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Numerous solutions have been proposed to the semantic paradoxes. Two that are frequently singled out and compared are the following. The first is that according to which paradoxical sentences are neither true nor false — as it is sometimes put, they are semantic gaps. The second is that according to which paradoxical sentences are both true and false — as it is sometimes put, they are semantic gluts (dialetheias). Calling the first of these a solution is, in fact, somewhat misleading: it is rather like calling an opening gambit a game of chess. For the solution runs into severe problems almost immediately, and so can be only the first of a series of (often ad hoc) moves made to defend the original weak opening. Nonetheless, the symmetry involved in the gap and glut solutions is obvious enough to make the comparison a natural one.
Logical consequence is a notion that every person who reasons must possess, at least implicitly. To give a precise and accurate characterization of this notion is the fundamental task of logic. In a similar way, the notion of effectivity is a concept that anyone with a basic training in mathematics possesses, and the most fundamental task of a theory of computability is to give a precise characterization of this notion. The problem concerning effectivity was solved (at least to the satisfaction of most people) in the 1930s, almost as soon as it was raised, by the work of Turing, Church, and others. By contrast, the correct and precise characterization of logical consequence has been hotly contested through the two and a half thousand-year history of logic.
Ontological arguments feature prominently in the history of Christian philosophy. An ontological argument is, roughly, one that tries to establish the existence of God from God’s nature, or definition which captures that nature. The aim of this paper is not to present a survey of such arguments. Rather, the point is to home in on what I take to be the central nerve of such arguments: the Characterization Principle – essentially, a principle to the effect that an object has those properties it is characterised as having. The principle interacts in important ways with two other notions: existence and necessity. They will also, therefore, fall within the ambit of the discussion. We will analyse matters by looking at ontological arguments as presented at various historical times. The earliest ontological argument for a Christian god was given by Anselm of Canterbury. We come to him in due course. We will start with early modern philosophy, where the nerve of the argument is at its most exposed. We will then turn back to Anselm. After that, we will move on to later modern philosophy.
Buddhism and Marxism may seem unlikely bedfellows, since they come from such different times and places, and appear to address such different concerns. But the two have at least this much in common: both say that life, as we find it, is unsatisfactory; both have a diagnosis of why this is; and both offer the hope of making it better. In this paper, I argue that aspects of each complement aspects of the other. In particular, Buddhism provides a stable ethical base that Marxism always lacked; and Marxism provides a sophisticated political philosophy, which Buddhism never had. I will explain those aspects of each of the two on which I wish to draw, and then explain how they are complementary.
The semantic notions of truth and logical validity in predicate logic, being dependent on what the correlates of our universal terms are, demand at least a certain semantic clarification of the issue of universals. Apparently, the primary issue concerning universals is ontological. It should be clear that these objective concepts are non-conventionally objective. It should also be clear that the laws of logic in the framework are supposed to be fundamentally different from the laws of psychology. For while the former are the laws of the logical relations among objective concepts, the latter are the laws of the causal relations among formal concepts. Thus, whereas logic can be normative, prescribing the laws of valid inference, cognitive psychology can only be descriptive, describing and perhaps explaining the psychological mechanisms that can make us prone to certain types of logical errors.
My topic concerns the martial arts – or at least the East Asian martial arts, such as karatedo, taekwondo, kendo, wushu. To what extent what I have to say applies to other martial arts, such as boxing, silat, capoeira, I leave as an open question. I will illustrate much of what I have to say with reference to karatedo, since that is the art with which I am most familiar; but I am sure that matters are much the same with other East Asian martial arts.
Towards Non-Being (Priest ) develops an account of the semantics of intentional predicates and operators. The account appeals to objects, both existent and non-existent, and worlds, both possible and impossible. This paper formulates replies to a number of the more interesting objections to the semantics that have been proposed since the book was published.
Towards NonBeing (Priest, 2005) gives a noneist account of the semantics of intentional operators and predicates. The semantics for intentional operators are modelled on those for the □ in normal modal logics. In this paper an alternative semantics, modelled on neighborhood semantics for □, is given and assessed.
1.1 In standard modal logics, the worlds are 2-valued in the following sense: there are 2 values (true and false) that a sentence may take at a world. Technically, however, there is no reason why this has to be the case. The worlds could be many-valued. This paper presents one simple approach to a major family of many-valued modal logics, together with an illustration of why this family is philosophically interesting.
The paper argues that the view that the particular quantifier is ‘existentially loaded’ is a relatively new one historically and that it has become entrenched in modern philosophical logic for less than happy reasons.
23.1.1 This chapter brings together the techniques of previous chapters, to look at a variety of logics that they may generate. The chapter also acts as a bridge between the basic system of relevant logic of the last chapter, First Degree Entailment, and the full relevant logics of the next.
23.1.2 By this stage of the book we have many independent techniques that may be employed in constructing the semantics of a logic: normal and non-normal worlds, constant and variable domains, different numbers of truth values, negation using many values and the * semantics, necessary and contingent identity. These techniques can be combined to produce a vast variety of logics, far too many to consider here. We will consider only some of the more notable ones.
23.1.3 We begin with the basic relevant logics N4 and N*, starting with the former. This will require an application of the matrix semantics employed for non-normal modal logics in chapter 18. The logics K4 and K* are then obtained as special cases. We will look only at the constant domain versions of the logics.
23.1.4 Identity for the logics in question is next on the agenda. We will concern ourselves only with necessary identity, though the behaviour of identity at non-normal worlds gives it something of the flavour of contingent identity.
23.1.5 There is then a philosophical interlude concerning one application of identity in relevant logic: relevant predication.
Part I of the book has explored, in various ways, a relevant account of conditionals. Such an account seems to me to be better than any of the other accounts that we have traversed in the course of Part I. This is, naturally, a contentious view. Logic is a contentious subject, and the conditional has been particularly so since the earliest years of the discipline. It was the Stoic logicians who first discussed conditionals explicitly, and they had at least four competing accounts. These accounts survived – one way or another – and others were added throughout the Middle Ages. Consensus might have been reached locally, but only locally.
The changes in logic at the beginning of the twentieth century were revolutionary. The power of the mathematical techniques employed by the founders of modern logic made anything before obsolete. (Which is not to say that there is not now a good deal to be learned from it – just that whatever is of value in it must be seen through radically new eyes.) It is perhaps not surprising, then, that their work established a very general consensus over the conditional. The view of the conditional as material became highly orthodox – though never universal, as C. I. Lewis bears witness.
Digesting the results of the revolution occupied logicians in the first half of the century. But the second half was quite different.
14.1.1 In this chapter we will start to look at quantified normal modal logics. These come in two varieties: constant domain (where the domain of quantification is the same in all worlds), and variable domain (where the domain may vary from world to world).
14.1.2 Where it is necessary to distinguish between the two, I will use the following notation. If S is any system of propositional modal logic, CS will denote the constant domain quantified version, and VS will denote the variable domain quantified version.
14.1.3 In this chapter we will look at the semantics and tableaux for constant domain logics, saving variable domains for the next.
14.1.4 For these two chapters we will take it that identity is not part of the language. We will turn to the topic of identity in modal logic in chapter 16.
14.1.5 We will also take a quick look at one of the major philosophical issues to which quantified modal logic gives rise: the issue of essentialism.
14.1.6 The chapter ends by showing how the semantic and tableau techniques of normal modal logic extend to tense logic.
Constant Domain K
14.2.1 The syntax of quantified modal logic augments the language of first order classical logic (12.2) with the operators □ and ◇, as propositional modal logic extends classical propositional logic (2.3.1, 2.3.2).
4.1.1 In this chapter we look at some systems of modal logic weaker than K (and so non-normal). These involve so-called non-normal worlds. Nonnormal worlds are worlds where the truth conditions of modal operators are different.
4.1.2 We are then in a position to return to the issue of the conditional, and have a look at an account of a modal conditional called the strict conditional.
4.2.1 Let us start by looking at the technicalities concerning non-normality. In due course we will be able to discuss what they mean.
4.2.2 A non-normal interpretation of a modal propositional language is a structure, 〈W, N, R, ν〉, where W, R and ν are as in previous chapters, and N ⊆ W. Worlds in N are called normal. Worlds in W – N (the worlds that are not normal) are called non-normal.
4.2.3 The truth conditions for the truth functions, ∧, ∨, ¬, etc. are the same as before (2.3.4). The truth conditions for □ and ◇ at normal worlds are also as before (2.3.5). But if w is non-normal:
νw(□A) = 0
νw(◇A) = 1
In a sense, at non-normal worlds, everything is possible, and nothing is necessary.
4.2.4 Note that at every world, w, ¬□A and ◇¬A still have the same truth value, as do ¬◇A and □¬A. We saw this to be the case for normal worlds in 2.3.9 and 2.3.10.