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Vector logic is a mathematical model of the propositional calculus in which the logical variables are represented by vectors and the logical operations by matrices. In this framework, many tautologies of classical logic are intrinsic identities between operators and, consequently, they are valid beyond the bivalued domain. The operators can be expressed as Kronecker polynomials. These polynomials allow us to show that many important tautologies of classical logic are generated from basic operators via the operations called Type I and Type I1 products. Finally, it is described a matrix version of the Fredkin gate that extends its properties to the many-valued domain, and it is proved that the filtered Fredkin operators are second degree Kronecker polynomials that cannot be generated by Type I or Type I1 products. MathematicsSubject Classification: 03B05, 03B50. '11 would like to thank Professor M. H. OTERO and my colleagues L. ACERENZA, F. ALVAREZ, ')=-mail: mizrsj@fcien.edu.uy J. HERN~NDEZ and L. SETARO for m y useful ~~S C U S S~O M . Eduardo Mizraii In 1858, CAYLEY published his description of matrix operators; some years later CHARLES PEIRCE suggested the interest of this formalism for the theory of logic, and in 1948 COPILOWICH [4] published an application of matrix algebra to the calculus of relations. The potentialities of matrices as operators emerge from the existence of a well-defined repertory of algebraic operations among them. Some years ago, matrix operators have been shown to be extremely well adapted to represent the properties of biological associative memories (ANDERSON [l], KO-HONEN [8]). Recently, during an investigation of the properties of context-dependent associative memories, a simple vector-matrix formalism to represent the propositional calculus has been found (MIZRAJI [lo, 111). The basic feature of this vector logic is the mapping of the truth values on the elements of a vector space and the use of Kroneker products to represent binary functions. In vector logic, complex expressions produce an order of operators and variables identical with the order obtained in the Polish notation; nevertheless, this operator formalism allows further factorizations due to the properties of the Kronecker products. The operators of the basic vector logic, defined from bivalent logic, are able to compute a kind of fuzzy truth values, generating a many-valued logic (MIZRAJI [Ill). This property allows the construction of truth-functional modal operators (Mizraji [12]).In classical logic, the tautologies involve operations that act on bivalent variables. In the present article we show that, in the framework of vector logic, some of these tautologies are intrinsic to operators and that they are not dependent on the bivalent character of the variables. We begin by defining the basic matrix operators of vector logic. Then we define the Kronecker polynomials, a matrix structure that embraces a large variety of logical operations. Finally, we describe the matrix operator that corresponds to the Fredkin gate and...

Several psychiatric and neurological conditions affect the semantic organization and content of a patient's speech. Specifically, the discourse of patients with schizophrenia is frequently characterized as lacking coherence. The evaluation of disturbances in discourse is often used in diagnosis and in assessing treatment efficacy, and is an important factor in prognosis.Measuring these deviations, such as "loss of meaning" and incoherence, is difficult and requires substantial human effort. Computational procedures can be employed to characterize the nature of the anomalies in discourse. We present a set of new tools derived from network theory and information science that may assist in empirical and clinical studies of communication patterns in patients, and provide the foundation for future automatic procedures. First we review information science and complex network approaches to measuring semantic coherence, and then we introduce a representation of discourse that allows for the computation of measures of disorganization. Finally we apply these tools to speech transcriptions from patients and a healthy participant, illustrating the implications and potential of this novel framework.