On the Problem of Extending Matrix Semantics Sdequate to Classical Implicative Logic to Matrix Xemantics Adequate to Classical Conjunctive-Implicative Logic

В (Попов 2019) дан перечень всех логических матриц, носитель каждой из которых есть {1, 1/2, 0} и выделенное множество каждой из которых есть {1}, адекватных классической импликативной логике. В частности, этому перечню принадлежат логические матрицы ⟨{1, 1/2, 0}, {1}, ⊃ (1, 0, 0, 1)⟩ и ⟨{1, 1/2, 0}, {1}, ⊃ (1/2, 0, 0, 1/2)⟩. Настоящая статья содержит построение бинарной операции & на {1, 1/2, 0} и доказательство того, что ⟨{1, 1/2, 0}, {1}, &, ⊃ (1, 0, 0, 1)⟩ есть L&⊃ -матрица, адекватная классической конъюнктивно-импликативной логике, а также доказательство того, что не существует операции ψ, для которой ⟨{1, 1/2, 0}, {1}, ψ, ⊃ (1/2, 0, 0, 1/2)⟩ есть L&⊃ -матрица, адекватная классической конъюнктивно-импликативной логике. In (Popov 2019), a list of all logical matrices is given, the carrier of each of which is {1, 1/2, 0} and the designated set of each of which is {1}, adequate to classical implicative logic. In particular, to this list belong the logical matrices ⟨{1, 1/2, 0}, {1}, ⊃ (1, 0, 0, 1)⟩ and ⟨{1, 1/2, 0}, {1}, ⊃ (1/2, 0, 0, 1/2)⟩. This article contains the construction of the binary operation & on {1, 1/2, 0} and the proof that ⟨{1, 1/2, 0}, {1}, &, ⊃ (1, 0, 0, 1)⟩ there is an L&⊃ -matrix adequate to the classical conjunctive-implicative logic, as well as a proof that there is no operation ψ for which ⟨{1, 1/2, 0}, {1}, ψ, ⊃ (1/2, 0, 0, 1/2)⟩ is an L&⊃ -matrix that is adequate to the classical conjunctive-implicative logic.

Novel Approach to Geological Modeling with Combination of Machine Learning

<p>This work shows that the traditional version of geological models of oil and gas fields obtained by a computer approach is not the only possible one and it prevents the development of modeling as a whole, since it is not truly mathematical.</p><p>Given that computers do not work with images, but with numbers, a novel approach is presented for the construction of truly mathematical geological models. The proposed model has an unusual appearance and is not intended for visual analysis, but it is more effective for forecasting. The mathematical basis of the novel approach is the cascades of fuzzy-logical matrices, which are formed from spatial coordinates and considered geological parameters.</p><p>Suppose that for each point in the geological grid there is a coordinate vector, in the simplest case these are the lateral coordinates X and Y, as well as the vertical coordinate Z. There is also a set of points (wells) at which the specified coordinates and the values of considered geological parameter, for example, porosity or oil saturation are determined. If some seismic parameter is added to them, which can be taken from grids constructed according to seismic data at the points of the wells, then four coordinates become available.</p><p>Preliminary, all considered geological parameters should be normalized in the range from -1.0 to + 1.0 in order to standardize and equalize them.</p><p>Four coordinates give six independent pairs. A matrix is constructed for each of these pairs. The matrix size can be different - from 100 per 100 to 1000 per 1000.</p><p>Next, the values of the considered geological parameter at the well points determined by four coordinates are applied to these matrices. Certainly, such points are much smaller than the points in the matrix, therefore, to fill the entire polygon of the matrix, the interpolation method is used, based on the idea of the lattice Boltzmann equations.</p><p>The number of fuzzy-logical matrices in one geological model can reach several hundreds.</p><p>Using the obtained matrices, one can construct membership functions and predict the values of the selected geological parameters, as well as the distribution of initial hydrocarbon reserves or the effectiveness of new drilling at the field.</p><p>The novel approach to geological modeling based on the cascades of fuzzy-logical matrices may seem complicated. However, the calculation of these cascades is carried out completely automatically, since they are the truly mathematical functions, and not the illustrations of the geological structure of the filed, and they are directly used in forecasting calculations.</p><p>The cascades of fuzzy-logical matrices can be considered as a new form of machine learning algorithms, for which it is advisable to use big data sets. It opens up the additional possibilities for the application of machine learning methods in geological modeling of oil and gas fields with conventional and unconventional reserves.</p>

Autoimmune Disease, Deceptional vs Logical Matrices & Informational Fidelity

The continuous interplay between concreteness & randomness, the incessant interchange of conceivable, perceivable, comprehensible with the unknown, the mysterious, the abstruse & the consequential fractals from which infinite geometrical nexuses of bonds, forms of connections, pathways & associations appear as mazy combinations in a single cluster of seconds, with the true values alongside the false values to get involved & entangled to alterate, exchange perpetually their bipolar products to emerged matrices of obscure data, in which their noisy action & omnidirectional behavior of the reverberating echo of the abstract result confuses, bias & warps into one the physio-logical with the abnormal, the real with the mirroring, the tangible with the imaginery, the rational with the aberrant, the conscious with the subconscious, the psyche with the somatic, developing as a result an unconscious field of causative potentiallity, a kymatic burst of unexplained interactions & interpretations, which swirl the embedded hidden elements of distortion & delusion, in the end this horrorous deceptive complicated morphoma comes as the ultimate form of abyssal functioning against the humble simplicity and substract expression of the luminous linear true path of taintless logic.

Modal logic with non-deterministic semantics: Part I—Propositional case

Dugundji proved in 1940 that most parts of standard modal systems cannot be characterized by a single finite deterministic matrix. In the eighties, Ivlev proposed a semantics of four-valued non-deterministic matrices (which he called quasi-matrices), in order to characterize a hierarchy of weak modal logics without the necessitation rule. In a previous paper, we extended some systems of Ivlev’s hierarchy, also proposing weaker six-valued systems in which the (T) axiom was replaced by the deontic (D) axiom. In this paper, we propose even weaker systems, by eliminating both axioms, which are characterized by eight-valued non-deterministic matrices. In addition, we prove completeness for those new systems. It is natural to ask if a characterization by finite ordinary (deterministic) logical matrices would be possible for all those Ivlev-like systems. We will show that finite deterministic matrices do not characterize any of them.

Discriminant Structures Associated to Matrix Semantics

In this paper we show a method to characterize logical matrices by means of a special kind of structures, called here discriminant structures for this purpose. Its definition is based on the discrimination of each truthvalue of a given (finite) matrix M = (A, D), w.r.t. its belonging to D. From this starting point, we define a whole class SM of discriminant structures. This class is characterized by a set of Boolean equations, as it is shown here. In addition, several technical results are presented, and it is emphasized the relation of the Discriminant Structures Semantics (D.S.S) with other related semantics such as Dyadic or Twist-Structure.

On the ‘classical’ operations in three-valued logics

The general aim of the present paper is to provide the analysis of the connection between proof-theoretical and functional properties of certain logical matrices. To be more precise, we consider the class of three-valued matrices that induce the classical consequence relation and show that their operations always constitute a subset of one of the maximal classes of functions, which preserve non-trivial equivalence relations. We use a matrix with the single designated value as a sample for in-depth analysis, and generalize the results to suit other cases. Furthermore, on the basis of obtained results we conclude the paper with methodological considerations concerning the nature and interpretation of the truth-values in logical matrices.

Equality of consequence relations in finite-valued logical matrices

In this paper the procedure is presented that allows to determine in finite number of steps if consequence relations in two finite-valued logical matrices for propostional language L are equal.