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The Concept of Logical Consequence

An Introduction to Philosophical Logic

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Matthew W. McKeon

The Concept of Logical Consequence is a critical evaluation of the model-theoretic and proof-theoretic characterizations of logical consequence that proceeds from Alfred Tarski’s characterization of the informal concept of logical consequence. This study evaluates and expands upon ideas set forth in Tarski’s 1936 article on logical consequence, and appeals to his 1935 article on truth. Classical logic, as well as extensions and deviations are considered. Issues in the philosophy of logic such as the nature of logical constants, the philosophical significance of completeness, and the metaphysical and epistemological implications of logic are discussed in the context of the examination of the concept of logical consequence.

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Chapter 1: Introduction ................................................................................... 1 Chapter 2: The Concept of Logical Consequence........................................... 5 Tarski’s Characterization of the Common Concept of Logical Consequence............................................................................. 5 Logical Constants .................................................................................... 14 Model-Theoretic and Deductive-Theoretic Approaches to Logic ........... 24 Chapter 3: Set-Theoretic and Linguistic Preliminaries ................................. 27 Set-Theoretic Preliminaries ..................................................................... 27 Linguistic Preliminaries........................................................................... 29 Syntax of M ............................................................................................. 29 Semantics for M....................................................................................... 32 Chapter 4: Model-Theoretic Consequence.................................................... 39 Truth in a Structure .................................................................................. 40 Model-Theoretic Consequence Defined .................................................. 45 The Metaphysics of Domains .................................................................. 49 Interpretations .......................................................................................... 71 The Adequacy of the Model-Theoretic Characterization of Logical Consequence............................................................................... 79 What is a Logical Constant? .................................................................. 108 Chapter 5: Deductive Consequence ............................................................ 113 Deductive System N .............................................................................. 114 The Deductive-Theoretic Characterization and the Common Concept of Logical Consequence .......................................................... 126 Inferentialism, Harmony, and the Justification of Inference Rules ....... 128 Tarski’s Criticism of the Deductive-Theoretic Characterization of Logical Consequence......................................................................... 150 Is N a Correct Deductive System? ....................................................... 152 Bibliography ............................................................................................... 161 Chapter 1 Introduction This book is an inquiry into the concept of logical consequence, arguably the central concept of logic. We take logical consequence to be a relation between a given set of sentences and the sentences that logically follow. One sentence is said to be a logical consequence of a set of sentences, if and only if, in virtue of logic alone, it is impossible for the sentences in the set to be all true without the other sentence being true as well. The central question to be investigated here is: what conditions must be met in order for a sentence to be a logical consequence of others? One historically significant...

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