Definition Of Logical Consequence Research Articles

On a Definition of Logical Consequence

Bilateralists, who accept that there are two primitive speech acts, assertion and denial, can offer an attractive definition of consequence: Y follows from X if and only if it is incoherent to assert all formulas X and to deny all formulas Y. The present paper argues that this definition has consequences many will find problematic, amongst them that truth coincides with assertibility. Philosophers who reject these consequences should therefore reject this definition of consequence.

Material Consequence and Formal Grounding

According to Alfred Tarski’s classical definition, logical consequence is necessary and formal. This paper focuses on the question: In what sense (if any) is material consequence a logical relation? For Tarski, material consequence has no modal force. Treating all terms (of a language with a fixed domain) as logical, he reduces logical consequence to material consequence. Thus, Tarskian material consequence seems to be a logical oxymoron designed to emphasize the importance of the distinction between logical and extra-logical terms for the definition of logical consequence. Historically, however, there have been different approaches to material consequences. This paper attempts to provide an investigation into the parallels between Tarski’s dichotomy of formal and material consequence and the modern

Abstract Logical Constants

A possibility of defining logical constants within abstract logical frameworks is discussed, in relation to abstract definition of logical consequence. We propose using duals as a general method of applying the idea of invariance under replacement as a criterion for logicality.

The Adventure of Reason, by Paolo Mancosu.

The first half of the twentieth century witnessed the birth and flourishing of a radically novel subdiscipline: mathematical logic. While mathematics and logic had been on friendly terms at least since the seventeenth century (see M. Mugnai, ‘Logic and Mathematics in the Seventeenth Century’, History and Philosophy of Logic, 31 (2011), pp. 297–314), and while there were nineteenth century precursors to the idea of applying mathematics to logic (e.g. Boole) and logic to mathematics (e.g. Frege), it is only in the first half of the twentieth century that mathematical logic became a fully mature subdiscipline/research programme. The Adventure of Reason offers a compelling narrative of this exciting chapter of the history of logic and mathematics. Paolo Mancosu is one of the world’s leading experts on these developments, and while many others have made important contributions to the topic, it is fair to say that Mancosu has gone a step beyond with his painstaking work on sources other than the canonical, printed versions of articles and books. He has done extensive research at a number of archives both in Europe and North America, examining documents such as letters, minutes of meetings, informal reports, and transcriptions of lectures — in short, unpublished material of various sorts. This approach has enabled him to produce new insights and at times to provide novel answers to questions of interpretation concerning some of the towering figures in this tradition (such as the controversy on whether Tarski did or did not accept domain variation in his definition of logical consequence, examined in Ch. 16).

Validity and Interpretation

This paper claims that there is a plausible sense in which validity is a matter of truth preservation relative to interpretations of the sentences that occur in an argument, although it is not the sense one might have in mind. §1 outlines three independent problems: the first is the paradox of the sorites, the second concerns the fallacy of equivocation, and the third arises in connection with the standard treatment of indexicals. §2 elucidates the claim about validity, while §§3–5 show how the three problems outlined can be handled in accordance with it. §6 explains how the claim squares with the traditional idea that validity is related to formality, and in particular with a broadly accepted definition based on that idea, the model-theoretic definition of logical consequence. Unlike other works on the subject, this paper does not focus on necessity. It is not its intention to provide a characterization of necessity that conforms to some ideal of rigour or to some pre-theoretical understanding of validity. What follows can be taken as conditional on the assumption that such a characterization can be provided.

Remarks on Second-Order Consequence

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Active logic semantics for a single agent in a static world

For some time we have been developing, and have had significant practical success with, a time-sensitive, contradiction-tolerant logical reasoning engine called the active logic machine (ALMA). The current paper details a semantics for a general version of the underlying logical formalism, active logic. Central to active logic are special rules controlling the inheritance of beliefs in general (and of beliefs about the current time in particular), very tight controls on what can be derived from direct contradictions ( P & ¬ P ), and mechanisms allowing an agent to represent and reason about its own beliefs and past reasoning. Furthermore, inspired by the notion that until an agent notices that a set of beliefs is contradictory, that set seems consistent (and the agent therefore reasons with it as if it were consistent), we introduce an “apperception function” that represents an agent's limited awareness of its own beliefs, and serves to modify inconsistent belief sets so as to yield consistent sets. Using these ideas, we introduce a new definition of logical consequence in the context of active logic, as well as a new definition of soundness such that, when reasoning with consistent premises, all classically sound rules remain sound in our new sense. However, not everything that is classically sound remains sound in our sense, for by classical definitions, all rules with contradictory premises are vacuously sound, whereas in active logic not everything follows from a contradiction.

Reduction and Tarski's Definition of Logical Consequence

In his classic 1936 paper Tarski sought to motivate his definition of logical consequence by appeal to the inference form: P(0), P(1), . . ., P(n), . . . therefore ∀nP(n). This is prima facie puzzling because these inferences are seemingly first-order and Tarski knew that Gödel had shown first-order proof methods to be complete, and because ∀nP(n) is not a logical consequence of P(0), P(1), . . ., P(n), . . . by Taski's proposed definition. An attempt to resolve the puzzle due to Etchemendy is considered and rejected. A second attempt due to Gómez-Torrente is accepted as far as it goes, but it is argued that it raises a further puzzle of its own: it takes the plausibility of Tarski's claim that his definition captures our common concept of logical consequence to depend upon our common concept being a reductive conception. A further interpretation of what Tarski had in mind when he offered the example is proposed, using materials well known to Tarski at the time. It is argued that this interpretation makes the motivating example independent of reductive definitions which take natural numbers to be higher-order set theoretic entities, and it also explains why he did not regard the distinction between defined and primitive terms as pressing, as was the distinction between logical and nonlogical terms.

Did Tarski commit “Tarski's fallacy”?

In his 1936 paper,On the Concept of Logical Consequence, Tarski introduced the celebrated definition oflogical consequence: “The sentenceσfollows logicallyfrom the sentences of the class Γ if and only if every model of the class Γ is also a model of the sentenceσ.” [55, p. 417] This definition, Tarski said, is based on two very basic intuitions, “essential for the proper concept of consequence” [55, p. 415] and reflecting common linguistic usage: “Consider any class Γ of sentences and a sentence which follows from the sentences of this class. From an intuitive standpoint it can never happen that both the class Γ consists only of true sentences and the sentenceσis false. Moreover, … we are concerned here with the concept of logical, i.e.,formal, consequence.” [55, p. 414] Tarski believed his definition of logical consequence captured the intuitive notion: “It seems to me that everyone who understands the content of the above definition must admit that it agrees quite well with common usage. … In particular, it can be proved, on the basis of this definition, that every consequence of true sentences must be true.” [55, p. 417] The formality of Tarskian consequences can also be proven. Tarski's definition of logical consequence had a key role in the development of the model-theoretic semantics of modern logic and has stayed at its center ever since.