Jose Saguillo

This paper discusses the history of the confusion and controversies over whether the definition of consequence presented in the 11-page 1936 Tarski consequence-definition paper is based on a monistic fixed-universe framework?like Begriffsschrift and Principia Mathematica. Monistic fixed-universe frameworks, common in pre-WWII logic, keep the range of the individual variables fixed as the class of all individuals. The contrary alternative is that the definition is predicated on a pluralistic multiple-universe framework?like the 1931 Gödel incompleteness paper. A pluralistic multiple-universe framework recognizes multiple universes of discourse serving as different ranges of the individual variables in different interpretations?as in post-WWII model theory. In the early 1960s, many logicians?mistakenly, as we show?held the ?contrary alternative? that Tarski 1936 had already adopted a Gödel-type, pluralistic, multiple-universe framework. We explain that Tarski had not yet shifted out of the monistic, Frege?Russell, fixed-universe paradigm. We further argue that between his Principia-influenced pre-WWII Warsaw period and his model-theoretic post-WWII Berkeley period, Tarski's philosophy underwent many other radical changes

Tarski's 1936 paper, “On the concept of logical consequence”, is a rather philosophical, non-technical paper that leaves room for conflicting interpretations. My purpose is to review some important issues that explicitly or implicitly constitute its themes. My discussion contains four sections: terminological and conceptual preliminaries, Tarski's definition of the concept of logical consequence, Tarski's discussion of omega-incomplete theories, and concluding remarks concerning the kind of conception that Tarski's definition was intended to explicate. The third section involves subsidiary issues, such as Tarski's discussion concerning the distinction between material and formal consequence and the important question ofthe criterion for distinguishing between logical and non-logical terms.§1. Preliminaries. In this paper an argument is a two-part system composed of a set of propositions P and a single proposition c. The expression ‘c is a [logical] consequence of P’ is used with the same meaning as the expression ‘c is [logically] implied by P’. The expressions ‘is a logical consequence of’ and the converse ‘implies’ are relational. Often, I shall be talking in the same sense of validity of an argument. Validity is a property of arguments; an argument with premise-set P and conclusion c is valid if and only if P implies c; i.e., c is a logical consequence of P. Notice that this notion of argument is strictly ontic; it does not involve any agent that thinks, determines or establishes that a given proposition is or is not a consequence of a given set of propositions.

This article discusses two coextensive concepts of logical consequence that are implicit in the two fundamental logical practices of establishing validity and invalidity for premise-conclusion arguments. The premises and conclusion of an argument have information content (they ?say? something), and they have subject matter (they are ?about? something). The asymmetry between establishing validity and establishing invalidity has long been noted: validity is established through an information-processing procedure exhibiting a step-by-step deduction of the conclusion from the premise-set. Invalidity is established by exhibiting a countermodel satisfying the premises but not the conclusion. The process of establishing validity focuses on information content; the process of establishing invalidity focuses on subject matter. Corcoran's information-theoretic concept of logical consequence corresponds to the former. Tarski's model-theoretic concept of logical consequence formulated in his famous 1936 no-countermodels definition corresponds to the latter. Both are found to be indispensable for understanding the rationale of the deductive method and each complements the other. This study discusses the ontic question of the nature of logical consequence and the epistemic question of the human capabilities presupposed by practical applications of these two concepts as they make validity and invalidity accessible to human knowledge

Each science has its own domain of investigation, but one and the same science can be formalized in different languages with different universes of discourse. The concept of the domain of a science and the concept of the universe of discourse of a formalization of a science are distinct, although they often coincide in extension. In order to analyse the presuppositions and implications of choices of domain and universe, this article discusses the treatment of omega arguments in three very different formalizations of arithmetic. In Peano's formalization the domain is a restricted class of individuals, while the universe of discourse is the unrestricted class of all individuals. In Gödel's formalization the domain is a restricted class of individuals as in Peano's formalization, but the universe of discourse coincides with the domain. In Whitehead-Russell's formalization the domain is a class of logical notions in Tarski's sense, that are necessarily not individuals, whereas the universe of discourse is the unrestricted class of individuals as in Peano's formalization. The present approach emphasizes the viewpoint that the universe of discourse of a given discourse is important in determining which propositions are expressed by which sentences

This is a survey paper of approaches to the concept of logical implication. Roughly stated the main motivation of these approaches is to provide a necessary and sufficient condition for a set of propositions to logically imply a single proposition. In regard to their affinities these approaches are grouped into two: the transformational conception and the informational conception. Some approaches in each conception are philosophical and some are mathematical in character, their common assumption being that they reflect a previous intuition – in some cases – already at work in classical mathematics. The first part of the paper is devoted to approaches within the transformational conception of logical implication emphasizing the contributions of Bolzano, Russell, Tarski, Quine and current model – theory. The second part of the paper discusses approaches within the informational conception of logical implication. Carnap and Bar Hillel’s extrinsic approach and Corcoran’s intrinsic approach developed in his information-theoretic logic are respectively compared and some of their philosophical underpinnings are brought to light. The paper concludes with some final remarks concerning the ontic question of the nature of logical implication and the epistemic question of the human access to this relation in both conceptions of logical implication. The article also attempts to connect the views of these representative thinkers with the current philosophical debate on the nature of logical consequence and the role of set-theoretic model theory in philosophy of logic

I discuss Putnam’s conception of logical truth as grounded in his picture of mathematical practice and ontology. i begin by comparing Putnam’s 1971 Philosophy of Logic with Quine’s homonymous book. Next, Putnam’s changing views on modality are surveyed, moving from the modal pre-formal to the de-modalized formal characterization of logical validity. Section three suggests a complementary view of Platonism and modalism underlying different stages of a dynamic mathematical practice. The final section argues for the pervasive platonistic conception of the working mathematician.

En este artículo una paradoja es un tipo de argumentación con respecto a un sujeto X (sea un individuo o una comunidad) en un determinado momento T. Muchas argumentaciones paradójicas tienen lugar en el desarrollo histórico y práctico de las ciencias. Algunas suponen grandes sorpresas acompañadas de profundas crisis, como ocurre con las llamadas antinomias. Solventar, y eventualmente resolver, una paradoja en este sentido supone avances revolucionarios que se obtienen al precio de rechazar creencias previamente asumidas o tenidas por verdaderas por la comunidad científica. El concepto de paradoja que se propone es relativo a sujetos y dinámico en el tiempo. Una argumentación que resulta paradójica para X pudiera no ser paradójica para Y, simplemente porque X e Y no tienen por qué compartir las mismas creencias. Asimismo, X puede descubrir que tiene una paradoja en T, y dejar de tenerla ulteriormente en virtud de un cambio de sus creencias.

This paper discusses the history of the confusion and controversies over whether the definition of consequence presented in the 11-page Tarski consequence-definition paper is based on a monistic fixed-universe framework—like Begriffsschrift and Principia Mathematica. Monistic fixed-universe frameworks, common in pre-WWII logic, keep the range of the individual variables fixed as ‘the class of all individuals’. The contrary alternative is that the definition is predicated on a pluralistic multiple-universe framework—like the Gödel incompleteness paper. A pluralistic multiple-universe framework recognizes multiple universes of discourse serving as different ranges of the individual variables in different interpretations—as in post-WWII model theory. In the early 1960s, many logicians—mistakenly, as we show—held the ‘contrary alternative’ that Tarski had already adopted a Gödel-type, pluralistic, multiple-universe framework. We explain that Tarski had not yet shifted out of the monistic, Frege-Russell, fixed-universe paradigm. We further argue that between his Principia-influenced pre-WWII Warsaw period and his model-theoretic post-WWII Berkeley period, Tarski’s philosophy underwent many other radical changes.