Franz Berto

Berto’s highly readable and lucid guide introduces students and the interested reader to Gödel’s celebrated _Incompleteness Theorem_, and discusses some of the most famous - and infamous - claims arising from Gödel's arguments. Offers a clear understanding of this difficult subject by presenting each of the key steps of the _Theorem_ in separate chapters Discusses interpretations of the _Theorem_ made by celebrated contemporary thinkers Sheds light on the wider extra-mathematical and philosophical implications of Gödel’s theories Written in an accessible, non-technical style.

A logic is called 'paraconsistent' if it rejects the rule called 'ex contradictione quodlibet', according to which any conclusion follows from inconsistent premises. While logicians have proposed many technically developed paraconsistent logical systems and contemporary philosophers like Graham Priest have advanced the view that some contradictions can be true, and advocated a paraconsistent logic to deal with them, until recent times these systems have been little understood by philosophers. This book presents a comprehensive overview on paraconsistent logical systems to change this situation. The book includes almost every major author currently working in the field. The papers are on the cutting edge of the literature some of which discuss current debates and others present important new ideas. The editors have avoided papers about technical details of paraconsistent logic, but instead concentrated upon works that discuss more 'big picture' ideas. Different treatments of paradoxes takes centre stage in many of the papers, but also there are several papers on how to interpret paraconistent logic and some on how it can be applied to philosophy of mathematics, the philosophy of language, and metaphysics.

I want to model a finite, fallible cognitive agent who imagines that p in the sense of mentally representing a scenario—a configuration of objects and properties—correctly described by p. I propose to capture imagination, so understood, via variably strict world quantifiers, in a modal framework including both possible and so-called impossible worlds. The latter secure lack of classical logical closure for the relevant mental states, while the variability of strictness captures how the agent imports information from actuality in the imagined non-actual scenarios. Imagination turns out to be highly hyperintensional, but not logically anarchic. Section 1 sets the stage and impossible worlds are quickly introduced in Sect. 2. Section 3 proposes to model imagination via variably strict world quantifiers. Section 4 introduces the formal semantics. Section 5 argues that imagination has a minimal mereological structure validating some logical inferences. Section 6 deals with how imagination under-determines the represented contents. Section 7 proposes additional constraints on the semantics, validating further inferences. Section 8 describes some welcome invalidities. Section 9 examines the effects of importing false beliefs into the imagined scenarios. Finally, Sect. 10 hints at possible developments of the theory in the direction of two-dimensional semantics.

It has been said that, when some paraconsistent logicians supporting dialetheism assert: "For some sentence α, both α and not-α are true", therefore claiming that the Law of Non-Contradiction (LNC) fails, we should wonder what "true" and "not" mean here. After surveying two classical paraconsistent approaches to negation (provided by da Costa's positive-plus systems and Graham Priest's Logic of Paradox), I describe a negation with the following features: (1) its definition does not make reference to the controversial concept truth; (2) it has strong pre-theoretical appeal and motivation, because it performs an indispensable expressive function in language and communication; and (3) it is accepted by dialetheists, too, since it is based on a very deep metaphysical intuition they also show to fully share: this intuition I call the one of material exclusion. If my characterization is sufficient to confer a determinate meaning to the negation in question, we can conveniently formulate via this negation a version of the LNC which I take, therefore, to be indisputable also from the dialetheist's point of view. Such a result, however, does not constitute a quick and easy success against supporters of true contradictions. It may simply show that the versions of the LNC dialetheists most convincingly attack are those that were not to be defended, and that supporters of consistency have been historically confused in assimilating them to the indisputable one. Quine's famous argument, that to change the logic is to change the subject, may be right to this extent: classical negation and non-classical negations have different meanings. But the substantial issue that Quine never addressed is why we should suppose that the meaning of the vernacular negation is classical. © 2011 Elsevier B.V., All rights reserved.