The Liar paradox raises foundational questions about logic, language, and truth. 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 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. (shrink)

Contrary to what often seems to be the implicit belief, Tarski's 1933 version of the theorem on the indefinability of truth did not mention semantic notions, either defined or intuitive. I state this version in a somewhat modernized form and explain briefly the self-imposed mathematico-philosophical constraints that led Tarski to formulate it as he did. I also point out that close attention to its content suggests a refined view of the exact contrast between Tarski's achievement and Gödel's achievement in his (...) earlier discovery of another version of the result. (shrink)

Dialetheism is the view that some contradictions are true. This is a view which runs against orthodoxy in logic and metaphysics since Aristotle, and has implications for many of the core notions of philosophy. Doubt Truth to Be a Liar explores these implications for truth, rationality, negation, and the nature of logic, and develops further the defense of dialetheism first mounted in Priest's In Contradiction, a second edition of which is also available.

In Spandrels of Truth, Beall concisely presents and defends a modest, so-called dialetheic theory of transparent truth.

A selective background -- Broadly classical approaches -- Paracompleteness -- More on paracomplete solutions -- Paraconsistent dialetheism.

Vagueness provides the first comprehensive examination of a topic of increasing importance in metaphysics and the philosophy of logic and language. Timothy Williamson traces the history of this philosophical problem from discussions of the heap paradox in classical Greece to modern formal approaches such as fuzzy logic. He illustrates the problems with views which have taken the position that standard logic and formal semantics do not apply to vague language, and defends the controversial realistic view that vagueness is a kind (...) of ignorance--that there really is a grain of sand whose removal turns a heap into a non-heap, but we cannot know which one it is. (shrink)

It is “the received wisdom” that any intuitively natural and consistent resolution of a class of semantic paradoxes immediately leads to other paradoxes just as bad as the first. This is often called the “revenge problem”. Some proponents of the received wisdom draw the conclusion that there is no hope of any natural treatment that puts all the paradoxes to rest: we must either live with the existence of paradoxes that we are unable to treat, or adopt artificial and ad (...) hoc means to avoid them. Others (“dialetheists”) argue that we can put the paradoxes to rest, but only by licensing the acceptance of some contradictions (presumably in a paraconsistent logic that prevents the contradictions from spreading everywhere). (shrink)

Tarski’s Indefinability Theorem can be generalized so that it applies to many-valued languages. We introduce a notion of strong semantic self-representation applicable to any (sufficiently rich) interpreted many-valued language L. A sufficiently rich interpreted many-valued language L is SSSR just in case it has a function symbol n(x) such that, for any f Sent(L), the denotation of the term n(“f”) in L is precisely ||f||L, the semantic value of f in L. By a simple diagonal construction (finding a sentence l (...) such that l is equivalent to n(“l”) T), it is shown that no such language strongly represents itself semantically. Hence, no such language can be its own metalanguage. (shrink)

A formal theory of truth, alternative to tarski's 'orthodox' theory, based on truth-value gaps, is presented. the theory is proposed as a fairly plausible model for natural language and as one which allows rigorous definitions to be given for various intuitive concepts, such as those of 'grounded' and 'paradoxical' sentences.

The paper offers a solution to the semantic paradoxes, one in which (1) we keep the unrestricted truth schema “True(A)↔A”, and (2) the object language can include its own metalanguage. Because of the first feature, classical logic must be restricted, but full classical reasoning applies in “ordinary” contexts, including standard set theory. The more general logic that replaces classical logic includes a principle of substitutivity of equivalents, which with the truth schema leads to the general intersubstitutivity of True(A) with A (...) within the language.The logic is also shown to have the resources required to represent the way in which sentences (like the Liar sentence and the Curry sentence) that lead to paradox in classical logic are “defective”. We can in fact define a hierarchy of “defectiveness” predicates within the language. Contrary to claims that any solution to the paradoxes just breeds further paradoxes (“revenge problems”) involving defectiveness predicates, there is a general consistency/conservativeness proof that shows that talk of truth and the various “levels of defectiveness” can all be made coherent together within a single object language. (shrink)

In this rigorous investigation into the logic of truth Anil Gupta and Nuel Belnap explain how the concept of truth works in both ordinary and pathological..

I ON THE PRIMITIVE TERM OF LOGISTICf IN this article I propose to establish a theorem belonging to logistic concerning some connexions, not widely known, ...

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. (shrink)

In this ingenious and powerfully argued book Tim Maudlin sets out a novel account of logic and semantics which allows him to deal with certain notorious paradoxes which have bedevilled philosophical theories of truth. All philosophers interested in logic and language will find this a stimulating read.

Awarded the 1988 Johnsonian Prize in Philosophy. Published with the aid of a grant from the National Endowment for the Humanities.

A Mathematical Introduction to Logic, Second Edition, offers increased flexibility with topic coverage, allowing for choice in how to utilize the textbook in a course. The author has made this edition more accessible to better meet the needs of today's undergraduate mathematics and philosophy students. It is intended for the reader who has not studied logic previously, but who has some experience in mathematical reasoning. Material is presented on computer science issues such as computational complexity and database queries, with additional (...) coverage of introductory material such as sets. Increased flexibility of the text, allowing instructors more choice in how they use the textbook in courses. Reduced mathematical rigour to fit the needs of undergraduate students. (shrink)

Thinking about truth can be more dangerous than it looks. Of course, our concept of truth is the source of one of the most frustrating and impenetrable paradoxes humans have ever contemplated, the liar paradox, but that is just the beginning of its treachery. In an effort to understand why one of the most beloved and revered members of our conceptual repertoire could cause us so much trouble, philosophers have for centuries proposed “solutions” to the liar paradox. However, it seems (...) that our concept of truth takes offense to our efforts to understand it because it appears to retaliate against those who propose “solutions” to the liar. It takes its revenge on us by creating new paradoxes from our own attempts to find resolution. That is, most proposed solutions to the liar paradox give rise to new, more insidious paradoxes—often called revenge paradoxes. For our attempts at understanding, truth rewards us with inconsistent theories, untenable logics, and a deep feeling of bewilderment. It is as if our concept of truth lashes out at us because it wants to remain a mystery. After a few run-ins with truth, many philosophers have the good sense to keep their distance. Far from being the serene, profound concept most people take it to be, those of us who think much about the liar paradox know truth to be a vengeful bully—a conceptual misanthrope. (shrink)

This book is about one of the most baffling of all paradoxes – the famous Liar paradox. Suppose we say: 'We are lying now'. Then if we are lying, we are telling the truth; and if we are telling the truth we are lying. This paradox is more than an intriguing puzzle, since it involves the concept of truth. Thus any coherent theory of truth must deal with the Liar. Keith Simmons discusses the solutions proposed by medieval philosophers and offers (...) his own solutions and in the process assesses other attempts to solve the paradox. Unlike such attempts, Simmons' 'singularity' solution does not abandon classical semantics and does not appeal to the kind of hierarchical view found in Barwise's and Etchemendy's The Liar. Moreover, Simmons' solution resolves the vexing problem of semantic universality – the problem of whether there are semantic concepts beyond the expressive reach of a natural language such as English. (shrink)

Jenkins (2007) charges that the language advanced in Beall (2007) is either expressively impoverished, or inconsistent. We argue that Jenkins’ objections are based on unreasonably strong constraints on formal theories of truth. Our primary concern is not to defend the ‘paranormal’ framework advanced in Beall, but to respond to a common – and implausible – ‘revenge’-style charge directed at a certain class of formal theories of truth and paradox.

Consider the sentence 'This sentence is not true'. It seems that the sentence can be neither true nor not true, on pain of contradiction. Certain notorious paradoxes like this have bedevilled philosophical theories of truth. Tim Maudlin presents an original account of logic and semantics which deals with these paradoxes, and allows him to set out a new theory of truth-values and the norms governing claims about truth. All philosophers interested in logic and language will find Truth and Paradox a (...) stimulating read. (shrink)

Truth and Paradox largely consists of three connected technical projects together with a more general account of the nature of truth. The first project is the most familiar: providing an account of how logically complex sentences get assigned truth values on the basis of the truth values assigned to the logically atomic sentences. The second is construction of valid, syntactically specifiable inference rules for a language that includes the familiar logical connectives and the truth predicate. The third is an account (...) of the normative rules governing the assertion of sentences. Let me sketch these in order. (shrink)