Linear reasoning. A new form of the Herbrand-Gentzen theorem

1957 ◽  
Vol 22 (3) ◽  
pp. 250-268 ◽  
Author(s):  
William Craig
Keyword(s):  
Normal Form ◽  
Distinctive Feature ◽  
First Order ◽  
Herbrand’S Theorem ◽  
Free Level ◽  
New Form ◽  
The Relationship ◽  
Herbrand's Theorem

In Herbrand's Theorem [2] or Gentzen's Extended Hauptsatz [1], a certain relationship is asserted to hold between the structures of A and A′, whenever A implies A′ (i.e., A ⊃ A′ is valid) and moreover A is a conjunction and A′ an alternation of first-order formulas in prenex normal form. Unfortunately, the relationship is described in a roundabout way, by relating A and A′ to a quantifier-free tautology. One purpose of this paper is to provide a description which in certain respects is more direct. Roughly speaking, ascent to A ⊃ A′ from a quantifier-free level will be replaced by movement from A to A′ on the quantificational level. Each movement will be closely related to the ascent it replaces.The new description makes use of a set L of rules of inference, the L-rules. L is complete in the sense that, if A is a conjunction and A′ an alternation of first-order formulas in prenex normal form, and if A ⊃ A′ is valid, then A′ can be obtained from A by an L-deduction, i.e., by applications of L-rules only. The distinctive feature of L is that each L-rule possesses two characteristics which, especially in combination, are desirable. First, each L-rule yields only conclusions implied by the premisses.

Algorithmic logic and its applications in the theory of programs I

1977 ◽  
Vol 1 (1) ◽  
pp. 1-17
Author(s):  
Grażyna Mirkowska
Keyword(s):  
Normal Form ◽  
Programming Language ◽  
Formal Proof ◽  
Herbrand’S Theorem ◽  
Algorithmic Logic ◽  
Herbrand's Theorem ◽  
Herbrand Theorem ◽  
Algorithmic Property

The paper presents tools for formalizing and proving properties of programs. The language of algorithmic logic constitutes an extension of a programming language by formulas that describe algorithmic properties. The paper contains two axiomatizations of algorithmic logic, which are complete. It can be proved that every valid algorithmic property possesses a formal proof. An analogue of Herbrand theorem and a theorem on the normal form of a program are proved. Results of meta-mathematical character are applied to theory of programs, e.g. Paterson’s theorem is an immediate corollary to Herbrand’s theorem.

scholarly journals HERBRAND’S THEOREM AND NON-EUCLIDEAN GEOMETRY

2015 ◽  
Vol 21 (2) ◽  
pp. 111-122 ◽  
Author(s):  
MICHAEL BEESON ◽  
PIERRE BOUTRY ◽  
JULIEN NARBOUX
Keyword(s):  
Euclidean Geometry ◽  
The Other ◽  
Basic Theorem ◽  
Very Old ◽  
First Order ◽  
Herbrand’S Theorem ◽  
Non Euclidean Geometry ◽  
Herbrand's Theorem

AbstractWe use Herbrand’s theorem to give a new proof that Euclid’s parallel axiom is not derivable from the other axioms of first-order Euclidean geometry. Previous proofs involve constructing models of non-Euclidean geometry. This proof uses a very old and basic theorem of logic together with some simple properties of ruler-and-compass constructions to give a short, simple, and intuitively appealing proof.

A strong version of Herbrand's theorem for introvert sentences

1998 ◽  
Vol 63 (2) ◽  
pp. 555-569 ◽  
Author(s):  
Tore Langholm
Keyword(s):  
Ground Term ◽  
New Members ◽  
First Order ◽  
Herbrand’S Theorem ◽  
Universal Sentence ◽  
Order Language ◽  
The Matrix ◽  
Image Position ◽  
Infinite Set ◽  
Herbrand's Theorem

A version of Herbrand's theorem tells us that a universal sentence of a first-order language with at least one constant is satisfiable if and only if the conjunction of all its ground instances is. In general the set of such instances is infinite, and arbitrarily large finite subsets may have to be inspected in order to detect inconsistency. Essentially, the reason that every member of such an infinite set may potentially matter, can be traced back to sentences like(1) Loosely put, such sentences effectively sabotage any attempt to build a model from below in a finite number of steps, since new members of the Herbrand universe are constantly brought to attention. Since they cause an indefinite expansion of the relevant part of the Herbrand universe, such sentences could quite appropriately be called expanding.When such sentences are banned, stronger versions of Herbrand's theorem can be stated. Define a clause (disjunction of literals) to be non-expanding if every non-ground term occurring in a positive literal also occurs (possibly as an embedded subterm) in a negative literal of the same clause. Written as a disjunction of literals, the matrix of (1) clearly fails this criterion. Moreover, say that a sentence is non-expanding if it is a universal sentence with a quantifier-free matrix that is a conjunction of non-expanding clauses. Such sentences do in a sense never reach out beyond themselves, and the relevant part of the Herbrand universe is therefore drastically reduced.

Nonlinear Analysis of Visual Evoked Potentials Elicited by Color Stimulation

1997 ◽  
Vol 36 (04/05) ◽  
pp. 315-318 ◽  
Author(s):  
K. Momose ◽  
K. Komiya ◽  
A. Uchiyama
Keyword(s):  
Visual System ◽  
Evoked Potentials ◽  
Visual Evoked Potentials ◽  
Normal Subjects ◽  
Second Order ◽  
First Order ◽  
The Relationship ◽  
Perceived Color ◽  
Second Order Nonlinearity ◽  
Visual Evoked

Abstract:The relationship between chromatically modulated stimuli and visual evoked potentials (VEPs) was considered. VEPs of normal subjects elicited by chromatically modulated stimuli were measured under several color adaptations, and their binary kernels were estimated. Up to the second-order, binary kernels obtained from VEPs were so characteristic that the VEP-chromatic modulation system showed second-order nonlinearity. First-order binary kernels depended on the color of the stimulus and adaptation, whereas second-order kernels showed almost no difference. This result indicates that the waveforms of first-order binary kernels reflect perceived color (hue). This supports the suggestion that kernels of VEPs include color responses, and could be used as a probe with which to examine the color visual system.

Explore the Possible Impact of BCG Vaccination Policy on the Morbidity, Mortality, and Recovery Rates due to COVID-19 Infection. (Preprint)

2020 ◽  
Author(s):  
Yue-Cune Chang
Keyword(s):  
Odds Ratio ◽  
Randomized Clinical Trials ◽  
Vaccination Policy ◽  
Bacille Calmette Guerin ◽  
Recovery Rates ◽  
Bcg Vaccination ◽  
The World ◽  
New Form ◽  
The Relationship ◽  
Disease Specific

BACKGROUND The Coronavirus Disease-19 (COVID-19) is the new form of an acute infectious respiratory disease and has quickly spread over most continents in the world. Recently, it has been shown that Bacille Calmette-Guerin (BCG) might protect against COVID-19. This study aims to investigate the possible correlation between BCG vaccination and morbidity/mortality/recovery rate associated with COVID-19 infection. OBJECTIVE Our findings confirm that the BCG vaccination might protect against COVID-19 virus infection. METHODS Data of COVID-19 confirmed cases, deaths, recoveries, and population were obtained from https://www.worldometers.info/coronavirus/ (Accessed on 12 June, 2020). To have meaningful comparisons among countries’ mortality and recovery rates, we only choose those countries with COVID-19 infected cases at least 200. The Poisson regression and logistic regression were used to explore the relationship between BCG vaccination and morbidity, mortality and recovery rates. RESULTS Among those 158 countries with at least 200 COVID-19 infected cases, there were 141 countries with BCG vaccination information available. The adjusted rates ratio of COVID-19 confirmed cases for Current BCG vaccination vs. non-Current BCG vaccination was 0.339 (with 95% CI= (0.338,0.340)). Moreover, the adjusted odds ratio (OR) of death and recovery after coronavirus infected for Current BCG vaccination vs. non-Current BCG vaccination were 0.258 (with 95% CI= (0.254,0.261)) and 2.151 (with 95% CI= (2.140,2.163)), respectively. CONCLUSIONS That data in this study show the BCG might provide the protection against COVID-19, with consequent less COVID-19 infection and deaths and more rapid recovery. BCG vaccine might bridge the gap before the disease-specific vaccine is developed, but this hypothesis needs to be further tested in rigorous randomized clinical trials. INTERNATIONAL REGISTERED REPORT RR2-https://doi.org/10.1101/2020.06.14.20131268

scholarly journals Extensions in graph normal form

2020 ◽  
Author(s):  
Michał Walicki
Keyword(s):  
Normal Form ◽  
Fixed Points ◽  
Propositional Logic ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Special Cases

Abstract Graph normal form, introduced earlier for propositional logic, is shown to be a normal form also for first-order logic. It allows to view syntax of theories as digraphs, while their semantics as kernels of these digraphs. Graphs are particularly well suited for studying circularity, and we provide some general means for verifying that circular or apparently circular extensions are conservative. Traditional syntactic means of ensuring conservativity, like definitional extensions or positive occurrences guaranteeing exsitence of fixed points, emerge as special cases.

scholarly journals On the correspondence between nested calculi and semantic systems for intuitionistic logics

2020 ◽  
Author(s):  
Tim Lyon
Keyword(s):  
Intuitionistic Logic ◽  
Structural Rules ◽  
First Order ◽  
Constant Domains ◽  
The Relationship ◽  
Intuitionistic Logics

Abstract This paper studies the relationship between labelled and nested calculi for propositional intuitionistic logic, first-order intuitionistic logic with non-constant domains and first-order intuitionistic logic with constant domains. It is shown that Fitting’s nested calculi naturally arise from their corresponding labelled calculi—for each of the aforementioned logics—via the elimination of structural rules in labelled derivations. The translational correspondence between the two types of systems is leveraged to show that the nested calculi inherit proof-theoretic properties from their associated labelled calculi, such as completeness, invertibility of rules and cut admissibility. Since labelled calculi are easily obtained via a logic’s semantics, the method presented in this paper can be seen as one whereby refined versions of labelled calculi (containing nested calculi as fragments) with favourable properties are derived directly from a logic’s semantics.

Relational Chase Procedures Interpreted as Resolution with Paramodulation1

1991 ◽  
Vol 15 (2) ◽  
pp. 123-138
Author(s):  
Joachim Biskup ◽  
Bernhard Convent
Keyword(s):  
Alternative Interpretation ◽  
Order Logic ◽  
First Order Logic ◽  
Inference Problem ◽  
First Order ◽  
Inference Problems ◽  
The One ◽  
The Relationship ◽  
Theoretical Foundations ◽  
Insight Into

In this paper the relationship between dependency theory and first-order logic is explored in order to show how relational chase procedures (i.e., algorithms to decide inference problems for dependencies) can be interpreted as clever implementations of well known refutation procedures of first-order logic with resolution and paramodulation. On the one hand this alternative interpretation provides a deeper insight into the theoretical foundations of chase procedures, whereas on the other hand it makes available an already well established theory with a great amount of known results and techniques to be used for further investigations of the inference problem for dependencies. Our presentation is a detailed and careful elaboration of an idea formerly outlined by Grant and Jacobs which up to now seems to be disregarded by the database community although it definitely deserves more attention.

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