and 1685 Ma. Amphibole ages constrain crystallization of small mafic plutons in the Kaipokok domain (reworked Archean foreland) to be no younger than 1670–1660 Ma. These ages are the oldest yet obtained for Labradorian plutonism in the Makkovik Province.

Second-order logic, philosophical issues in

2018 ◽  
Author(s):  
Stewart Shapiro
Keyword(s):  
Mathematical Practice ◽  
Expressive Power ◽  
Deductive System ◽  
Second Order ◽  
Order Logic ◽  
Mathematical Structures ◽  
First Order ◽  
Correct Reasoning ◽  
Second Order Logic

Typically, a formal language has variables that range over a collection of objects, or domain of discourse. A language is ‘second-order’ if it has, in addition, variables that range over sets, functions, properties or relations on the domain of discourse. A language is third-order if it has variables ranging over sets of sets, or functions on relations, and so on. A language is higher-order if it is at least second-order. Second-order languages enjoy a greater expressive power than first-order languages. For example, a set S of sentences is said to be categorical if any two models satisfying S are isomorphic, that is, have the same structure. There are second-order, categorical characterizations of important mathematical structures, including the natural numbers, the real numbers and Euclidean space. It is a consequence of the Löwenheim–Skolem theorems that there is no first-order categorical characterization of any infinite structure. There are also a number of central mathematical notions, such as finitude, countability, minimal closure and well-foundedness, which can be characterized with formulas of second-order languages, but cannot be characterized in first-order languages. Some philosophers argue that second-order logic is not logic. Properties and relations are too obscure for rigorous foundational study, while sets and functions are in the purview of mathematics, not logic; logic should not have an ontology of its own. Other writers disqualify second-order logic because its consequence relation is not effective – there is no recursively enumerable, sound and complete deductive system for second-order logic. The deeper issues underlying the dispute concern the goals and purposes of logical theory. If a logic is to be a calculus, an effective canon of inference, then second-order logic is beyond the pale. If, on the other hand, one aims to codify a standard to which correct reasoning must adhere, and to characterize the descriptive and communicative abilities of informal mathematical practice, then perhaps there is room for second-order logic.

scholarly journals On characterizations of the basic feasible functionals, Part I

2001 ◽  
Vol 11 (1) ◽  
pp. 117-153 ◽  
Author(s):  
ROBERT J. IRWIN ◽  
JAMES S. ROYER ◽  
BRUCE M. KAPRON
Keyword(s):  
Polynomial Time ◽  
Theoretic Characterization

We introduce a typed programming formalism, type-2 inflationary tiered loop programs or ITLP2, that characterizes the type-2 basic feasible functionals. ITLP2 is based on Bellantoni and Cook's (1992) and Leivant's (1995) type-theoretic characterization of polynomial-time, and turns out to be closely related to Kapron and Cook's (1991; 1996) machine-based characterization of the type-2 basic feasible functionals.

Barack Obama, being sharp

Gesture ◽  
2011 ◽  
Vol 11 (3) ◽  
pp. 241-270 ◽  
Author(s):  
Michael Lempert
Keyword(s):  
Information Structure ◽  
Precision Grip ◽  
Barack Obama ◽  
Second Order ◽  
First Order ◽  
Political Oratory ◽  
The Way

Gesture in political oratory and debate is renowned for its nonreferential indexical functions, for the way it purportedly can indicate qualities of speaker and materialize acts of persuasion — functions famously addressed in Quintilian’s classic writings but understudied today. I revisit this problematic through a case study of precision-grip (especially thumb to tip of forefinger) in Barack Obama’s debate performances (2004–2008). Cospeech gesture can index valorized attributes of speaker — not directly but through orders of semiotic motivation. In terms of first-order indexicality, precision-grip highlights discourse in respect of information structure, indicating focus. In debate, precision grip has undergone a degree of conventionalization and has reemerged as a second-order pragmatic resource for performatively “making a ‘sharp’, effective point.” Repetitions and parallelisms of precision grip in debate can, in turn, exhibit speaker-attributes, such as being argumentatively ‘sharp’, and from there may even partake in candidate branding.

A feasible theory for analysis

1994 ◽  
Vol 59 (3) ◽  
pp. 1001-1011 ◽  
Author(s):  
Fernando Ferreira
Keyword(s):  
Polynomial Time ◽  
Order Theory ◽  
Second Order ◽  
First Order ◽  
Weak König’S Lemma ◽  
König’S Lemma ◽  
Computable Functions

AbstractWe construct a weak second-order theory of arithmetic which includes Weak König's Lemma (WKL) for trees defined by bounded formulae. The provably total functions (with -graphs) of this theory are the polynomial time computable functions. It is shown that the first-order strength of this version of WKL is exactly that of the scheme of collection for bounded formulae.

scholarly journals הכל הבל in Ecclesiastes 1:2 and 12:8 – Descriptive metaphysics of properties as comparative-philosophical supplement

2021 ◽  
Vol 77 (1) ◽  
Author(s):  
Jaco Gericke
Keyword(s):  
Second Order ◽  
Property Theory ◽  
First Order ◽  
Conceptual Clarification ◽  
Order Language ◽  
The World ◽  
Original Contribution ◽  
The Way

In this article, a supplementary yet original contribution is made to the ongoing attempts at refining ways of comparative-philosophical conceptual clarification of Qohelet’s claim that הבל הכל in 1:2 (and 12:8). Adopting and adapting the latest analytic metaphysical concerns and categories for descriptive purposes only, a distinction is made between הבל as property of הכל and the properties of הבל in relation to הכל. Involving both correlation and contrast, the second-order language framework is hereby extended to a level of advanced nuance and specificity for restating the meaning of the book’s first-order language on its own terms, even if not in them.Contribution: By considering logical, ontological, mereological and typological aspects of property theory in dialogue with appearances of הכל and of הבל in Ecclesiastes 1:2 and 12:8 and in-between, a new way is presented in the quest to explain why things in the world of the text are the way they are, or why they are at all.

Delineating classes of computational complexity via second order theories with weak set existence principles. I

1995 ◽  
Vol 60 (1) ◽  
pp. 103-121 ◽  
Author(s):  
Aleksandar Ignjatović
Keyword(s):  
Computational Complexity ◽  
Polynomial Time ◽  
Second Order ◽  
Complexity Classes ◽  
Bounded Arithmetic ◽  
Recursive Functions ◽  
Speed Up ◽  
Bounded Complexity ◽  
Polynomial Time Hierarchy ◽  
Comprehension Axiom

AbstractIn this paper we characterize the well-known computational complexity classes of the polynomial time hierarchy as classes of provably recursive functions (with graphs of suitable bounded complexity) of some second order theories with weak comprehension axiom schemas but without any induction schemas (Theorem 6). We also find a natural relationship between our theories and the theories of bounded arithmetic (Lemmas 4 and 5). Our proofs use a technique which enables us to “speed up” induction without increasing the bounded complexity of the induction formulas. This technique is also used to obtain an interpretability result for the theories of bounded arithmetic (Theorem 4).

scholarly journals Betweenness of partial orders

2020 ◽  
Vol 54 ◽  
pp. 7
Author(s):  
Bruno Courcelle
Keyword(s):  
Partial Order ◽  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Second Order ◽  
Partial Orders ◽  
First Order ◽  
Betweenness Relations

We construct a monadic second-order sentence that characterizes the ternary relations that are the betweenness relations of finite or infinite partial orders. We prove that no first-order sentence can do that. We characterize the partial orders that can be reconstructed from their betweenness relations. We propose a polynomial time algorithm that tests if a finite relation is the betweenness of a partial order.

scholarly journals Characterizations of Model Manifolds by Means of Certain Differential Systems

2012 ◽  
Vol 55 (3) ◽  
pp. 632-645 ◽  
Author(s):  
S. Pigola ◽  
M. Rimoldi
Keyword(s):  
Vector Fields ◽  
Second Order ◽  
Differential Systems ◽  
Space Forms ◽  
Complete Manifolds ◽  
The Way

AbstractWe prove metric rigidity for complete manifolds supporting solutions of certain second order differential systems, thus extending classical works on a characterization of space-forms. Along the way, we also discover new characterizations of space-forms. We next generalize results concerning metric rigidity via equations involving vector fields.

Empathy Revisited: Who’s in Narrative Control?

2019 ◽  
pp. 207-228
Author(s):  
Jeanne Gaakeer
Keyword(s):  
Second Order ◽  
Legal Practice ◽  
Judicial Decision ◽  
Daily Lives ◽  
First Order ◽  
Narrative Control ◽  
The Way

Part III deals with what Cardozo called “the perplexities of judges” that become “the scholar’s opportunity”. Chapter 11 revisits the topic of empathy by distinguishing between forms of empathy and the way in which they are triggered, in works of fiction as much as in our daily lives. It argues that all forms of empathy are connected to emotion(s), first-order emotion such as anger of grief, and second-order emotion as the reaction to another person’s first-order emotion. It then asks what the cognitive turn in narratology means for legal practice, i.e. who is in narrative control of the situation, in court or in other legal surroundings? The judge, the prosecutor or a party? The story of Ian McEwan’s fictional judge Fiona Maye in The Children Act exemplifies the pitfalls of a first-order empathy, triggered as it may be by parties in a case by means of deliberate narratological strategies aimed at influencing the judicial decision.

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