Table Of Content“This is the perfect book for coverage of classic debates in mainstream philosophy
of logic. It’s also the perfect source for exceptionally clear reviews of standard
logical machinery (e.g., standard modal machinery, quantifier machinery, higher-
order machinery, etc.). Very user-friendly, clear, and accurate on all of the topics
that it covers, this is my new required text for classic debates in the philosophy of
logic.”
Jc Beall, University of Notre Dame
“John MacFarlane displays his usual lively and engaging writing style, and is neutral
on controversial issues, giving the arguments employed by both sides. It is an
excellent overview of some key topics in the field.”
Stewart Shapiro, Ohio State University
Philosophical Logic
Introductory logic is generally taught as a straightforward technical discipline.
In this book, John MacFarlane helps the reader think about the limitations of,
presuppositions of, and alternatives to classical first-order predicate logic, making
this an ideal introduction to philosophical logic for any student who already has
completed an introductory logic course.
The book explores the following questions. Are there quantificational idioms that
cannot be expressed with the familiar universal and existential quantifiers? How
can logic be extended to capture modal notions like necessity and obligation?
Does the material conditional adequately capture the meaning of ‘if’—and if not,
what are the alternatives? Should logical consequence be understood in terms of
models or in terms of proofs? Can one intelligibly question the validity of basic
logical principles like Modus Ponens or Double Negation Elimination? Is the fact
that classical logic validates the inference from a contradiction to anything a flaw,
and if so, how can logic be modified to repair it? How, exactly, is logic related to
reasoning? Must classical logic be revised in order to be applied to vague language,
and if so how? Each chapter is organized around suggested readings and includes
exercises designed to deepen the reader’s understanding.
Key Features:
• An integrated treatment of the technical and philosophical issues comprising
philosophical logic
• Designed to serve students taking only one course in logic beyond the introduc-
tory level
• Provides tools and concepts necessary to understand work in many areas of
analytic philosophy
• Includes exercises, suggested readings, and suggestions for further exploration
in each chapter
John MacFarlane is Professor of Philosophy and a member of the Group in Logic
and the Methodology of Science at the University of California, Berkeley. He is
the author of Assessment Sensitivity: Relative Truth and Its Applications (2014).
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Philosophical Logic
A Contemporary Introduction
John MacFarlane
First published 2021
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© 2021 John MacFarlane
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Publisher’s Note
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Contents
List of Exercises xii
Preface xv
Acknowledgements xix
1 Fundamentals 1
1.1 Propositional logic 1
1.1.1 Grammar 1
1.1.2 Semantics 2
1.1.3 Proofs 6
1.1.4 Proof strategy 13
1.1.5 The relation of semantics and proofs 14
1.2 Predicate logic 15
1.2.1 Grammar 16
1.2.2 Scope 17
1.2.3 Semantics 17
1.2.4 Proofs 21
1.3 Identity 26
1.3.1 Grammar 28
1.3.2 Semantics 28
1.3.3 Proofs 28
1.4 Use and mention 29
2 Quantifiers 35
2.1 Beyond ∀and ∃ 35
2.1.1 What is a quantifier? 35
2.1.2 Semantics of binary quantifiers 37
2.1.3 Most: an essentially binary quantifier 37
2.1.4 Unary quantifiers beyond ∀and ∃ 38
2.1.5 Generalized quantifiers 39
2.2 Definite descriptions 39
viii Contents
2.2.1 Terms or quantifiers? 39
2.2.2 Definite descriptions and scope 41
2.2.3 Russell’s theory of descriptions 41
2.2.4 Proofs 43
2.3 Second-order quantifiers 44
2.3.1 Standard semantics for monadic second-order logic 46
2.3.2 Expressive limitations of first-order logic 47
2.3.3 Set theory in sheep’s clothing? 50
2.3.4 Boolos’s plural interpretation 52
2.3.5 Beyond monadic second-order logic 54
2.4 Substitutional quantifiers 57
2.4.1 Objectual and substitutional quantification 57
2.4.2 Nonexistent objects 58
2.4.3 Quantifying into attitude reports 59
2.4.4 Sentence quantifiers 60
2.4.5 Quantifying into quotes 61
2.4.6 Defining truth 61
2.4.7 Quantifying into quotes and paradox 62
2.4.8 The circularity worry 64
3 Modal Logic 67
3.1 Modal propositional logic 67
3.1.1 Grammar 67
3.1.2 Semantics 68
3.1.3 Modal logics from K to S5 70
3.1.4 Proofs 74
3.2 Modal predicate logic 80
3.2.1 Opaque contexts 80
3.2.2 Opaque contexts and quantification 81
3.2.3 The number of planets argument 82
3.2.4 Smullyan’s reply 83
3.3 The slingshot argument 85
3.3.1 Applications of slingshot arguments 87
3.3.2 The Gödel slingshot 87
3.3.3 Critique of the slingshot 88
3.4 Kripke’s defense of de re modality 90
3.4.1 Kripke’s strategy 90
3.4.2 The contingent a priori 91
3.4.3 The necessary a posteriori 93
3.4.4 Epistemic and alethic modals 94
Contents ix
4 Conditionals 97
4.1 The material conditional 97
4.1.1 Indicative vs. counterfactual 97
4.1.2 Entailments between indicatives and material conditionals 99
4.1.3 Thomson against the “received opinion” 100
4.2 No truth conditions? 101
4.2.1 Arguments for the material conditional analysis 102
4.2.2 Arguments against the material conditional analysis 102
4.2.3 Rejecting Or-to-if 104
4.2.4 Edgington’s positive view 105
4.2.5 Against truth conditions 107
4.3 Stalnaker’s semantics and pragmatics 109
4.3.1 Propositions, assertion, and the common ground 109
4.3.2 Semantics 110
4.3.3 Reasonable but invalid inferences 111
4.3.4 Contraposition and Hypothetical Syllogism 113
4.3.5 The argument for fatalism 114
4.4 Is Modus Ponens valid? 115
4.4.1 The intuitive counterexamples 116
4.4.2 McGee’s counterexamples as seen by Edgington 117
4.4.3 McGee’s counterexamples as seen by Stalnaker 119
4.4.4 Modus Ponens vs. Exportation 120
5 Logical Consequence via Models 123
5.1 Informal characterizations of consequence 123
5.1.1 In terms of necessity 123
5.1.2 In terms of proof 126
5.1.3 In terms of counterexamples 128
5.2 Tarski’s account of logical consequence 132
5.2.1 Tarski’s aim 132
5.2.2 Why proof-based approaches won’t work 132
5.2.3 Criteria of adequacy 135
5.2.4 The insufficiency of (F) 136
5.2.5 The semantic definition 137
5.2.6 Satisfying the criteria of adequacy 138
5.2.7 Logical constants 139
5.3 Interpretational and representational semantics 140
6 Logical Consequence via Proofs 145
6.1 Introduction rules as self-justifying 145
