Truth is one of the oldest and most central topics in philosophy. Formal theories explore the connections between truth and logic, and they address truth-theoretic paradoxes such as the Liar. Three leading philosopher-logicians now present a concise overview of the main issues and ideas in formal theories of truth. Beall, Glanzberg, and Ripley explain key logical techniques on which such formal theories rely, providing the formal and logical background needed to develop formal theories of truth. They examine the most important truth-theoretic paradoxes, including the Liar paradoxes. They explore approaches that keep principles of truth simple while relying on nonclassical logic; approaches that preserve classical logic but do so by complicating the principles of truth; and approaches based on substructural logics that change the shape of the target consequence relation itself. Finally, inconsistency and revision theories are reviewed, and contrasted with the approaches previously discussed. For any reader who has a basic grounding in logic, this book offers an ideal guide to formal theories of truth.

Deflationist accounts of truth are widely held in contemporary philosophy: they seek to show that truth is a dispensable concept with no metaphysical depth. However, logical paradoxes present problems for deflationists that their work has struggled to overcome. In this volume of fourteen original essays, a distinguished team of contributors explore the extent to which, if at all, deflationism can accommodate paradox. The volume will be of interest to philosophers of logic, philosophers of language, and anyone working on truth.
Contributors include Bradley Armour-Garb, Jody Azzouni, JC Beall, Hartry Field, Christopher Gauker, Michael Glanzberg, Dorothy Grover, Anil Gupta, Volker Halbach, Leon Horsten, Paul Horwich, Graham Priest, Greg Restall, and Alan Weir

Truth: The Basics is a concise and engaging introduction to philosophical theories about the nature of truth. The two authors – leading philosophers in this field – build the book around a single question: what, if anything, is common to all truths, which makes them true? The book explores five important answers (‘theories’) to the given question: correspondence, semantic, verifiability, transparency, and plurality. For each given theory the following questions are addressed: • What is the theory’s answer to the central question? • What is the basic motivation behind that answer? • What is a precise argument for that answer? • What are the biggest objections to that answer? • What are a few good resources for understanding more about the theory? An additional chapter provides an extensive introduction to the notorious liar paradox. Truth: The Basics is an ideal starting point for anyone seeking a lively and accessible introduction to the rich and complex philosophical study of truth. Key Features:> Written in a clear and concise fashion. > Clearly explains five major theories of truth for an uninitiated readership of undergraduate students and general readers. > Prepares the reader to tackle more advanced work in truth studies. > Makes connections between truth and other areas of philosophy, including the philosophy of language, semantics, metaphysics, logic and epistemology. > Includes technical appendices for more advanced readers.

The Liar paradox raises foundational questions about logic, language, and truth (and semantic notions in general). A simple Liar sentence like 'This sentence is false' appears to be both true and false if it is either true or false. For if the sentence is true, then what it says is the case; but what it says is that it is false, hence it must be false. On the other hand, if the statement is false, then it is true, since it says (only) that it is false.
How, then, should we classify Liar sentences? Are they true or false? A natural suggestion would be that Liars are neither true nor false; that is, they fall into a category beyond truth and falsity. This solution might resolve the initial problem, but it beckons the Liar's revenge. A sentence that says of itself only that it is false or beyond truth and falsity will, in effect, bring back the initial problem. The Liar's revenge is a witness to the hydra-like nature of Liars: in dealing with one Liar you often bring about another.
JC Beall presents fourteen new essays and an extensive introduction, which examine the nature of the Liar paradox and its resistance to any attempt to solve it. Written by some of the world's leading experts in the field, the papers in this volume will be an important resource for those working in truth studies, philosophical logic, and philosophy of language, as well as those with an interest in formal semantics and metaphysics.

The Liar paradox raises foundational questions about logic, language, and truth (and semantic notions in general). A simple Liar sentence like 'This sentence is false' appears to be both true and false if it is either true or false. For if the sentence is true, then what it says is the case; but what it says is that it is false, hence it must be false. On the other hand, if the statement is false, then it is true, since it says (only) that it is false.
How, then, should we classify Liar sentences? Are they true or false? A natural suggestion would be that Liars are neither true nor false; that is, they fall into a category beyond truth and falsity. This solution might resolve the initial problem, but it beckons the Liar's revenge. A sentence that says of itself only that it is false or beyond truth and falsity will, in effect, bring back the initial problem. The Liar's revenge is a witness to the hydra-like nature of Liars: in dealing with one Liar you often bring about another.
JC Beall presents fourteen new essays and an extensive introduction, which examine the nature of the Liar paradox and its resistance to any attempt to solve it. Written by some of the world's leading experts in the field, the papers in this volume will be an important resource for those working in truth studies, philosophical logic, and philosophy of language, as well as those with an interest in formal semantics and metaphysics.

Logic is fundamental to thought and language. But which logical principles are correct? The paradoxes play a crucial role in answering that question. The so-called Liar and Heap paradoxes challenge our basic ideas about logic; at the very least, they teach us that the correct logical principles are not as obvious as common sense would have it. The essays in this volume, written by leading figures in the field, discuss novel thoughts about the paradoxes.

Truth: The Basics is a concise and engaging introduction to philosophical theories about the nature of truth. The two authors – leading philosophers in this field – build the book around a single question: what, if anything, is common to all truths, which makes them true? The book explores five important answers (‘theories’) to the given question: correspondence, semantic, verifiability, transparency, and plurality. For each given theory the following questions are addressed: • What is the theory’s answer to the central question? • What is the basic motivation behind that answer? • What is a precise argument for that answer? • What are the biggest objections to that answer? • What are a few good resources for understanding more about the theory? An additional chapter provides an extensive introduction to the notorious liar paradox. Truth: The Basics is an ideal starting point for anyone seeking a lively and accessible introduction to the rich and complex philosophical study of truth. Key Features:> Written in a clear and concise fashion. > Clearly explains five major theories of truth for an uninitiated readership of undergraduate students and general readers. > Prepares the reader to tackle more advanced work in truth studies. > Makes connections between truth and other areas of philosophy, including the philosophy of language, semantics, metaphysics, logic and epistemology. > Includes technical appendices for more advanced readers.

The Law of Non-Contradiction -- that no contradiction can be true -- has been a seemingly unassailable dogma since the work of Aristotle, in Book G of the Metaphysics. It is an assumption challenged from a variety of angles in this collection of original papers. Twenty-three of the world's leading experts investigate the "law," considering arguments for and against it and discussing methodological issues that arise whenever we question the legitimacy of logical principles. The result is a balanced inquiry into a venerable principle of logic, one that raises questions at the very center of logic itself.
The aim of this volume is to present a comprehensive debate about the Law of Non-Contradiction, from discussions as to how the law is to be understood, to reasons for accepting or re-thinking the law, and to issues that raise challenges to the law, such as the Liar Paradox, and a "dialetheic" resolution of that paradox. The editors contribute an introduction which surveys the issues and serves to frame the debate, and a useful bibliography offering a guide to further reading.
This volume will be of interest to anyone working on philosophical logic, and to anyone who has ever wondered about the status of logical laws and about how one might proceed to mount arguments for or against them.

Deflationist accounts of truth are widely held in contemporary philosophy: they seek to show that truth is a dispensable concept with no metaphysical depth. However, logical paradoxes present problems for deflationists, which their work has struggled to overcome. In this volume of fourteen original essays, a distinguished team of contributors explore the extent to which, if at all, deflationism can accommodate paradox. The volume will be of interest to philosophers of logic, philosophers of language, and anyone working on truth.

Building on his paradigm-shifting work on the incarnation in The Contradictory Christ (OUP, 2021), Jc Beall extends a robust contradictory theology with an account of the trinity. Throughout the history of the Christian church, heretics, apophatics, mystics, atheists, and many others have long proclaimed that the doctrine of the trinity - one of the central doctrines of the Christian faith - is contradictory. In this work, Beall agrees; however, as Beall convincingly argues, one needn't abandon orthodoxy, play language games, inflate one's metaphysics, nor abandon the standard faith in the face of such divine contradiction. Instead, one can accept central axioms of the trinity at face value and, with a suitable account of logical entailment, accept the 'contradictory truths' thereby entailed. With the clarity and precision that only a logician could provide, Beall provided theology and the Christian church in general with a very simple and viable (and arguably correct) model of divine reality. Unlike the vast number of theologians and philosophers before him, Beall rejects the quest for a logically consistent account of divine reality. The triune god (viz., God) is truly and fully described only via contradiction. As such, attempts to remove the contradiction are attempts to remove truths of God.

Consequence is at the heart of logic; an account of consequence, of what follows from what, offers a vital tool in the evaluation of arguments. Since philosophy itself proceeds by way of argument and inference, a clear view of what logical consequence amounts to is of central importance to the whole discipline. In this book, JC Beall and Greg Restall present and defend what thay call logical pluralism, arguing that the notion of logical consequence doesn't pin down one deductive consequence relation; it allows for many of them. In particular, they argue that broadly classical, intuitionistic, and relevant accounts of deductive logic are genuine logical consequence relations; we should not search for one true logic, since there are many. Their conclusions have profound implications for many linguists as well as for philosophers.

Semantic and soritical paradoxes challenge entrenched, fundamental principles about language - principles about truth, denotation, quantification, and, among others, 'tolerance'. Study of the paradoxes helps us determine which logical principles are correct. So it is that they serve not only as a topic of philosophical inquiry but also as a constraint on such inquiry: they often dictate the semantic and logical limits of discourse in general. Sixteen specially written essays by leading figures in the field offer new thoughts and arguments about the paradoxes.

Extensively classroom-tested, this text provides an accessible and carefully structured introduction to modal and many-valued logic. The authors cover the basic formal frameworks, as well as considering a variety of philosophical issues surrounding 'possibilities and paradox'.

In order to aid understanding, each chapter provides the following features: exercises to give students hands-on experience, examples to demonstrate the application of concepts and a list of further readings.

Logic: The Basics is an accessible introduction to several core areas of logic. The first part of the book features a self-contained introduction to the standard topics in classical logic, such as: * mathematical preliminaries * propositional logic * quantified logic (first monadic, then polyadic) * English and standard 'symbolic translations' * tableau procedures. Alongside comprehensive coverage of the standard topics, this thoroughly revised second edition also introduces several philosophically important nonclassical logics, free logics, and modal logics, and gives the reader an idea of how they can take their knowledge further. With its wealth of exercises (solutions available in the encyclopedic online supplement), Logic: The Basics is a useful textbook for courses ranging from the introductory level to the early graduate level, and also as a reference for students and researchers in philosophical logic.

In this ground-breaking study, Jc Beall shows that the fundamental "problem" of Christology is simple to see from the role that Christ occupies: the Christ figure is to have the divine and essentially limitless properties of the one and only God but Christ is equally to have the human, essentially limit-imposing properties involved in human nature, limits essentially involved in being human. The role that Christ occupies thereby appears to demand a contradiction: all of the limitlessness of God, and all of the limits of humans. This book lays out Beall's contradictory account of Jesus Christ - and thereby a contradictory Christian theology.

Among the various conceptions of truth is one according to which 'is true' is a transparent, entirely see-through device introduced for only practical (expressive) reasons. This device, when introduced into the language, brings about truth-theoretic paradoxes (particularly, the notorious Liar and Curry paradoxes). The options for dealing with the paradoxes while preserving the full transparency of 'true' are limited. In Spandrels of Truth, Beall concisely presents and defends a modest, so-called dialetheic theory of transparent truth.

Among the various conceptions of truth is one according to which 'is true' is a transparent, entirely see-through device introduced for only practical (expressive) reasons. This device, when introduced into the language, brings about truth-theoretic paradoxes (particularly, the notorious Liar and Curry paradoxes). The options for dealing with the paradoxes while preserving the full transparency of 'true' are limited. In Spandrels of Truth, Beall concisely presents and defends a modest, so-called dialetheic theory of transparent truth.

Consequence is at the heart of logic; an account of consequence, of what follows from what, offers a vital tool in the evaluation of arguments. Since philosophy itself proceeds by way of argument and inference, a clear view of what logical consequence amounts to is of central importance to the whole discipline. In this book JC Beall and Greg Restall present and defend what thay call logical pluralism, arguing that the notion of logical consequence doesn't pin down one deductive consequence relation; it allows for many of them. In particular, they argue that broadly classical, intuitionistic, and relevant accounts of deductive logic are genuine logical consequence relations; we should not search for one true logic, since there are many. Their conclusions have profound implications for many linguists as well as for philosophers.

Truth is one of the oldest and most central topics in philosophy. Formal theories explore the connections between truth and logic, and they address truth-theoretic paradoxes such as the Liar. Three leading philosopher-logicians now present a concise overview of the main issues and ideas in formal theories of truth. Beall, Glanzberg, and Ripley explain key logical techniques on which such formal theories rely, providing the formal and logical background needed to develop formal theories of truth. They examine the most important truth-theoretic paradoxes, including the Liar paradoxes. They explore approaches that keep principles of truth simple while relying on nonclassical logic; approaches that preserve classical logic but do so by complicating the principles of truth; and approaches based on substructural logics that change the shape of the target consequence relation itself. Finally, inconsistency and revision theories are reviewed, and contrasted with the approaches previously discussed. For any reader who has a basic grounding in logic, this book offers an ideal guide to formal theories of truth.

Logic: The Basics is an accessible introduction to several core areas of logic. The first part of the book features a self-contained introduction to the standard topics in classical logic, such as:

· mathematical preliminaries

· propositional logic

· quantified logic (first monadic, then polyadic)

· English and standard ‘symbolic translations’

· tableau procedures.

Alongside comprehensive coverage of the standard topics, this thoroughly revised second edition also introduces several philosophically important nonclassical logics, free logics, and modal logics, and gives the reader an idea of how they can take their knowledge further. With its wealth of exercises (solutions available in the encyclopedic online supplement), Logic: The Basics is a useful textbook for courses ranging from the introductory level to the early graduate level, and also as a reference for students and researchers in philosophical logic.

Table of Contents

I BACKGROUND IDEAS

1 Consequences

1.1 Relations of support

1.2 Logical consequence: the basic recipe

1.3 Valid arguments and truth

1.4 Summary, looking ahead, and reading

2 Models, Modeled, and Modeling

2.1 Models

2.2 Models in science

2.3 Logic as modeling

2.4 A note on notation, metalanguages, etc.

2.5 Summary, looking ahead, and reading

3 Language, Form, and Logical Theories

3.1 Language and formal languages

3.2 Languages: syntax and semantics

3.3 Atoms, connectives, and molecules

3.4 Connectives and form

3.5 Validity and form

3.6 Logical theories: rivalry

3.7 Summary, looking ahead, and reading

4 Set-theoretic Tools

4.1 Sets

4.2 Ordered sets: pairs and n-tuples

4.3 Relations

4.4 Functions

4.5 Sets as tools

4.6 Summary and looking ahead

II THE BASIC CLASSICAL THEORY

5 Basic Classical Syntax and Semantics

5.1 Cases: complete and consistent

5.2 Classical ‘truth conditions’

5.3 Basic classical consequence

5.4 Motivation: precision

5.5 Formal picture

5.6 Defined connectives

5.7 Some notable valid forms

5.8 Summary and looking ahead

6 Basic Classical Tableaux

6.1 What are tableaux?

6.2 Tableaux for the Basic Classical Theory

6.3 Summary and looking ahead

7 Basic Classical Translations

7.1 Atoms, Punctuation, and Connectives

7.2 Syntax, altogether

7.3 Semantics

7.4 Consequence

7.5 Summary and Looking Ahead

III FIRST-ORDER CLASSICAL THEORY

8 Atomic Innards: Unary

8.1 Atomic innards: names and predicates

8.2 Truth and falsity conditions for atomics

8.3 Cases, domains, and interpretation functions

8.4 Classicality

8.5 A formal picture

8.6 Summary and looking ahead

9 Everything and Something

9.1 Validity involving quantifiers

9.2 Quantifiers: an informal sketch

9.3 Truth and falsity conditions

9.4 A formal picture

9.5 Summary and looking ahead.

10 First-Order Language with Any-Arity Innards

10.1 Truth and falsity conditions for atomics

10.2 Cases, domains, and interpretation functions

10.3 Classicality

10.4 A formal picture

10.5 Summary and looking ahead

11 Identity

11.1 Logical expressions, forms, sentential forms

11.2 Validity involving identity

11.3 Identity: informal sketch

11.4 Truth conditions: informal sketch

11.5 Formal picture

11.6 Summary and looking ahead

12 Tableaux for First-Order Logic with Identity

12.1 A Few Reminders

12.2 Tableaux for Polyadic First-Order Logic

12.3 Summary and looking ahead

13 First-Order Translations

13.1 Basic Classical Theory with Innards

13.2 First-Order Classical Theory

13.3 Polyadic Innards

13.4 Examples in the polyadic language

13.5 Adding Identity

13.6 Summary and Looking Ahead

IV NONCLASSICAL THEORIES

14 Alternative Logical Theories

14.1 Apparent unsettledness

14.2 Apparent overdeterminacy

14.3 Options

14.4 Cases

14.5 Truth and falsity conditions

14.6 Logical Consequence

14.7 Summary, looking ahead, and reading

15 Nonclassical Sentential Logics

15.1 Syntax

15.2 Semantics, Broadly

15.3 Defined connectives

15.4 Some notable forms

15.5 Summary and looking ahead

16 Nonclassical First-order Theories

16.1 An Informal Gloss

16.2 A formal picture

16.3 Summary and looking ahead

17 Nonclassical Tableaux

17.1 Closure Conditions

17.2 Tableaux for Nonclassical First-Order Logics

17.3 Summary and looking ahead

18 Nonclassical Translations

18.1 Syntax and Semantics

18.2 Consequence

18.3 Summary and looking ahead

V FREEDOM, NECESSITY AND BEYOND

19 Speaking Freely

19.1 Speaking of non-existent ‘things’

19.2 Existential import

19.3 Freeing our terms, expanding our domains

19.4 Truth conditions: an informal sketch

19.5 Formal picture

19.6 Summary and looking ahead

20 Possibilities

20.1 Possibility and necessity

20.2 Towards truth and falsity conditions

20.3 Cases and consequence

20.4 Formal picture

20.5 Remark on going beyond possibility

20.6 Summary and looking ahead

21 Free and Modal Tableaux

21.1 Free Tableaux

21.2 Modal Tableaux

21.3 Summary and looking ahead

22 Glimpsing Different Logical Roads

22.1 Other conditionals

22.2 Other negations

22.3 Other alethic modalities: actuality

22.4 Same connectives, different truth conditions

22.5 Another road to difference: consequence

22.6 Summary and looking behind and ahead

References

The Law of Non-Contradiction -- that no contradiction can be true -- has been a seemingly unassailable dogma since the work of Aristotle, in Book G of the Metaphysics. It is an assumption challenged from a variety of angles in this collection of original papers. Twenty-three of the world's leading experts investigate the "law," considering arguments for and against it and discussing methodological issues that arise whenever we question the legitimacy of logical principles. The result is a balanced inquiry into a venerable principle of logic, one that raises questions at the very center of logic itself.
The aim of this volume is to present a comprehensive debate about the Law of Non-Contradiction, from discussions as to how the law is to be understood, to reasons for accepting or re-thinking the law, and to issues that raise challenges to the law, such as the Liar Paradox, and a "dialetheic" resolution of that paradox. The editors contribute an introduction which surveys the issues and serves to frame the debate, and a useful bibliography offering a guide to further reading.
This volume will be of interest to anyone working on philosophical logic, and to anyone who has ever wondered about the status of logical laws and about how one might proceed to mount arguments for or against them.