Benjamin Alan Burgis - Logic Without Gaps or Gluts: How to Solve the Paradoxes Without Sacrificing Classical Logic (Synthese Library, 458)
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Logic Without Gaps or Gluts: How to Solve the Paradoxes Without Sacrificing Classical Logic (Synthese Library, 458): summary, description and annotation
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This book offers a defense against non-classical approaches to the paradoxes. The author argues that, despite appearances, the paradoxes give no reason at all to reject classical logic. In fact, he believes classical solutions fare better than non-classical ones with respect to key tests like Currys Paradox, a Liar-like paradox that dialetheists are forced to solve in a way totally disjoint from their solution to the Liar.
Graham Priests In Contradiction was the first major work that advocated the use of non-classical approaches. Since then, these views have moved into the philosophical mainstream. Much of this movement is fueled by a widespread sense that these logically heterodox solutions get to the real nub of the issue. They lack the ad hoc feel of many other solutions to the paradoxes. The author believes that its long past time for a response to these attacks against classical orthodoxy. He presents a non-logically-revisionary solution to the paradoxes.
This title offers a literal way of cashing out the disquotation metaphor. While the details of the view are novel, the idea has a pre-history in the relevant literature. The author examines objections in detail. He rejects each in turn and concludes by comparing the virtues of his logically orthodox approach with those of the paraconsistent and paracomplete competition.
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The aim of Synthese Library is to provide a forum for the best current work in the methodology and philosophy of science and in epistemology, all broadly understood. A wide variety of different approaches have traditionally been represented in the Library, and every effort is made to maintain this variety, not for its own sake, but because we believe that there are many fruitful and illuminating approaches to the philosophy of science and related disciplines.
Special attention is paid to methodological studies which illustrate the interplay of empirical and philosophical viewpoints and to contributions to the formal (logical, set-theoretical, mathematical, information-theoretical, decision-theoretical, etc.) methodology of empirical sciences. Likewise, the applications of logical methods to epistemology as well as philosophically and methodologically relevant studies in logic are strongly encouraged. The emphasis on logic will be tempered by interest in the psychological, historical, and sociological aspects of science. In addition to monographs Synthese Library publishes thematically unified anthologies and edited volumes with a well-defined topical focus inside the aim and scope of the book series. The contributions in the volumes are expected to be focused and structurally organized in accordance with the central theme(s), and should be tied together by an extensive editorial introduction or set of introductions if the volume is divided into parts. An extensive bibliography and index are mandatory.
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The Liar and related paradoxes create a prima facie conflict between two equally important principles of classical logicthe Law of the Excluded Middle (LEM) and the Law of Non-Contradiction (LNC). Dialetheists such as Graham Priest and JC Beall solve the paradoxes by accepting the negations of some instances of the LNC and rejecting classical logic in favor of a paraconsistent logical framework in which contradictions do not entail every conclusion. Hartry Field advocates the mirror image of this solution, rejecting certain instances of the LEM and thus rejecting classical logic in favor of a paracomplete framework in which not every LEM instance is entailed by every premise. Some paradox-solvers, eager to avoid either of these amputations of classical logic, reject paradoxical instances of the T-Schema. This is an easy way out, but the search for an adequate solution is best conducted elsewhere. The rest of the book will be spent on the project of resisting the paraconsistent and paracomplete solutions to the paradoxes and finally proposing a positive solution that retains classical logic without sacrificing the T-Schema.
Sentence (1) is not true.
- (i)
(1) is true
- (ii)
(1) is not true
- (iii)
(1) is both true and not true.
To fully appreciate the problem all of this poses for classical orthodoxy, let us take a moment to unpack that familiar reasoning. Tarskis Biconditional Truth Schema (henceforth, the T-Schema) tells us that for every proposition , Tr < > (). Since (1) just is Tr < L>, when we feed it into the T-Schema, we get Tr < L > Tr < L>. There are multiple roads from here to contradictionthe inference is clear enough that classical logicians often talk loosely and say that a sentence of the form is a contradictionbut all of them involve the LEM.
The most obvious way to solve the paradox without sacrificing classical logic would be to reject (some instances of) the T-Schema, so that we could assert either (i) or (ii) without being stuck with (iii). Gilbert Harman takes this line in his 1986 classic
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