Concatenation as basis for a complete system of arithmetic

1953 ◽  
Vol 18 (1) ◽  
pp. 1-6 ◽  
Author(s):  
M. H. Löb
Keyword(s):  
Complete System ◽  
Universal Quantifier ◽  
Class Membership ◽  
Classical Mathematics ◽  
Universal Quantification ◽  
Membership Relation ◽  
Pair Function

In [3] Myhill has constructed a complete system K which allows in it the development of a large and important section of classical mathematics. Completeness is achieved essentially by sacrificing universal quantification and introducing instead the proper ancestral as a primitive idea.In the following we are presenting a system K0 which will be shown to be equivalent to K (i.e. the primitive operators of both systems are mutually definable in terms of one another). K0 is also complete and covers the same ground as K. K0, however, differs from K by the introduction of the limited universal quantifier instead of the proper ancestral and of concatenation instead of the ordered-pair function as primitive operators. By a further reduction K0 will be shown to be equivalent to the system K1 not containing the abstraction-operator and the class-membership relation.

On second order intuitionistic propositional logic without a universal quantifier

2009 ◽  
Vol 74 (1) ◽  
pp. 157-167 ◽  
Author(s):  
Konrad Zdanowski
Keyword(s):  
Propositional Logic ◽  
Second Order ◽  
Universal Quantifier ◽  
Intuitionistic Propositional Logic ◽  
Glivenko’S Theorem ◽  
Free Variables ◽  
Universal Quantification

AbstractWe examine second order intuitionistic propositional logic, IPC2. Let ℱ∃ a be the set of formulas with no universal quantification. We prove Glivenko's theorem for formulas in ℱ∃ that is, for φ ∈ ℱ∃, φ is a classical tautology if and only if ┐┐φ is a tautology of IPC2. We show that for each sentence φ ∈ ℱ∃ (without free variables), φ is a classical tautology if and only if φ is an intuitionistic tautology. As a corollary we obtain a semantic argument that the quantifier ∀ is not definable in IPC2 from ⊥, ⋁, ⋀, →, ∃.

scholarly journals Expressive Power of Abstract Meaning Representations

2016 ◽  
Vol 42 (3) ◽  
pp. 527-535 ◽  
Author(s):  
Johan Bos
Keyword(s):  
Expressive Power ◽  
Order Logic ◽  
First Order Logic ◽  
Simple Extension ◽  
Universal Quantifier ◽  
Proper Treatment ◽  
First Order ◽  
Current Definition ◽  
Universal Quantification ◽  
Definition Of

The syntax of abstract meaning representations (AMRs) can be defined recursively, and a systematic translation to first-order logic (FOL) can be specified, including a proper treatment of negation. AMRs without recurrent variables are in the decidable two-variable fragment of FOL. The current definition of AMRs has limited expressive power for universal quantification (up to one universal quantifier per sentence). A simple extension of the AMR syntax and translation to FOL provides the means to represent projection and scope phenomena.

Variation in the expression of universal quantification and free choice

2008 ◽  
Vol 8 ◽  
pp. 179-232 ◽  
Author(s):  
Malte Zimmermann
Keyword(s):  
Free Choice ◽  
Generalized Quantifiers ◽  
Universal Quantifier ◽  
Alternative Analysis ◽  
Semantic Variation ◽  
Universal Quantification

I argue that the interpretation of expressions consisting of disjunction marker and wh-element (wh-disj expressions), which varies across languages, constitutes a case of semantic variation. In Hausa, these expressions denote universal generalized quantifiers, which give rise to free choice effects in intensional contexts (Giannakidou 2001). The universal meaning is derived in compositional fashion, where the disjunction marker expresses set union over the wh-domain. The free choice effects follow from the scopal interaction of universal quantifier and intensional operator. The account relates to Giannakidou & Cheng’s (2006) analysis of (quasi)universal FCIs, but it does not extend to Japanese and Malayalam wh-disj expressions, which are interpreted with existential force and should be analyzed as indeterminate pronouns (Jayaseelan 2001; Kratzer & Shimoyama 2002). Motivated by the analysis of FCIs in Menendéz-Benito (2005), we finally consider an alternative analysis of koo-wh expressions as selective indeterminate pronouns, which is rejected on conceptual and empirical grounds.

What hinders child semantic computation: children's universal quantification and the development of cognitive control

2011 ◽  
Vol 39 (5) ◽  
pp. 919-956 ◽  
Author(s):  
UTAKO MINAI ◽  
NOBUYUKI JINCHO ◽  
NAOTO YAMANE ◽  
REIKO MAZUKA
Keyword(s):  
Cognitive Control ◽  
Eye Tracking ◽  
Semantic Knowledge ◽  
Universal Quantifier ◽  
Universal Quantification ◽  
Semantic Computation

ABSTRACTRecent studies on the acquisition of semantics have argued that knowledge of the universal quantifier is adult-like throughout development. However, there are domains where children still exhibit non-adult-like universal quantification, and arguments for the early mastery of relevant semantic knowledge do not explain what causes such non-adult-like interpretations. The present study investigates Japanese four- and five-year-old children's atypical universal quantification in light of the development of cognitive control. We hypothesized that children's still-developing cognitive control contributes to their atypical universal quantification. Using a combined eye-tracking and interpretation task together with a non-linguistic measure of cognitive control, we revealed a link between the achievement of adult-like universal quantification and the development of flexible perspective-switch. We argue that the development of cognitive control is one of the factors that contribute to children's processing of semantics.

A complete theory of natural, rational, and real numbers

1950 ◽  
Vol 15 (3) ◽  
pp. 185-196 ◽  
Author(s):  
John R. Myhill
Keyword(s):  
Number Theory ◽  
Complete Theory ◽  
Future Research ◽  
Present System ◽  
Basic Logic ◽  
Real Numbers ◽  
Mathematical System ◽  
Classical Mathematics ◽  
Universal Quantification ◽  
Range Of Values

The purpose of the present paper is to construct a fragment of number theory not subject to Gödel incompletability. Originally the system was designed as a metalanguage for classical mathematics (see section 10); but it now appears to the author worthwhile to present it as a mathematical system in its own right, to serve however rather as an instrument of computation than of proof. Its resources in the latter respect seem very extensive, sufficient apparently for the systematic tabulation of every function used in any but the most recondite physics. The author intends to pursue this topic in a later paper; the present one will simply present the system, along with proofs of consistency and completeness and a few metatheorems which will be used as lemmas for future research.Completeness is achieved by sacrificing the notions of negation and universal quantification customary in number-theoretic systems; the losses consequent upon this are made good in part by the use of the ancestral as a primitive idea. The general outlines of the system follow closely the pattern of Fitch's “basic logic”; however the latter system uses combinatory operators in place of the variables used in the present paper, and if variables are introduced into Fitch's system by definition their range of values will be found to be much more extensive than that of my variables. The present system K is thus a weaker form of Fitch's system.It is apparently not known whether or not Fitch's system is complete.

Denitrification in a Carbon and Nitrogen Removal System for Leachate Treatment: Performance of a Upflow Sludge Blanket (USB) Reactor

1999 ◽  
Vol 40 (8) ◽  
pp. 145-151 ◽  
Author(s):  
Liliana Borzacconi ◽  
Gisela Ottonello ◽  
Elena Castelló ◽  
Heber Pelaez ◽  
Augusto Gazzola ◽  
...  
Keyword(s):  
Complete System ◽  
Carbon Sources ◽  
Uasb Reactor ◽  
Leachate Treatment ◽  
N Removal ◽  
Start Up ◽  
Aerobic Reactor ◽  
C And N ◽  
Reactor Operating ◽  
Removal System

The performance of a bench scale upflow sludge bed (USB) denitrifying reactor was evaluated in order to integrate it into a C and N removal system for Sanitary Landfill Leachate. The raw leachate used presented COD and NH4-N average values of 30000 mg/l and 1000 mg/l, respectively. The complete system comprises in addition an UASB reactor and a nitrifying RBC. A portion of the aerobic reactor effluent was recycled into the denitrification stage and some raw leachate was also added as an additional C source. In order to obtain operating parameters the denitrifying reactor was operated alone. Sludge from an aerobic reactor (RBC) treating raw leachate was used as inoculum. Shortly after the start up, good granulation of the sludge bed was observed. Using raw leachate and UASB outlet as carbon sources with COD/NO3-N ratios of 4 and 12, respectively, denitrification efficiencies of about 90% were reached. A sludge yield of 0.16 gVSS/gCODremoved was obtained operating with raw leachate. For the anoxic reactor operating in the complete system, denitrification efficiencies of 90% were also achieved. A nitrogen gas recycle was a successful way to avoid frequently observed sludge bed rising problems.

scholarly journals Interactions of C-reactive protein with the complement system. III. Complement-dependent passive hemolysis initiated by CRP.

1975 ◽  
Vol 142 (5) ◽  
pp. 1065-1077 ◽  
Author(s):  
A.P. Osmand ◽  
R.F. Mortensen ◽  
Joan Siegel ◽  
H. Gewurz
Keyword(s):  
Guinea Pig ◽  
Human Serum ◽  
Complete System ◽  
Gel Filtration ◽  
C Reactive Protein ◽  
Inflammatory Reactions ◽  
Reactive Protein ◽  
Rabbit Igg ◽  
The Complement System ◽  
Pig Serum

Interactions of CRP with various substrates in the presence of human serum have been shown to result in efficient activation of C components C1-C5. We now report the ability of CRP to initiate C-dependent hemolysis. For this purpose CRP was isolated by affinity chromatography using pneumococcal CPS and gel filtration; its purity was established by several criteria. Erythrocytes were coated with CPS (E-CPS) and passively sensitized with CRP. C-dependent lysis of these cells was observed upon the addition of suitably absorbed human serum, and the efficiency of hemolysis compared favorably with that initiated by rabbit IgG anti-CPS antibody. CRP also sensitized E-CPS for lysis by guinea pig C; partial lysis was seen when C4-deficient guinea pig serum was used, suggesting that CRP also shares with antibody the ability of CRP to fully activate the C system and provide further evidence for a role for CRP similar to that of antibody in the initiation and modulation of inflammatory reactions via the complete system.

scholarly journals Hybrid structures applied to ideals in near-rings

2021 ◽  
Author(s):  
B. Elavarasan ◽  
G. Muhiuddin ◽  
K. Porselvi ◽  
Y. B. Jun
Keyword(s):  
Set Theory ◽  
Fuzzy Set ◽  
Real World ◽  
Rough Set Theory ◽  
Wide Spectrum ◽  
Hybrid Structure ◽  
Soft Set ◽  
Hybrid Structures ◽  
Classical Mathematics ◽  
Classical Means

AbstractHuman endeavours span a wide spectrum of activities which includes solving fascinating problems in the realms of engineering, arts, sciences, medical sciences, social sciences, economics and environment. To solve these problems, classical mathematics methods are insufficient. The real-world problems involve many uncertainties making them difficult to solve by classical means. The researchers world over have established new mathematical theories such as fuzzy set theory and rough set theory in order to model the uncertainties that appear in various fields mentioned above. In the recent days, soft set theory has been developed which offers a novel way of solving real world issues as the issue of setting the membership function does not arise. This comes handy in solving numerous problems and many advancements are being made now-a-days. Jun introduced hybrid structure utilizing the ideas of a fuzzy set and a soft set. It is to be noted that hybrid structures are a speculation of soft set and fuzzy set. In the present work, the notion of hybrid ideals of a near-ring is introduced. Significant work has been carried out to investigate a portion of their significant properties. These notions are characterized and their relations are established furthermore. For a hybrid left (resp., right) ideal, different left (resp., right) ideal structures of near-rings are constructed. Efforts have been undertaken to display the relations between the hybrid product and hybrid intersection. Finally, results based on homomorphic hybrid preimage of a hybrid left (resp., right) ideals are proved.

Sign in / Sign up

Export Citation Format

Share Document