Chapter 8 is the first of two chapters on the phenomenon of revenge paradoxes, paradoxes which, r... more Chapter 8 is the first of two chapters on the phenomenon of revenge paradoxes, paradoxes which, roughly speaking, are constructed out of the very terms of a purported solution. The chapter begins by exploring the difficulties that revenge presents for Kripke’s theory of truth, in either of two versions: a version that admits truth value gaps, and a paracomplete version which rejects the law of excluded middle. The chapter goes on to critically examine Field’s paracomplete theory of truth and its treatment of revenge, arguing that Field’s theory is couched in terms that are artificial and too far removed from natural language. The chapter concludes with a critical discussion of Priest’s dialetheist approach to the Liar paradox, according to which there are true contradictions. It is argued that Priest’s theory is itself subject to revenge paradoxes.
Chapter 7 puts the singularity theory to work on a number of semantic paradoxes that have intrins... more Chapter 7 puts the singularity theory to work on a number of semantic paradoxes that have intrinsic interest of their own. These include a transfinite paradox of denotation, and variations on the Liar paradox, including the Truth-Teller, Curry’s paradox, and paradoxical Liar loops. The transfinite paradox of denotation shows the need to accommodate limit ordinals. The Truth-Teller, like the Liar, exhibits semantic pathology-but, unlike the Liar, it does not produce a contradiction. The distinctive challenge of the Curry paradox is that it seems to allow us to prove any claim we like (for example, the claim that 2+2=5). Paradoxical Liar loops, such as the Open Pair paradox, extend the Liar paradox beyond single self-referential sentences. The chapter closes with the resolution of paradoxes that do not exhibit circularity yet still generate contradictions. These include novel versions of the definability paradoxes and Russell’s paradox, and Yablo’s paradox about truth.
Chapter 5 moves beyond the simple paradoxes discussed in Chapters 2-4. The chapter applies the si... more Chapter 5 moves beyond the simple paradoxes discussed in Chapters 2-4. The chapter applies the singularity approach to the traditional paradoxes of definability (or denotation), associated with Berry, Richard, and König. The chapter goes on to argue that there are two settings for Russell’s paradox, one in terms of the mathematical notion of set, and the other in terms of the logico-semantic notion of extension. The chapter then applies the singularity approach to Russell’s paradox for extensions. The chapter moves on to the case of truth, and applies the singularity approach to various versions of the Liar paradox, paying particular attention to the so-called strengthened Liar.
... KEITH SIMMONS The University of North Carolina ... In correspondence with Frege, Russell insi... more ... KEITH SIMMONS The University of North Carolina ... In correspondence with Frege, Russell insisted that Mont Blanc, "despite all its snowfields" (Frege [1979], p. 169), is a part of the proposition expressed by "Mont Blanc is more than 4000 metres high". ...
Dialetheism is the view that there are true contradictions. The strongest argument in favour of d... more Dialetheism is the view that there are true contradictions. The strongest argument in favour of dialetheism is that it alone allows us to solve semantic paradoxes like the liar paradox. This chapter presents two main criticisms of dialetheism. First, it argues that semantic pathology spreads to the dialetheist theory itself, putting into question the acceptability of the theory. Second, it argues that, even though dialetheism admits true contradictions, it is nevertheless subject to a revenge liar.
I argue for an account of semantic paradox that requires minimal logical revision. I first consid... more I argue for an account of semantic paradox that requires minimal logical revision. I first consider a phenomenon that is common to the paradoxes of definability, Russell’s paradox and the Liar. The phenomenon—which I call Repetition—is this: given a paradoxical expression, we can go on to produce a semantically unproblematic expression composed of the very same words. I argue that Kripke’s and Field’s theories of truth make heavy weather of Repetition, and suggest a simpler contextual account. I go on to outline a ‘singularity’ theory of semantical predicates in the spirit of remarks of Gödel. According to this theory, ‘denotes’, ‘extension’ and ‘true’ are context-sensitive expression that apply almost everywhere on a given occasion of use, except for certain singular points. I then turn to revenge paradoxes, and argue that even the dialetheist is subject to revenge. I then examine how the singularity theory responds to revenge.
In Thought and World, Chris Hill offers us the most sophisticated version of minimalism to date. ... more In Thought and World, Chris Hill offers us the most sophisticated version of minimalism to date. Hill's starting point is Horwich's minimal theory of truth (MT), according to which propositions - or thoughts in Hill's preferred terminology - are the truth-bearers, and the axioms of the theory are the thoughts expressed by sentences like "The thought that aardvarks amble is true if and only if aardvarks amble". But Hill's theory is superior to Horwich's in a number of crucial respects. For example, while Horwich gives up on a finite formulation of MT, taking it that an infinity of axioms is needed, Hill's basic theory (simple substitutionalism or SS) is composed of just one axiom. And unlike MT, SS can accommodate certain generalizations about truth (for example, the generalization that only thoughts are true, and general propositions about the truth-conditional properties of logical connectives.) Moreover, Hill provides a detailed deflationary theory not only of truth, but also of related semantic notions such as reference and concept-denotation, notions that have received relatively little attention in the literature on deflationism; he takes correspondence intuitions much more seriously than most deflationists, and is at pains to show how those intuitions may be accommodated within his framework; he provides the most developed deflationary account available of indexical concepts and indexical thoughts; and unlike many deflationists, he is not content to merely set aside problems concerning bi valence and the Liar paradox. Those of us interested in the proper treatment of our semantic concepts, whether we are deflationists or not, are in Hill's debt.
This book is about one of the most baffling of all paradoxes--the famous Liar paradox. Suppose we... more 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 ...
In 1905, Richard discovered his paradox of definability, and in a letter written that year he pre... more In 1905, Richard discovered his paradox of definability, and in a letter written that year he presented both the paradox and a solution to it.Soon afterwards, Poincaré endorsed a variant of Richard’s solution.In this paper, I critically examine Richard’s and Poincaré’s ways out.I draw on an objection of Peano’s, and argue that their stated solutions do not work.But I also
... 254-260); E. J. Ashworth, Language and logic in the post-medieval period (Dordrecht, Reidel, ... more ... 254-260); E. J. Ashworth, Language and logic in the post-medieval period (Dordrecht, Reidel, 1974), 104; Francesco Bottin, Le antinomie semantiche nella logica medievale (Padua, 1976), 173-175. ... 26 For example, Paul Spade and E. J. Ashworth encourage such a view. ...
The paradoxes of definability-Berry's paradox, Richard's paradox, and König's paradoxall exhibit ... more The paradoxes of definability-Berry's paradox, Richard's paradox, and König's paradoxall exhibit some form of circularity, or self-reference understood in a suitably broad sense. Consider Berry's paradox, for example. There are denumerably many positive integers, but only finitely many phrases of English with fewer than twenty eight syllables. So there is an integer which is the least positive integer not denoted by an English phrase with fewer than twenty eight syllables. Now consider the English phrase formed by the last sixteen words of the previous sentence. This phrase-call it the Berry phrase-denotes a positive integer. But the Berry phrase has twenty seven syllables. So the least positive integer which cannot be denoted by a phrase with less than twenty eight syllables is denoted by a phrase with twenty seven syllablescontradiction. Observe that the Berry phrase involves quantification over a certain domain of English phrases which contains the Berry phrase itself. Here is the circularity-or self-reference, in the sense that the Berry phrase makes indirect reference to itself. We may construct new versions of the definability paradoxes where the self-reference is more explicit. For example, consider the paradox generated by the following phrases written on the board in room 101: A. The ratio of the circumference of a circle to its diameter. B. The positive square root of 36. C. The sum of the numbers denoted by expressions on the board in room 101. 1 Are there definability paradoxes without self-reference? We can certainly say that there
This book aims to provide a solution to the semantic paradoxes. It argues for a unified solution ... more This book aims to provide a solution to the semantic paradoxes. It argues for a unified solution to the paradoxes generated by the concepts of reference or denotation, predicate extension, and truth. The solution makes two main claims. The first is that our semantic expressions ‘denotes’, ‘extension’, and ‘true’ are context-sensitive. The second, inspired by a brief, tantalizing remark of Gödel’s, is that these expressions are significant everywhere except for certain singularities, in analogy with division by zero. A formal theory of singularities is presented and applied to a wide variety of versions of the definability paradoxes, Russell’s paradox, and the Liar paradox. The book argues that the singularity theory satisfies the following desiderata: it recognizes that the proper setting of the semantic paradoxes is natural language, not regimented formal languages; it minimizes any revision to our semantic concepts; it respects as far as possible Tarski’s intuition that natural la...
Truth pluralism is a metaphysical theory of the nature of truth. The pluralist rejects the deflat... more Truth pluralism is a metaphysical theory of the nature of truth. The pluralist rejects the deflationist claim that truth is at best a ‘shallow’, insubstantial property. Indeed, the pluralist embraces a plurality of truth properties (such as correspondence, superassertibility, coherence), each appropriate to a different domain (or domains) of discourse. On the face of it, the pluralist will inherit all the main problems of the various traditional substantivist theories of truth. In addition, a strong pluralist, who only recognizes a plurality of truth properties (so that truth emerges not as one but only many), faces a number of problems, such as the problem of mixed discourse. But the moderate pluralism, who acknowledges that there is, in addition to diverse truth properties, a single property of truth (so that truth emerges as both one and many), faces problems of her own. After raising specific difficulties we see with moderate pluralism, we propose a less extravagant way to prese...
Chapter 8 is the first of two chapters on the phenomenon of revenge paradoxes, paradoxes which, r... more Chapter 8 is the first of two chapters on the phenomenon of revenge paradoxes, paradoxes which, roughly speaking, are constructed out of the very terms of a purported solution. The chapter begins by exploring the difficulties that revenge presents for Kripke’s theory of truth, in either of two versions: a version that admits truth value gaps, and a paracomplete version which rejects the law of excluded middle. The chapter goes on to critically examine Field’s paracomplete theory of truth and its treatment of revenge, arguing that Field’s theory is couched in terms that are artificial and too far removed from natural language. The chapter concludes with a critical discussion of Priest’s dialetheist approach to the Liar paradox, according to which there are true contradictions. It is argued that Priest’s theory is itself subject to revenge paradoxes.
Chapter 7 puts the singularity theory to work on a number of semantic paradoxes that have intrins... more Chapter 7 puts the singularity theory to work on a number of semantic paradoxes that have intrinsic interest of their own. These include a transfinite paradox of denotation, and variations on the Liar paradox, including the Truth-Teller, Curry’s paradox, and paradoxical Liar loops. The transfinite paradox of denotation shows the need to accommodate limit ordinals. The Truth-Teller, like the Liar, exhibits semantic pathology-but, unlike the Liar, it does not produce a contradiction. The distinctive challenge of the Curry paradox is that it seems to allow us to prove any claim we like (for example, the claim that 2+2=5). Paradoxical Liar loops, such as the Open Pair paradox, extend the Liar paradox beyond single self-referential sentences. The chapter closes with the resolution of paradoxes that do not exhibit circularity yet still generate contradictions. These include novel versions of the definability paradoxes and Russell’s paradox, and Yablo’s paradox about truth.
Chapter 5 moves beyond the simple paradoxes discussed in Chapters 2-4. The chapter applies the si... more Chapter 5 moves beyond the simple paradoxes discussed in Chapters 2-4. The chapter applies the singularity approach to the traditional paradoxes of definability (or denotation), associated with Berry, Richard, and König. The chapter goes on to argue that there are two settings for Russell’s paradox, one in terms of the mathematical notion of set, and the other in terms of the logico-semantic notion of extension. The chapter then applies the singularity approach to Russell’s paradox for extensions. The chapter moves on to the case of truth, and applies the singularity approach to various versions of the Liar paradox, paying particular attention to the so-called strengthened Liar.
... KEITH SIMMONS The University of North Carolina ... In correspondence with Frege, Russell insi... more ... KEITH SIMMONS The University of North Carolina ... In correspondence with Frege, Russell insisted that Mont Blanc, "despite all its snowfields" (Frege [1979], p. 169), is a part of the proposition expressed by "Mont Blanc is more than 4000 metres high". ...
Dialetheism is the view that there are true contradictions. The strongest argument in favour of d... more Dialetheism is the view that there are true contradictions. The strongest argument in favour of dialetheism is that it alone allows us to solve semantic paradoxes like the liar paradox. This chapter presents two main criticisms of dialetheism. First, it argues that semantic pathology spreads to the dialetheist theory itself, putting into question the acceptability of the theory. Second, it argues that, even though dialetheism admits true contradictions, it is nevertheless subject to a revenge liar.
I argue for an account of semantic paradox that requires minimal logical revision. I first consid... more I argue for an account of semantic paradox that requires minimal logical revision. I first consider a phenomenon that is common to the paradoxes of definability, Russell’s paradox and the Liar. The phenomenon—which I call Repetition—is this: given a paradoxical expression, we can go on to produce a semantically unproblematic expression composed of the very same words. I argue that Kripke’s and Field’s theories of truth make heavy weather of Repetition, and suggest a simpler contextual account. I go on to outline a ‘singularity’ theory of semantical predicates in the spirit of remarks of Gödel. According to this theory, ‘denotes’, ‘extension’ and ‘true’ are context-sensitive expression that apply almost everywhere on a given occasion of use, except for certain singular points. I then turn to revenge paradoxes, and argue that even the dialetheist is subject to revenge. I then examine how the singularity theory responds to revenge.
In Thought and World, Chris Hill offers us the most sophisticated version of minimalism to date. ... more In Thought and World, Chris Hill offers us the most sophisticated version of minimalism to date. Hill's starting point is Horwich's minimal theory of truth (MT), according to which propositions - or thoughts in Hill's preferred terminology - are the truth-bearers, and the axioms of the theory are the thoughts expressed by sentences like "The thought that aardvarks amble is true if and only if aardvarks amble". But Hill's theory is superior to Horwich's in a number of crucial respects. For example, while Horwich gives up on a finite formulation of MT, taking it that an infinity of axioms is needed, Hill's basic theory (simple substitutionalism or SS) is composed of just one axiom. And unlike MT, SS can accommodate certain generalizations about truth (for example, the generalization that only thoughts are true, and general propositions about the truth-conditional properties of logical connectives.) Moreover, Hill provides a detailed deflationary theory not only of truth, but also of related semantic notions such as reference and concept-denotation, notions that have received relatively little attention in the literature on deflationism; he takes correspondence intuitions much more seriously than most deflationists, and is at pains to show how those intuitions may be accommodated within his framework; he provides the most developed deflationary account available of indexical concepts and indexical thoughts; and unlike many deflationists, he is not content to merely set aside problems concerning bi valence and the Liar paradox. Those of us interested in the proper treatment of our semantic concepts, whether we are deflationists or not, are in Hill's debt.
This book is about one of the most baffling of all paradoxes--the famous Liar paradox. Suppose we... more 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 ...
In 1905, Richard discovered his paradox of definability, and in a letter written that year he pre... more In 1905, Richard discovered his paradox of definability, and in a letter written that year he presented both the paradox and a solution to it.Soon afterwards, Poincaré endorsed a variant of Richard’s solution.In this paper, I critically examine Richard’s and Poincaré’s ways out.I draw on an objection of Peano’s, and argue that their stated solutions do not work.But I also
... 254-260); E. J. Ashworth, Language and logic in the post-medieval period (Dordrecht, Reidel, ... more ... 254-260); E. J. Ashworth, Language and logic in the post-medieval period (Dordrecht, Reidel, 1974), 104; Francesco Bottin, Le antinomie semantiche nella logica medievale (Padua, 1976), 173-175. ... 26 For example, Paul Spade and E. J. Ashworth encourage such a view. ...
The paradoxes of definability-Berry's paradox, Richard's paradox, and König's paradoxall exhibit ... more The paradoxes of definability-Berry's paradox, Richard's paradox, and König's paradoxall exhibit some form of circularity, or self-reference understood in a suitably broad sense. Consider Berry's paradox, for example. There are denumerably many positive integers, but only finitely many phrases of English with fewer than twenty eight syllables. So there is an integer which is the least positive integer not denoted by an English phrase with fewer than twenty eight syllables. Now consider the English phrase formed by the last sixteen words of the previous sentence. This phrase-call it the Berry phrase-denotes a positive integer. But the Berry phrase has twenty seven syllables. So the least positive integer which cannot be denoted by a phrase with less than twenty eight syllables is denoted by a phrase with twenty seven syllablescontradiction. Observe that the Berry phrase involves quantification over a certain domain of English phrases which contains the Berry phrase itself. Here is the circularity-or self-reference, in the sense that the Berry phrase makes indirect reference to itself. We may construct new versions of the definability paradoxes where the self-reference is more explicit. For example, consider the paradox generated by the following phrases written on the board in room 101: A. The ratio of the circumference of a circle to its diameter. B. The positive square root of 36. C. The sum of the numbers denoted by expressions on the board in room 101. 1 Are there definability paradoxes without self-reference? We can certainly say that there
This book aims to provide a solution to the semantic paradoxes. It argues for a unified solution ... more This book aims to provide a solution to the semantic paradoxes. It argues for a unified solution to the paradoxes generated by the concepts of reference or denotation, predicate extension, and truth. The solution makes two main claims. The first is that our semantic expressions ‘denotes’, ‘extension’, and ‘true’ are context-sensitive. The second, inspired by a brief, tantalizing remark of Gödel’s, is that these expressions are significant everywhere except for certain singularities, in analogy with division by zero. A formal theory of singularities is presented and applied to a wide variety of versions of the definability paradoxes, Russell’s paradox, and the Liar paradox. The book argues that the singularity theory satisfies the following desiderata: it recognizes that the proper setting of the semantic paradoxes is natural language, not regimented formal languages; it minimizes any revision to our semantic concepts; it respects as far as possible Tarski’s intuition that natural la...
Truth pluralism is a metaphysical theory of the nature of truth. The pluralist rejects the deflat... more Truth pluralism is a metaphysical theory of the nature of truth. The pluralist rejects the deflationist claim that truth is at best a ‘shallow’, insubstantial property. Indeed, the pluralist embraces a plurality of truth properties (such as correspondence, superassertibility, coherence), each appropriate to a different domain (or domains) of discourse. On the face of it, the pluralist will inherit all the main problems of the various traditional substantivist theories of truth. In addition, a strong pluralist, who only recognizes a plurality of truth properties (so that truth emerges not as one but only many), faces a number of problems, such as the problem of mixed discourse. But the moderate pluralism, who acknowledges that there is, in addition to diverse truth properties, a single property of truth (so that truth emerges as both one and many), faces problems of her own. After raising specific difficulties we see with moderate pluralism, we propose a less extravagant way to prese...
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