On a fallacy attributed to tarski

In this article I describe and evaluate the debate that surrounds the proper interpretation of Tarski’s account of logical consequence given in his classic 1936 article ‘On the concept of logical consequence’. In the late 1980s Etchemendy argued that the familiar model theoretic account of logical consequence is not to be found in Tarski’s original article. Whereas the contemporary account of logical consequence is a variable‐domain conception – in that it calls for a reinterpretation of the domain of variation of the quantifiers when evaluating logical consequence –, no such reinterpretation is found in Tarski’s original account, which was rather based on a ‘fixed‐domain’ conception. Etchemendy’s claims have sparked a debate on Tarski’s conception of logical consequence with important contributions by, among others, Bach, Bays, Corcoran, Gómez‐Torrente, Mancosu, Ray, Sagüillo, and Sher.

The overgeneration argument attempts to show that accepting second-order validity as a sound formal counterpart of logical truth has the unacceptable consequence that the Continuum Hypothesis is either a logical truth or a logical falsehood. The argument was presented and vigorously defended in John Etchemendy’s The Concept of Logical Consequence and it has many proponents to this day. Yet it is nothing but a seductive fallacy. I demonstrate this by considering five versions of the argument; as I show, each is either unsound or lacks a troubling conclusion.

Debates about logical consequence are typically motivated by the desire to capture ‘the intuitive concept of logical consequence’. The authors of One True Logic believe that there is a mistake here from the start. There is no single, well-behaved target for our precise definitions of logical consequence but rather a historical tangle of intuitions such as formality, necessity, and normativity. The chapter explores this history, and these features in particular, in order to get a clearer view of the target. A number of notable figures in the history of logic are considered: Philo, Bolzano, Quine, and especially Tarski, who receives extensive discussion.

In his classic 1936 essay “On the Concept of Logical Consequence”, Alfred Tarski used the notion of satisfaction to give a semantic characterization of the logical properties. Tarski is generally credited with introducing the model-theoretic characterization of the logical properties familiar to us today. However, in his book, The Concept of Logical Consequence, Etchemendy argues that Tarski's account is inadequate for quite a number of reasons, and is actually incompatible with the standard model-theoretic account. Many of his criticisms are meant to apply to the model-theoretic account as well.

Parametric logic is a framework that generalises classical first-order logic. A generalised notion of logical consequence—a form of preferential entailment based on a closed world assumption—is defined as a function of some parameters. A concept of possible knowledge base—the counterpart to the consistent theories of first-order logic—is introduced. The notion of compactness is weakened. The degree of weakening is quantified by a nonnull ordinal—the larger the ordinal, the more significant the weakening. For every possible knowledge base T, a hierarchy of sentences that are generalised logical consequences of T is built. The first layer of the hierarchies corresponds to sentences that can be obtained by a deductive inference, characterised by the compactness property. The second layer of the hierarchies corresponds to sentences that can be obtained by an inductive inference, characterised by the property of weak compactness quantified by 1. Weaker forms of compactness—quantified by nonnull ordinals—determine higher layers in the hierarchies, corresponding to more complex inferences. The naturalness of the hierarchies built over the possible knowledge bases is attested by fundamental connections with notions from Learning theory and from topology. The naturalness of the hierarchies built over the possible knowledge bases is attested by fundamental connections with notions from Learning theory—classification in the limit, with or without a bounded number of mind changes—and from topology—in reference to the Borel and the difference hierarchies. In this paper, we introduce the key model-theoretic aspects of Parametric logic, justify the concept of the knowledge base, define the hierarchies of generalised logical consequences and illustrate their relevance to Nonmonotonic reasoning. More specifically, we show that the degree of nonmonotonicity that is required to infer a sentence can be characterised by the least nonnull ordinal that quantifies the weakening of compactness used to locate the inferred sentence in the hierarchies.

My long-standing interest in the notion of logical consequence became urgent when I was working on my recent book on higher-order logic (1991). The most common complaint is that second-order logic is out of bounds. as logic, because its consequence relation is not effective. Even though higher order logic is squarely within the prevailing model-theoretic tradition, its set of logical truths is not recursively enumerable-it is not even in the Kleene hierarchy.

Stefania Centrone's book Studien zu Bolzano, published in 2015, is a collection of six German written articles dealing mostly with Bolzano's logic and previously published in several journals. The number of texts is expanded to seven by a new study on Bolzano's concept of logical consequence. Bolzano's Contributions are discussed in the first essay right from the beginning. For Bolzano, the proper method of mathematics is the method of proper scientific treatment of all objective regions. However, there are ambivalences, weaker points. Centrone points out that Bolzano's logic is not able to satisfy the principles according to which consequences should be founded in their grounds. Bolzano states the principles mentioned earlier only after he attempts to prove that there are first grounding propositions which necessarily have simple concepts as subjects and predicates. Centrone points out that Bolzano considers "clearness" as an internal property of the ideas, while for Leibniz, it consists of a relation between an idea and its object.

The Concept of Logical Consequence

Jean Buridan developed his theory of consequence within a semantical framework compatible with what we now call token-based semantics. In his Treatise on Consequences and Sophismata, Buridan showed that the notion of consequence accepted by the logicians of his time (that the antecedent cannot be true without the consequent being true) fails when we consider inferences such as: (1) ‘No proposition is negative, therefore, no ass is running’; and (2) ‘Every proposition is affirmative, therefore, no proposition is negative’. Our analysis of Buridan’s semantic and logical theory shows that: (a) for Buridan, time was a fundamental needed concept to adequately define the notion of consequence; (b) sentences like ‘No proposition is negative’ — appearing in (1) and (2) — are temporal indexicals; (c) these indexicals appealed to Buridan because they suggest a new type of validity, that has been called contextual validity by Blackburn and Jørgensen, and which was originally discussed by Kamp and Kaplan in the twentieth century. Finally, we offer a reconstruction of Buridan’s theory of consequence in terms of a contextual hybrid logic, a logical-symbolic apparatus that allows us to assess Buridan’s logical and philosophical contributions in contemporary terms and contribute to the current discussion on the topic.

The supervaluationist theory of vagueness provides a notion of logical consequence that is akin to classical consequence. In the absence of a definitely operator, supervaluationist consequence coincides with classical consequence. In the presence of ‘definitely’, however, supervaluationist logic gives raise to counterexamples to classically valid patterns of inference. Foes of supervaluationism emphasize the last result to argue against the supervaluationist theory. This paper shows a way in which we might obtain systems of deduction adequate for supervaluationist consequence based on systems of deduction adequate for classical consequence. Deductions on the systems obtained this way adopt a completely classical form with the exception of a single step. The paper reviews (at least part of) the discussion on the non-classicality of supervaluationist logic under the light of this result.KeywordsVaguenessSupervaluationismGlobal ValidityDeductive SystemsLogical Consequence

One of the main functions of legal reasoning is to determine the truth-value of legal statements, i.e., of statements that assert that certain actions are legally required or permitted. In general, the notion of legal system provides a useful framework, which allows us to distinguish between legal norms and other socially relevant norms. Two concepts of legal system are often employed in legal discourse. On the one hand, it is usually claimed that the material provided by a certain authority has systematic structure, i.e. the norms are structured by means of specific relationships. e.g. hierarchical relationships. In this sense, law is an institutional system. On the other hand, it is frequently asserted that legal norms can be reconstructed systematically starting from the notion of logical consequence. In this sense, law can be regarded as a deductive system. Legal reasoning, then, would use in an explicit or implicit way both notions of normative system in the process of establishing the truth conditions of legal statements. However, this traditional conception of legal reasoning and normative systems has been object of important challenges. In this paper. I consider two groups of arguments against the systematic reconstruction of law:

Can Tarski's definition of truth be taken to be an analysis of truth – i.e., a method of defining formal truth predicates that are capable of playing the role of truth in all theoretical contexts in which that notion is needed? It is argued that although Tarski's truth predicates can play many of the roles demanded of truth, they cannot play the role of truth in theories of meaning and interpretation. Crucial to the argument is the observation that there is a conceptual connection between our ordinary notions of truth and meaning, in virtue of which statements of the truth conditions of sentences provide some information about their meanings. Since there is no corresponding connection between Tarski's notion of truth and the notion of meaning, statements of the Tarski‐truth conditions of sentences provide no information about meaning. The chapter closes with a detailed defense of Tarski's model‐theoretic analysis of logical truth and logical consequence against John Etchemendy's objections to it.

With parametric logic, we propose a unified approach to deduction and induction, both viewed as particular instances of a generalized notion of logical consequence. This generalized notion of logical consequence is Tarskian, in the sense that if a sentence φ is a generalized logical consequence of a set T of premises, then the truth of T implies the truth of φ. So in particular, if φ can be induced from T, then the truth of T implies the truth of φ. The difference between deductive and inductive consequences lies in the process of deriving the truth of φ from the truth of T.

In “Logical consequence: A defense of Tarski” (Journal of Philosophical Logic, vol. 25, 1996, pp. 617–677), Greg Ray defends Tarski"s account of logical consequence against the criticisms of John Etchemendy. While Ray"s defense of Tarski is largely successful, his attempt to give a general proof that Tarskian consequence preserves truth fails. Analysis of this failure shows that de facto truth preservation is a very weak criterion of adequacy for a theory of logical consequence and should be replaced by a stronger absence-of-counterexamples criterion. It is argued that the latter criterion reflects the modal character of our intuitive concept of logical consequence, and it is shown that Tarskian consequence can be proved to satisfy this criterion for certain choices of logical constants. Finally, an apparent inconsistency in Ray"s interpretation of Tarski"s position on the modal status of the consequence relation is noted.

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