Fixed‐ versus Variable‐domain Interpretations of Tarski’s Account of Logical Consequence

Mancosu, Paolo

doi:10.1111/j.1747-9991.2010.00324.x

Cited by 18 publications

(10 citation statements)

References 17 publications

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“…In the early 1960s, after the English had been available for years, nevertheless Corcoran and many other logicians still concurred with the Church interpretation. Now to the contrary, it is widely held that Tarski had not yet fully shifted out of the monistic Frege-Russell paradigm (Mancosu 2010b). We too support the monistic, one-universe interpretation of the 1936 Tarski consequence-definition paper.…”

Section: Introductionsupporting

confidence: 49%

The Absence of Multiple Universes of Discourse in the 1936 Tarski Consequence-Definition Paper

Corcoran

Sagüillo²

2011

History and Philosophy of Logic

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Dedicated to Professor Roberto Torretti, philosopher of science, historian of mathematics, teacher, friend, collaborator-on his eightieth birthday. 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 Principiainfluenced pre-WWII Warsaw period and his model-theoretic post-WWII Berkeley period, Tarski's philosophy underwent many other radical changes.

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Section: Introductionsupporting

confidence: 49%

The Absence of Multiple Universes of Discourse in the 1936 Tarski Consequence-Definition Paper

Corcoran

Sagüillo²

2011

History and Philosophy of Logic

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“…Even if you think a belief, say, the belief that NY won, lacks any intrinsic structure, you 4 Though, see Etchemendy (1988) for a famous argument that Tarski did not introduce the standard, modern notion of a model with a varying domain. See Mancosu (2010) for a recent assessment of the ensuing debate.…”

Section: Characterizing Interpretational Validitymentioning

confidence: 99%

“…We already argued that alternative re-interpretations of my beliefs do not seem to help explain the value of a valid inference. But, then, what new option is there for explaining the value of a valid inference in terms of interpretational validity, except by somehow including the extra claim that an interpretationally 24 See Mancosu (2011) for general discussion of explanation within mathematics. valid inference is a substitutionally valid inference?…”

Section: Why Substitutional Validity Provides the Best Explanationmentioning

confidence: 99%

Why Is a Valid Inference a Good Inference?

Dogramaci

2015

Philos Phenomenol Research

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True beliefs and truth-preserving inferences are, in some sense, good beliefs and good inferences. When an inference is valid though, it is not merely truth-preserving, but truthpreserving in all cases. This motivates my question: I consider a Modus Ponens inference, and I ask what its validity in particular contributes to the explanation of why the inference is, in any sense, a good inference. I consider the question under three different definitions of 'case', and hence of 'validity': (i) the orthodox definition given in terms of interpretations or models, (ii) a metaphysical definition given in terms of possible worlds, and (iii) a substitutional definition defended by Quine. I argue that the orthodox notion is poorly suited to explain what's good about a Modus Ponens inference. I argue that there is something good that is explained by a certain kind of truth across possible worlds, but the explanation is not provided by metaphysical validity in particular; nothing of value is explained by truth across all possible worlds. Finally, I argue that the substitutional notion of validity allows us to correctly explain what is good about a valid inference.

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“…This view no doubt stems from work of Tarski, whether it is Tarski's original view or not. For a review of some of the historical work on this and related issues, see Mancosu (2010).…”

mentioning

confidence: 99%

Logical Consequence and Natural Language

Glanzberg¹

2015

Foundations of Logical Consequence

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One of the great successes of the past fifty or so years of the study of language has been the application of formal methods. This has yielded a flood of results in many areas, both of linguistics and philosophy, and has spawned fruitful research programs with names like 'formal semantics' or 'formal syntax' or 'formal pragmatics'.1 'Formal' here often means the tools and methods of formal logic are used (though other areas of mathematics have played important roles as well). The success of applying logical methods to natural language has led some to see the connection between the two as extremely close. To put the idea somewhat roughly, logic studies various languages, and the only special feature of the study of natural language is its focus on the languages humans happen to speak.This idea, I shall argue, is too much of a good thing. To make my point, I shall focus on consequence relations. Though they hardly constitute the full range of issues, tools, or techniques studied in logic, a consequence relation is the core feature of a logic. Thus, seeing how consequence relations relate to natural language is a good way to measure how closely related logic and natural language are. I shall argue here that what we find in natural language is not really logical consequence. In particular, I shall argue that studying the semantics of a natural language is not to study a genuinely logical consequence relation. There is indeed a lot we can glean * Thanks to Jc Beall for valuable discussions of the topics explored in this paper. Versions of this material were presented at the Conference on the Foundations of Logical Consequence sponsored by Arché, St. Andrews,

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Fixed‐ versus Variable‐domain Interpretations of Tarski’s Account of Logical Consequence

Cited by 18 publications

References 17 publications

The Absence of Multiple Universes of Discourse in the 1936 Tarski Consequence-Definition Paper

The Absence of Multiple Universes of Discourse in the 1936 Tarski Consequence-Definition Paper

Why Is a Valid Inference a Good Inference?

Logical Consequence and Natural Language

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