Completeness

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bu teoremlerin meta-mantıksal yorumlamaları incelenecektir.İlk
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Ardından sağlamlık ve tutarlılık teoremlerine geçilerek, türetilebilirlik
vegeçerlilik arasındaki ilişkiye değinilecektir. İkinci
bölümde, önermeler mantığı dizgeleri içerisinde eksiksizlik teoremleri
sunulacak ve ardından bu teoremlerin meta mantıksal
yorumlamalarına değinilecektir.

OpenStreetMap (OSM) is a free map of the world which can be edited by global volunteers. Existing studies have showed that completeness of OSM road data in some developing countries (e.g. China) is much lower, resulting in concern in utilizing the data in various applications. But very few have focused on investigating what types of road are still poorly mapped. This study aims not only to investigate the completeness of OSM road datasets in China but also to investigate what types of road (called omission roads) have not been mapped, which is achieved by referring to both Street Map and Street View. Sixteen prefecture-level divisions in the urban areas of Hubei (China) were used as study areas. Results showed that: (1) the completeness for most prefecture-level divisions was at a low-to-medium level; most roads (in the Street Map), however, with traffic conditions had already been mapped well. (2) Most of the omission OSM roads were either private roads, or public roads not having yet been named and with only one single lane, indicating their minor importance in the urban road network. We argue that although the OSM road datasets in China are incomplete, they may still be used for several applications such as traffic flow analysis, road map representation, and backbone structure detection.

1972. Weak and Strong Completeness in Sentential Logic, Logique et Analyse 59/60, 429–34.  MR0337476 (49 #2245)

This is another study illustrating the fruitfulness of thinking of “logics” as three-part systems composed of a language, a semantics [or model theory] and a deductive system [or proof theory]. For simplicity we take a fixed language for sentential logic [or propositional logics] with a fixed standard two-valued semantics, but we “vary” deductive systems. In the 1960s when this paper was written, it was still widely assumed that a “logic” was “determined” by its theorems, by what it could “prove”. The vague assumption was that once the theorems were fixed, it was somehow automatic which conclusions were derivable from given premises.
The purpose of this paper is to give a method which weakens deriving power of a sentential logic while leaving proving power unchanged. In particular if the method is applied to a strongly complete logic the resultant will not have strong completeness but it will retain weak completeness. Moreover, because of the nature of the method two added facts obtain.
First, weak completeness of the resultant is always an obvious corollary to weak completeness of the operand, so that no new completeness proof need be constructed. Second, failure of strong completeness in the resultant is always obvious, so no multi-valued matrices need be constructed. The method applies to all sentential logics formulated using only axiom schemes and schematically statable rules. These results reaffirm, generalize, and simplify a result of Hiz [Journal of Symbolic Logic, 24 (1959), pp. 193-202].

Errata in Henkin’s 1950 type-theory completeness paper.
  The first paragraph of Leon Henkin’s influential and widely-read article [1] reads as follows.

The first order functional calculus was proved complete by Gödel in 1930. Roughly speaking, this proof demonstrates that each formula of the calculus is a formal theorem which becomes a true sentence under every one of a certain intended class of interpretations of the formal system.

  As it is written the second sentence says that each formula is a provable logical truth: a patent absurdity. Of course, what was intended was more like following, which moves the clause beginning ‘which becomes’ so it modifies ‘formula of the calculus’.

Roughly speaking, this proof demonstrates that each formula of the calculus which becomes a true sentence under every one of a certain intended class of interpretations of the formal system is a formal theorem.

  Taking more advantage of the disclaimer ‘roughly speaking’, he could say:

Roughly speaking, this proof demonstrates that each formula which is true under every interpretation is a theorem.

  Almost 20 years later, Jaakko Hintikka [2] reprinted [1] along with other important articles. Hintikka noted errata in another Henkin article but not in [1]. There are other mistakes in [1]. Moreover, it contains several passages that, if not actual mistakes, are regrettable. We list and analyse glitches in [1] in order to facilitate study of this landmark article. We also discuss the old question of whether, as logicians including Burton Dreben have charged [personal communication], [1]’s title is misleading in that it seems to contradict an easy consequence of Gödel’s 1931 incompleteness paper. The consequence in question is explicitly mentioned in the second paragraph of [1].
[1] LEON HENKIN, Completeness in the theory of types. Journal of Symbolic Logic, vol. 15 (1950) pp. 81–94.
[2] JAAKKO HINTIKKA, Philosophy of Mathematics, Oxford UP, 1969.
Acknowledgements: Mark Brown, William Demopoulos, Thomas Drucker, Rodolfo Ertola, William Frank, Richard Grandy, Idris Samawi Hamid, Allen Hazen, Leonard Jacuzzo, Miguel León, Kristo Miettinen, Joaquin Miller, Frango Nabrasa, David Plache, Sriram Nambiar, José Miguel Sagüillo, Michael Scanlan, Stewart Shapiro, James Smith, Roberto Torretti, Alasdair Urquhart, Albert Visser, George Weaver, George Williams, Stanley Ziewacz, and others—some of which disagree with some of my points or with my way of making them.

In particular, distinguished and accomplished logicians including Allen Hazen, Christopher Menzel, Alasdair Urquhart, and Albert Visser think that Henkin’s expression of completeness is correct as it is—awkwardness notwithstanding. To be clear, they believe and argue that I am wrong to imply that this is a mistake on Henkin’s part.
WHAT IS YOUR VIEW?

We describe a data structure for three-dimensional Nef complexes, algorithms for boolean operations on them, and our implementation of data structure and algorithms. Nef polyhedra were introduced by W. Nef in his seminal 1978 book on polyhedra. They are the closure of half-spaces under boolean operations and can represent non-manifold situations, open and closed boundaries, and mixed dimensional complexes. Our focus lies on the generality of the data structure, the completeness of the algorithms, and the exactness and efficiency of the implementation. In particular, all degeneracies are handled.

Questions of completeness of grasp or certainty are being viewed with suspicion, while human action, mirrored in complete reality, confronts us with limits. Dealing with these issues (as with all others) is tinted by the selected categorial systems through which the empirical phenomena are being judged. The categorial structure of the currently paradigmatic approaches does indeed not allow certainty, especially not when dealing with completeness; this is why the 'modern' formulations of laws are not essentialist, but appear increasingly in stochastic terms. But they do not allow an access to the core of singularities (individual cases), and within these approaches it can not be decided whether the totality is inaccessible per se. - An approach is proposed that does not have these weaknesses. It operates through strictly universally applicable fundamental concepts, in which perspectivity and completeness of grasp can become compatible, and it provides also a clarification of the foundation of the presently paradigmatic approaches, which they cannot afford themselves (issue of ultimate justification). The result is a structure of precisely balanced categorial concepts with heuristic content, which complements the 'modern' formulations of laws and causation (ie reached on an external-empirical basis).
German original:
Fragen um Ganzheitserfassung oder Gewißheit werden beargwöhnt, während menschliches Tun, gespiegelt an der ganzen Wirklichkeit, uns vor Grenzen stellt. Der Umgang mit diesen Fragen ist (wie mit allen anderen) gefärbt von den gewählten Kategoriensystemen, durch die die empirischen Phänomene beurteilt werden. Die Kategorien der derzeit paradigmatischen Ansätze erlauben in der Tat keine Gewißheit, vor allem nicht im Umgang mit der Ganzheit; deshalb sind die 'modernen' Formulierung von Gesetzmäßigkeiten nicht mehr essenzialistisch, sondern zunehmend stochastisch geworden. Sie erlauben aber keine Einsicht in den Kern der Singularitäten (Einzelfälle) und es kann innerhalb dieser Ansätze nicht endgültig entschieden werden, ob die Ganzheit per se unzugänglich sei. — Wir legen einen Ansatz vor, der diese Schwächen nicht hat. Er verläuft in streng universell anwendbaren Grundbegriffen, in denen Perspektivität und Ganzheitserfassung kompatibel ist, und bietet auch eine Klärung der Basis der nun paradigmatischen Ansätze, die diese selbst nicht leisten können (Letztbegründungs-Problematik). Es resultiert eine Struktur von präzis ausgewogenen Leit-Ordnungen mit heuristischem Gehalt, welche die 'modernen' (d.h. auf äußerlich-empirischer Basis erreichten) Formulierungen von Gesetzmäßigkeiten und Ursächlichkeiten komplementär ergänzen.

In this paper we argue that hybrid logic is the deductive setting most natural for Kripke semantics. We do so by investigating hybrid axiomatics for a variety of systems, ranging from the basic hybrid language (a decidable system with the same complexity as orthodox propositional modal logic) to the strong Priorean language (which offers full first-order expressivity). We show that hybrid logic offers a genuinely first-order perspective on Kripke semantics: it is possible to define base logics which extend automatically to a wide ...

We investigate a lattice of conditional logics described by a Kripke type semantics, which was suggested by Chellas and Segerberg – Chellas–Segerberg (CS) semantics – plus 30 further principles. We (i) present a non-trivial frame-based completeness result, (ii) a translation procedure which gives one corresponding trivial frame conditions for arbitrary formula schemata, and (iii) non-trivial frame conditions in CS semantics which correspond to the 30 principles.

Statistical models defined by imposing restrictions on marginal distributions of contingency tables have received considerable attention recently. This paper introduces a general definition of marginal log-linear parameters and describes conditions for a marginal log-linear parameter to be a smooth parameterization of the distribution, and to be variation independent. Statistical models defined by imposing affine restrictions on the marginal log-linear parameters are investigated. These models generalize ordinary log-linear and multivariate logistic models. Sufficient conditions for a log-affine marginal model to be nonempty, and to be a curved exponential family are given. Standard large sample theory is shown to apply to maximum likelihood estimation of log-affine marginal models for a variety of sampling procedures.

Binod CHANDRA TRIPATHY1 and AYHAN ESI2 1Mathematical Sciences Division; Institute of Advanced Study in Science and Technology; Paschim Boragaon; GARCHUK; GUWAHATI-781 035, INDIA 2Inonu University, Science and Art Faculty in Adiyaman, 02040, Adıyaman, ...

Life amazes us by continuous changes out of its material structures themselves. As a phenomenon, the capacity to grow according to an incorporated orderliness and to react according to internal criteria, thus to perform a certain degree of autonomy, distinguishes alive structures from structures of inanimate matter, whose changes are determined fully by external forces. What the changes are attributed to (genes, vital force, eidos, etc.) is another question. The life sciences seek forms of life and thus presuppose a principle of what life is, which is conceptually not clarified yet. Grasping adequately this principle requires a totally secure conceptual basis; our approach is an attempt in this sense. The means for achieving this is a non-compromised analysis of the laws that determine conceptual structures in referring to the natural forms of processual order (laws and forces). The basis is Aristotelian, and the focus on strict completeness, processuality, and agency. The mental process of this analysis yields a set of four fundamental categories that are universally applicable and allow thus in principle a full coverage of the phenomena: 'law', 'force', 'disequilibriabilty of force structures', and 'equilibrium of all forces in a force structure'. They cover in one continuum all processes, from those in material matter to those in consciousness, and are thus relevant for more fields than the life sciences can cover as yet. Following no idée fixe about 'objects', they show the origin of life as not limited to first aminoacid reactions or first living cells.

The (European) Human Sciences are currently in a dilemma: on the one hand their subject matter calls for a complete understanding, while on the other hand the methodology of the mainstream limits overwhelmingly this access. What development is now possible at all? The question is therefore whether universality (completeness of grasp and certainty) is possible on principle or not. Here a systematic proof is provided for this possibility, paving the way for allowing (Europe’s) integrative tradition to continue unfolding.

WHAT ARE MATHEMATICAL LOGICS?

Let ML= (G, D, S) be a mathematical logic with G for grammar, D, for deductive system and S or semantics. The grammar G is often studied separately from D and S and purely grammatical theorems are proved –e.g., that every sentence is uniquely decomposable according to a certain mode of decomposition, that sentencehood in G is decidable and so on.
When the deductive system D is studied apart from the semantics one gets purely proof-theoretic results - e.g., that every deduction is equivalent to a deduction which does not use a given rule of inference, that not all sentences are provable, and so on. When the semantics S is studied apart from the deductive system, one gets purely model-theoretic or semantic results - e.g., that every infinite interpretation is equivalent to a countable one, that certain sets of sentences are or are not satisfiable, that all interpretations satisfying a given set of sentences are isomorphic and so on.
Results which relate the deductive system to the semantics have been called bridge results, by Addison and other the members of the Berkeley School (but this terminology has not caught on). Examples of bridge results are completeness and soundness theorems. A deductive system can only be complete or sound relative to a semantics. It is only a logic as a whole that can be complete or sound. Corcoran 1973 [“Gaps”] pages 28-29
TRUNCATIONS
Semi-logics DL = (G, D), SL = (G, S), and the relational systems RS studied in “universal logic” cannot be regarded as adequate for most contemporary purpose because they lack bridge phenomena. Consider RS = (P A) where P is a set of propositions and A is a set of premise-conclusion arguments whose component propositions are taken from P.
EXPANDSIONS
For bridge theorems we need two sets of arguments, e.g. R2S = (P, A, B) where P is a set of propositions , A is the set of DEDUCIBLE premise-conclusion arguments whose component propositions are taken from P, and B is the set of VALID premise-conclusion arguments whose component propositions are taken from P. In fact even the full relational systems derived from mathematical logics are not sufficient for contemporary purposes: three or more sets of arguments are required.

This paper will investigate the general class theory of polycalculii, and theory originally developed and introduced on the Quora platform as a kind of philosophy of logic and mathematics. The ostensible goal at minima will be to demonstrate the exclusion of an array of class types which edicate the theorem of the polycalculus. The polycalculus is a general method for deriving equivalent calculus’s other than the traditional intelligent calculus. The domain of this discussion is not only mathematics, but also logic, philosophy, and philosophy of science, as well as philosophy of the foundations of mathematics.

We present an L2 discretization of the helium atom using a non-orthogonal Laguerre function basis. The frozen-core approximation is used to calculate the helium atom Hamiltonian. The resulting three-term recurrence relation is a special case of the recurrence relation of the Pollaczek polynomials which is a set of orthogonal polynomials having a non-empty continuous spectrum in addition to an infinite discrete spectrum. The completeness of the helium atom wave functions obtained is studied in terms of weights of the Gauss quadrature.

In this article we introduce the notion of statistical convergence difference sequences of fuzzy real numbers, ¯ c F (∆). We study some properties of the statistically convergent and sta-tistically null difference sequence spaces of fuzzy real numbers, like completeness, solidness, sequence algebra, symmetricity, con-vergence free, nowhere denseness and some inclusion results.

Toda tradicion culinaria define categorias sean estas sociales u ontologicas. La comida y su preparacion son, por lo tanto, importantes signos de diferenciacion y marcadores de alteridad. A partir de los contextos de preparacion e ingesta es posible plantear una distincion basica y rudimentaria entre la comida cotidiana y la ritual; es decir, entre situaciones de simple comensalidad y contextos de comensalidad y ofrenda. En ocasion de los rituales de costumbre de los totonacos de la Huasteca, situados en el extremo norte del estado de Puebla (Mexico), seremos testigos de la manera en que tres clases diferentes de tamales vehiculan relaciones de reciprocidad gracias al uso de un codigo numero preciso. La interrelacion entre este codigo (que supone las nociones de fraccion y completitud) con los animales usados en la preparacion y los contextos de comensalidad y ofrenda de los tamales, sera la base etnografica para proponer la existencia de un triangulo culinario del tamal totonaco.