Rolando Chuaqui

Pragmatic Truth and Approximation to Truth

There are several conceptions of truth, such as the classical correspondence conception, the coherence conception and the pragmatic conception. The classical correspondence conception, or Aristotelian conception, received a mathematical treatment in the hands of Tarski (cf. Tarski [1935] and [1944]), which was the starting point of a great progress in logic and in mathematics. In effect, Tarskis semantic ideas, especially his semantic characterization of truth, have exerted a major influence on various disciplines, besides logic and mathematics; for instance, linguistics, the philosophy of science, and the theory of knowledge. The importance of the Tarskian investigations derives, among other things, from the fact that they constitute a mathematical, formal mark to serve as a reference for the philosophical (informal) conceptions of truth. Today the philosopher knows that the classical conception can be developed and that it is free from paradoxes and other difficulties, if certain precautions are taken. We believe that is not an exaggeration if we assert that Tarskis theory should be considered as one of the greatest accomplishments of logic and mathematics of our time, an accomplishment which is also of extraordinary relevance to philosophy, as we have already remarked. In this paper we show that the pragmatic conception of truth, at least in one of its possible interpretations, has also a mathematical formulation, similar in spirit to that given by Tarski to the classical correspondence conception.

Free-variable axiomatic foundations of infinitesimal analysis: a fragment with finitary consistency proof

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An equational deductive system for the differential and integral calculus

This article presents an outline of a system of equational deductions in the calculus that can be the basis of a computer program. We intend that our system be as faithful as possible to the usual (Leibniz) calculus notation and inferences. The language, however, is restricted mainly to equations, so the system is not sufficient to provide a basis for proving theorems, but we think it is sufficient for solving the usual exercises in the calculus of one variable. (Concerning theorem proving in the calculus see [4].) Another limitation of the system proposed, is that it does not include the algebraic techniques for numerical solutions of systems of equations. Thus, after deducing an equation or system of equations in the system, the usual algebraic techniques should be used for solving them. Although the incorporation of these methods into the derivation system presents no essential difficulty, we decided not to include them, since i i is well known how they can be implemented, and it would complicate our system unduly. The work reported here is what may be called an excercise in “descriptive logic.” Instead of translating mathematical sentences into an artificial logical. langlage, we are trying to reproduce as faithfully as possible the usual calculus equations. In order to prove the soundness of the system we provide a translation into set theory. The main motivation for the system presented here is the need to develop a system of derivations cjf equation which is sound and copes with the peculiarities of the usual calculus notation. These Peculiarities make it different from the usual logical systems. In this introduction, we will discuss these features mainly through examples. In a normal algebraic situation, if we have an equation t = s, we can deduce without any problem, say, u + t = u + s. This is an example of the rule of replacement used in equational logic. In the calculus, however, we cannot always deduce dt/dz = ds/dx from t = s. For instance, if we assume x = 1, and we deduce dx/dx = dl/&, we obtain the contradiction 1 = O . It is perfectly consistent, however, to differentiate both sides of the equation x = 1 with respect to a different variable t . We then obtain, dz/dt = O. This only means that x is a constant function of t (see Example 1, below). The following example shows the need for the rule that allows for the differentiation of two sides of an equation. When we define the function f(s) = &?-f i , for all x, we should be able to deduce that f‘(z) = O, for all z. We can easily see that we cannot obtain f‘(0) = O , by employing the usual rules for derivatives (the rule for differentiating a sum, and the chain rule, for instance), since the derivative sf 0 does not exist at O. We can obtain f’(0) = O, however, by differentiating both sides

An elementary geometric nonstandard proof of the Jordan curve theorem

We give an elementary proof, using nonstandard analysis, of the Jordan curve theorem. We also give a nonstandard generalization of the theorem. The proof is purely geometrical in character, without any use of topological concepts and is based on a discrete finite form of the Jordan theorem, whose proof is purely combinatorial.Some familiarity with nonstandard analysis is assumed. The rest of the paper is self-contained except for the proof a discrete standard form of the Jordan theorem. The proof is based on hyperfinite approximations to regions on the plane.

A Semantical definition of Probability

Publisher Summary This chapter presents a new definition of a probability measure in semantical terms, based on the interpretation of probability relating to truth. The chapter aims to attain a definition of a probability measure, satisfying the axioms of the calculus that can account for almost all of its applications in current scientific and statistical practice. The chapter presents a general description of the procedures involved in probability measure and some mathematical preliminaries and the formal definition of the simple probability structures and the corresponding equiprobability relation—notions that arise when there are no sequences of outcomes.. The chapter explains the necessary and sufficient conditions for the existence of a probability measure compatible with an equivalence relation having the same properties as the equal likelihood relation previously defined and the necessary and sufficient conditions for the existence of a measure on a field of sets, invariant under a group of transformations. Compound probability structures and ways of defining a probability measure for them are discussed and examples and methods for applying the probability models are provided in the chapter.

Foundations of Statistical Methods Using A Semantical Definition of Probability

The purpose of this paper is to construct models for the main methods of statistical inference and estimation using the semantical definition of Probability introduced in the authors A semantical definition of Probability appearing in Non-classical Logics, Model Theory and Computability, North-Holland, 1977, pp. 135–167. Using the simple probability structures introduced there as building blocks, a new type of compound probability structures (cps) is defined. Different forms of cps give an account of the classical frequency methods, Bayesian methods, and stochastic processes.

Random Sequences and Hypotheses Tests

This presents a characterization of random sequences, in the sense of Kolmogorov, Chaitin and Martin-Lof, using the notion of hypotheses tests developed in (Chuaqui 1991).

The geometry of legal principles

We discuss several possible legal principles from the standpoint of Bayesian decision theory. In particular, we show that a compelling legal principle implies compatibility with decisions based on maximizing the expected utility.

Axiomatizations for σ-Additive Measurement Structures

In this paper, extensive measurement structures in the sense of Krantz, Luce, Suppes and Tversky, Foundations of Measurement I , Academic Press 1971 are discussed. In particular, conditions are obtained for the representation, preserving least upper bounds, of these structures. The proofs use some theorems of Cardinal Algebras obtained previously by Chuaqui. In the last section, an improvement of the main theorems of Villegas, On Qualitative Probability σ - Algebras , Ann. Math. Stat. 35 (1964) 1787–1796 is presented.