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An introduction to substructural logics
, 2000
"... Abstract: This is a history of relevant and substructural logics, written for the Handbook of the History and Philosophy of Logic, edited by Dov Gabbay and John Woods. 1 1 ..."
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Abstract: This is a history of relevant and substructural logics, written for the Handbook of the History and Philosophy of Logic, edited by Dov Gabbay and John Woods. 1 1
A revengeimmune solution to the semantic paradoxes
 Journal of Philosophical Logic
"... 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 “ord ..."
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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. 1
The semantic paradoxes and the paradoxes of vagueness
, 2003
"... Both in dealing with the semantic paradoxes and in dealing with vagueness and indeterminacy, there is some temptation to weaken classical logic: in particular, to restrict the law of excluded middle. The reasons for doing this are somewhat different in the two cases. In the case of the semantic para ..."
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Both in dealing with the semantic paradoxes and in dealing with vagueness and indeterminacy, there is some temptation to weaken classical logic: in particular, to restrict the law of excluded middle. The reasons for doing this are somewhat different in the two cases. In the case of the semantic paradoxes, a weakening of classical logic (presumably involving a restriction of excluded middle) is required if we are to preserve the naive theory of truth without inconsistency. In the case of vagueness and indeterminacy, there is no worry about inconsistency; but a central intuition is that we must reject the factual status of certain sentences, and it hard to see how we can do that while claiming that the law of excluded middle applies to those sentences. So despite the different routes, we have a similar conclusion in the two cases. There is also some temptation to connect up the two cases, by viewing the semantic paradoxes as due to something akin to vagueness or indeterminacy in semantic concepts like ‘true’. The thought is that the notion of truth is introduced by a schema that might initially appear to settle its extension uniquely:
Solving the Paradoxes, escaping revenge
 THE REVENGE OF THE LIAR
"... 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 h ..."
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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). I think the received wisdom is incorrect. In my effort to rebut it, I will focus on a certain type of solution to the paradoxes. This type of solution has the advantage of keeping the full Tarski truth schema (T) True(hAi) ↔ A (and more generally, a full satisfaction schema). This has a price, namely that
Maudlin’s truth and paradox
 Philosophy and Phenomenological Research
, 2005
"... Tim Maudlin’s Truth and Paradox is terrific. In some sense its solution to the paradoxes is familiar—the book advocates an extension of what’s called the KripkeFeferman theory (although the definition of validity it employs disguises this fact). Nonetheless, the perspective it casts on that solutio ..."
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Tim Maudlin’s Truth and Paradox is terrific. In some sense its solution to the paradoxes is familiar—the book advocates an extension of what’s called the KripkeFeferman theory (although the definition of validity it employs disguises this fact). Nonetheless, the perspective it casts on that solution is completely novel, and Maudlin uses this perspective to try to make the prima facie unattractive features of this solution seem palatable, indeed inescapable. Moreover, the book deals with many important issues that most writers on the paradoxes never deal with, including issues about the application of the Gödel theorems to powerful theories that include our theory of truth. The book includes intriguing excursions into general metaphysics, e.g. on the nature of logic, facts, vagueness, and much more; and it’s lucid and lively, a pleasure to read. It will interest a wide range of philosophers. 1. A virtue of Maudlin’s book is that it isn’t concerned only with the semantics of languages with truth predicates, but also with their inferential structure. Let λ be a name of a sentence that asserts its own untruth: the sentence it names is ¬Tλ. (‘T ’ abbreviates ‘True’.) In the first chapter Maudlin considers the following argument (where <A> is the name of sentence A): 1. Tλ is T<¬Tλ>, so by “Downward Tinference ” (the rule that allows the inference from
Where the Paths Meet: Remarks on Truth and Paradox∗
"... The study of truth is often seen as running on two separate paths: the nature path and the logic path. The former concerns metaphysical questions about the ‘nature’, if any, of truth. The latter concerns itself largely with logic, particularly logical issues arising from the truththeoretic paradoxe ..."
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The study of truth is often seen as running on two separate paths: the nature path and the logic path. The former concerns metaphysical questions about the ‘nature’, if any, of truth. The latter concerns itself largely with logic, particularly logical issues arising from the truththeoretic paradoxes. Where, if at all, do these two paths meet? It may seem, and it is all too often assumed, that they do not meet, or at best touch in only incidental ways. It is often assumed that work on the metaphysics of truth need not pay much attention to issues of paradox and logic; and it is likewise assumed that work on paradox is independent of the larger issues of metaphysics. Philosophical work on truth often includes a footnote anticipating some resolution of the paradox, but otherwise tends to take no note of it. Likewise, logical work on truth tends to have little to say about metaphysical presuppositions, and simply articulates formal theories, whose strength may be measured, and whose properties may be discussed. In practice, the paths go their own ways. Our aim in this paper is somewhat modest. We seek to illustrate one point
Reaching transparent truth
"... This paper presents and defends a way to add a transparent truth predicate to classical logic, such that T 〈A 〉 and A are everywhere intersubstitutable, where all Tbiconditionals hold, and where truth can be made compositional. A key feature of our framework, called STT (for StrictTolerant Truth), ..."
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This paper presents and defends a way to add a transparent truth predicate to classical logic, such that T 〈A 〉 and A are everywhere intersubstitutable, where all Tbiconditionals hold, and where truth can be made compositional. A key feature of our framework, called STT (for StrictTolerant Truth), is that it supports a nontransitive relation of consequence. At the same time, it can be seen that the only failures of transitivity STT allows for arise in paradoxical cases. 1
Midwest Studies in Philosophy, XXXII (2008) Where the Paths Meet: Remarks on Truth and Paradox*
"... The study of truth is often seen as running on two separate paths: the nature path and the logic path.The former concerns metaphysical questions about the “nature, ” if any, of truth. The latter concerns itself largely with logic, particularly logical issues arising from the truththeoretic paradoxe ..."
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The study of truth is often seen as running on two separate paths: the nature path and the logic path.The former concerns metaphysical questions about the “nature, ” if any, of truth. The latter concerns itself largely with logic, particularly logical issues arising from the truththeoretic paradoxes. Where, if at all, do these two paths meet? It may seem, and it is all too often assumed, that they do not meet, or at best touch in only incidental ways. It is often assumed that work on the metaphysics of truth need not pay much attention to issues of paradox and logic; and it is likewise assumed that work on paradox is independent of the larger issues of metaphysics. Philosophical work on truth often includes a footnote anticipating some resolution of the paradox, but otherwise tends to take no note of it. Likewise, logical work on truth tends to have little to say about metaphysical presuppositions, and simply articulates formal theories, whose strength may be measured, and whose properties may be discussed. In practice, the paths go their own ways. Our aim in this paper is somewhat modest. We seek to illustrate one point of intersection between the paths. Even so, our aim is not completely modest, as the point of intersection is a notable one that often goes unnoticed. We argue that the “nature ” path impacts the logic path in a fairly direct way. What one can and must
c © B.E Eva This work is licensed under the Creative Commons Attribution License. Towards a Paraconsistent Quantum Set Theory
"... In this paper, we will attempt to establish a connection between quantum set theory, as developed by Ozawa, Takeuti and Titani (see, for example, [13], [12], [10]), and topos quantum theory, as developed by Isham, Butterfield and Döring, amongst others (see, for example, [8], [6], [3]). Towards thi ..."
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In this paper, we will attempt to establish a connection between quantum set theory, as developed by Ozawa, Takeuti and Titani (see, for example, [13], [12], [10]), and topos quantum theory, as developed by Isham, Butterfield and Döring, amongst others (see, for example, [8], [6], [3]). Towards this end, we will study algebraic valued settheoretic structures whose truth values correspond to the clopen subobjects of the spectral presheaf of an orthomodular lattice of projections onto a given Hilbert space. In particular, we will attempt to recreate, in these new structures, Takeuti’s original isomorphism between the set of all Dedekind real numbers in a suitably constructed model of set theory and the set of all self adjoint operators on a chosen Hilbert space. 1