In Philosophical Logic, the Liar Paradox has been used to motivate the introduction of both truth value gaps and truth value gluts. Moreover, in the light of “revenge Liar” arguments, also higher-order combinations of generalized truth values have been suggested to account for so-called hyper-contradictions. In the present paper, Graham Priest's treatment of generalized truth values is scrutinized and compared with another strategy of generalizing the set of classical truth values and defining an entailment relation on the resulting sets of (...) higher-order values. This method is based on the concept of a multilattice. If the method is applied to the set of truth values of Belnap's “useful four-valued logic”, one obtains a trilattice, and, more generally, structures here called Belnap-trilattices. As in Priest's case, it is shown that the generalized truth values motivated by hyper-contradictions have no effect on the logic. Whereas Priest's construction in terms of designated truth values always results in his Logic of Paradox, the present construction in terms of truth and falsity orderings always results in First Degree Entailment. However, it is observed that applying the multilattice-approach to Priest's initial set of truth values leads to an interesting algebraic structure of a “bi-and-a-half” lattice which determines seven-valued logics different from Priest's Logic of Paradox. (shrink)

What is the rational response when confronted with a set of propositions each of which we have some reason to accept, and yet which taken together form an inconsistent class? This was, in a nutshell, the problem addressed by the Jaina logicians of classical India, and the solution they gave is, I think, of great interest, both for what it tells us about the relationship between rationality and consistency, and for what we can learn about the logical basis of philosophical (...) pluralism. The Jainas claim that we can continue to reason in spite of the presence of inconsistencies, and indeed construct a many-valued logical system tailored to the purpose. My aim in this paper is to offer a new interpretation of that system and to try to draw out some of its philosophical implications. (shrink)

In a paper on the logical work of the Jains, Graham Priest considers a consequence relation, semantically characterized, which has a natural analogue in modal logic. Here we give a syntactic/axiomatic description of the modal formulas which are consequences of the empty set by this relation, which is to say: those formulas which are, for every model, true at some point in that model.

Let be the class of frames satisfying the condition.

In this paper a propositional logic of viewpoints is presented. The language of this logic consists of the usual modal operatorsL (of necessity) andM (of possibility) as well as of two new operatorsA andR. The intuitive interpretations ofA andR are from all viewpoints and from some viewpoint, respectively. Semantically the language is interpreted by using Kripke models augmented with sets of viewpoints and with a new alternativeness relation for the operatorA. Truth values of formulas are evaluated with respect to a (...) world and a viewpoint. Various axiomatizations of the logic of viewpoints are presented and proved complete. Finally, some applications are given. (shrink)

This is the first introductory textbook on non-classical propositional logics.

For nearly two thousand years Buddhism has mystified and captivated both lay people and scholars alike. Seen alternately as a path to spiritual enlightenment, an system of ethical and moral rubrics, a cultural tradition, or simply a graceful philosophy of life, Buddhism has produced impassioned followers the world over. The Buddhist saint Nagarjuna, who lived in South India in approximately the first century CE, is undoubtedly the most important, influential, and widely studied Mahayana Buddhist philosopher. His many works include texts (...) addressed to lay audiences, letters of advice to kings, and a set of penetrating metaphysical and epistemological treatises. His greatest philosophical work, the Mulamadhyamikakarika--read and studied by philosophers in all major Buddhist schools of Tibet, China, Japan, and Korea--is one of the most influential works in the history of Indian philosophy. Now, in The Foundations of the Philosophy of the Middle Way, Jay L. Garfield provides a clear and and eminently readable translation of Nagarjuna's seminal work, offering those with little of no prior knowledge of Buddhist philosophy a view into the profound logic of the Mulamadhyamikakarika. Translated from the Tibetan, the tradition through which Nagarjuna's philosophical influence has largely been transmitted, Garfield presents a superb translation of Mulamadhyamikakarika in its entirety. Illuminating the systematic character of Nagarjuna's reasoning, as well as the works profundity, Garfield shows how Nagarjuna develops his doctrine that all phenomena are empty of inherent existence and essenceless. But, he argues, phenomena nonetheless exist conventionaly, and that indeed conventional existence and ultimate emptiness are in fact the same thing. This represents the radical understanding of the Buddhist doctrine of the two truths, or two levels of reality. Nagarjuna reinterprets all of Buddhist metaphysics and epistemology through this analytical framework--"a systematic and beautifully elegant philosophical dissection of reality." In turn, Garfield goes on to offer the only verse-by-verse commentary based upon the Indo-Tibetan Prasangika-Madhyamika reading of Nagarjuna, the school most influential in the development of Mahayana philosophy in Tibet, China, Korea, and Japan. Written specifically for the Western reader, the commentary explains Nagarjuna's positions and arguments in the language of Western metaphysics and epistemology, and connects Nagarjuna's concerns tho those of Western philosophers such as Sextus, Hume, and Wittgenstein. A fascinating and accessible translation of the foundational text for all Mahayana Buddhism text, The Fundamental Wisdom of the Middle Way will enlighten all those in search of the essence of reality. (shrink)

This revised and considerably expanded 2nd edition brings together a wide range of topics, including modal, tense, conditional, intuitionist, many-valued, paraconsistent, relevant, and fuzzy logics. Part 1, on propositional logic, is the old Introduction, but contains much new material. Part 2 is entirely new, and covers quantification and identity for all the logics in Part 1. The material is unified by the underlying theme of world semantics. All of the topics are explained clearly using devices such as tableau proofs, and (...) their relation to current philosophical issues and debates are discussed. Students with a basic understanding of classical logic will find this book an invaluable introduction to an area that has become of central importance in both logic and philosophy. It will also interest people working in mathematics and computer science who wish to know about the area. (shrink)

Original in content and approach, Philosophy in Classical India focuses on the rational principles of Indian philosophical theory, rather than the mysticism usually associated with it. Ganeri explores the philosophical projects of a number of major Indian philosophers and looks into the methods of rational inquiry deployed within these projects. In so doing, he illuminates a network of mutual reference and criticism, influence and response, in which reason is simultaneously used constructively and to call itself into question.

Graham Priest has asked whether the consequence relation associated with the Anderson–Belnap system of Tautological Entailment,1 in the language with connectives ¬, ∧, ∨, and countably many propositional variables as tomic formulas, maximal amongst the substitution-invariant relevant consequence relations on this language. Here a consequence relation is said to be relevant just in case whenever for a set of formulas Γ and formula B, we have Γ B only if some propositional variable occurring in B occurs in at least one (...) formula in Γ. (It follows that relevant consequence relations are atheorematic in the sense that whenever Γ B for some such consequence relation , Γ = ∅.) Here I write up in more detail the upshot of the conversation – returning an affirmative answer to Priest’s question – about this in the common room that Greg Restall and I were participating in last Friday [ = October 6, 2006], dotting some “i”s and crossing some “t”s (and adding the odd further reflection). (shrink)