arXiv subjects
Publications and source records attributed to Eduardo Mizraji.
In this brief article, we present the formula created by Rafael Barrett in 1903 in a note to Henri Poincaré, which remained unknown for decades. Discovered in the 1930s by a Uruguayan mathematician, this formula was published and analyzed in a journal published in Montevideo in 1935. In this study, we present Barrett's formula and analyze a challenge it could pose.
In 1929 Jan Lukasiewicz used, apparently for the first time, his Polish notation to represent the operations of formal logic. This is a parenthesis-free notation, which also implies that logical functions are operators preceding the variables on which they act. In the 1980s, within the framework of research into mathematical models on the parallel processing of neural systems, a group of operators emerged -- neurally inspired and based on matrix algebra -- which computed logical operations automatically. These matrix operators reproduce the order of operators and variables of Polish notation. These logical matrices can also generate a three-valued logic with broad similarities to Lukasiewicz's three-valued logic. In this paper, a parallel is drawn between relevant formulas represented in Polish notation, and their counterparts in terms of neurally based matrix operators. Lukasiewicz's three-valued logic, shown in Polish notation has several points of contact with what matrices produce when they process uncertain truth vectors. This formal parallelism opens up scientific and philosophical perspectives that deserve to be further explored.
The square root of Not is a logical operator of importance in quantum computing theory and of interest as a mathematical object in its own right. In physics, it is a square complex matrix of dimension 2. In the present work it is a complex square matrix of arbitrary dimension. The introduction of linear algebra into logical theory has been enhanced in recent decades by the researches in the field of neural networks and quantum computing. Here we will make a brief description of the representation of logical operations through matrices and we show how general expressions for the two square roots of the Not operator are obtained. Then, we explore two topics. First, we study an extension to a non-quantum domain of a short form of Deutsch's algorithm. Then, we assume that a root of Not is a matrix extension of the imaginary unit i, and under this idea we obtain fully matrix versions for the Euler expansions and for the representations of circular functions by complex exponentials.
In this work we investigate the representation of counterfactual conditionals using the vector logic, a matrix-vectors formalism for logical functions and truth values. Inside this formalism, the counterfactuals can be transformed in complex matrices preprocessing an implication matrix with one of the square roots of NOT, a complex matrix. This mathematical approach puts in evidence the virtual character of the counterfactuals. This happens because this representation produces a valuation of a counterfactual that is the superposition of the two opposite truth values weighted, respectively, by two complex conjugated coefficients. This result shows that this procedure gives an uncertain evaluation projected on the complex domain. After this basic representation, the judgment of the plausibility of a given counterfactual allows us to shift the decision towards an acceptance or a refusal. This shift is the result of applying for a second time one of the two square roots of NOT.
A variety of problems emerged investigating electronic circuits, computer devices and cellular automata motivated a number of attempts to create a differential and integral calculus for Boolean functions. In the present article, we extend this kind of calculus in order to include the semantic of classical logical operations. We show that this extension to logics is strongly helped if we submerge the elementary logical calculus in a matrix-vector formalism that naturally includes a kind of fuzzy-logic. In this way, guided by the laws of matrix algebra, we can construct compact representations for the derivatives and the integrals of logical functions. Inside this semantic-algebraic calculus, we obtain expressions for the derivatives of some of the basic logical operations and show the general way to obtain the derivatives of any well-formed formula of propositional calculus. We show that some of the basic tautologies (Excluded middle, Modus ponens, Hypothetical syllogism) are members of a kind of hierarchical system linked by the differentiation algorithm. In addition using the logical derivatives we show that relatively complex formulas can collapse in simple expressions that reveal clearly their hidden logical meaning. The search for the antiderivatives produces naturally an integral calculus. Within this logical formalism an indefinite integral can always be found for any logical expression. Moreover, particular integrals can be constructed based on detachment properties that lead to logical expressions of growing complexity. We show that these particular integrals have some similarities with the 'generalizing deduction' procedures investigated by Lukasiewicz.
Natural languages can express some logical propositions that humans are able to understand. We illustrate this fact with a famous text that Conan Doyle attributed to Holmes: 'It is an old maxim of mine that when you have excluded the impossible, whatever remains, however improbable, must be the truth'. This is a subtle logical statement usually felt as an evident truth. The problem we are trying to solve is the cognitive reason for such a feeling. We postulate here that we accept Holmes' maxim as true because our adult brains are equipped with neural modules that naturally perform modal logical computations.
Latent Semantic Analysis (LSA) is a well known method for information retrieval. It has also been applied as a model of cognitive processing and word-meaning acquisition. This dual importance of LSA derives from its capacity to modulate the meaning of words by contexts, dealing successfully with polysemy and synonymy. The underlying reasons that make the method work are not clear enough. We propose that the method works because it detects an underlying block structure (the blocks corresponding to topics) in the term by document matrix. In real cases this block structure is hidden because of perturbations. We propose that the correct explanation for LSA must be searched in the structure of singular vectors rather than in the profile of singular values. Using Perron-Frobenius theory we show that the presence of disjoint blocks of documents is marked by sign-homogeneous entries in the vectors corresponding to the documents of one block and zeros elsewhere. In the case of nearly disjoint blocks, perturbation theory shows that if the perturbations are small the zeros in the leading vectors are replaced by small numbers (pseudo-zeros). Since the singular values of each block might be very different in magnitude, their order does not mirror the order of blocks. When the norms of the blocks are similar, LSA works fine, but we propose that when the topics have different sizes, the usual procedure of selecting the first k singular triplets (k being the number of blocks) should be replaced by a method that selects the perturbed Perron vectors for each block.