Dialetheists’ Lies About the Liar

Liar-like paradoxes are typically arguments that, by using very intuitive resources of natural language, end up in contradiction. Consistent solutions to those paradoxes usually have difficulties either because they restrict the expressive power of the language, or else because they fall prey to extended versions of the paradox. Dialetheists, like Graham Priest, propose that we should take the Liar at face value and accept the contradictory conclusion as true. A logical treatment of such contradictions is also put forward, with the Logic of Paradox (LP), which should account for the manifestations of the Liar. In this paper we shall argue that such a formal approach, as advanced by Priest, is unsatisfactory. In order to make contradictions acceptable, Priest has to distinguish between two kinds of contradictions, internal and external, corresponding, respectively, to the conclusions of the simple and of the extended Liar. Given that, we argue that while the natural interpretation of LP was intended to account for true and false sentences, dealing with internal contradictions, it lacks the resources to tame external contradictions. Also, the negation sign of LP is unable to represent internal contradictions adequately, precisely because of its allowance of sentences that may be true and false. As a result, the formal account suffers from severe limitations, which make it unable to represent the contradiction obtained in the conclusion of each of the paradoxes.

Negation

The paper is concerned with negation in artificial and natural languages. "Negation" is an ambiguous word. It can mean three different things: An operation(negating), an operator (a sign of negation), the result of an operation. The threethings, however, are intimately linked. An operation such as negation, is realizedthrough an operator of negation, i.e. consists in adding a symbol of negation to an entity to obtain an entity of the same type; and which operation it is dependson what it applies to and on what results from its application. I argue that negation is not an operation on linguistic acts but rather anoperation on the objects of linguistic acts, namely sentences. And I assume that the negation of a sentence is a sentence that contradicts it. If so, the negation of a sentence may be obtained, in case the sentence is molecular, by applying the operation of negation not to the sentence itself but to a constituent sentence. To put it in a succinct and paradoxically sounding way we could say that in order to negate a sentence it is sufficient but not necessary to negate it. However that negation applies to sentences is true only for artificial languages, in which the sign of negation is a monadic sentential connective. In natural language, negation applies to expressions other than sentences, namely word sand non-sentential phrases. Still words and not sentential phrases are interesting and valuable only as ultimate or immediate constituents of sentences, as a means of saying (something that can be true or false) and the concern with negation is ultimately the concern with the negation of sentences. So the problem is what sub-sentential and non sentential expressions negation should apply to in order to obtain the negation of the containing sentence. The standard answer is that the negation of a natural language sentence is equivalent to the negation of its predicate. Yet, I argue, predicate negation is necessary but not sufficient, due to the existence of molecular sentences. Finally I notice that if to apply negation to an artificial sentence is to put the negation sign in front of it, to negate the predicate of a natural language sentencemay or may not be to put the negation sign in front of it.

The Development and Evaluation of Pictographic Symbols

The premise adopted in this study is that representatives of the audience for whom a symbol is intended should be participants in its evolution as well as subjects in its evaluation. Several situations in need of product misuse warnings were supplied by a manufacturer of ovenware products. Symbol design possibilities were first generated for each message category and then design input was obtained from a sample of potential product users. New design candidates were developed on the basis of subject recommendations. Study generated symbols proved to be significantly more effective than designs used by the manufacturer for the same message categories as assessed by differences in reaction time and error rate. The relative effectiveness of different negation sign designs was also evaluated. Differences in both reaction time and subjective ranks of communicativeness suggest that a thin black cross is more effective in conveying negation than a thin black slash, a partial slash or cross, and a contour slash or cross. Significant differences were not found in the extent that the designs interfere with symbol recognition.

Systems of modal logic which are not unreasonable in the sense of Halldén

Halldén, in [1], has recently pointed out that it is highly undesirable, in a system of sentential calculus, for there to exist two formulas α and β such that: (i) α and β contain no variable in common; (ii) neither α nor β is provable; (iii) α ∨ β is provable. We shall call a system unreasonable (in the sense of Halldén) if there exists a pair of formulas α and β having properties (i), (ii), and (iii). Halldén shows (in [1]) that the Lewis systems S1 and S3 are unreasonable in this sense; and that the same is true of any system which is between S1 and S3, as well as of every system which is stronger than S3 but weaker than both S4 and S7. In the present note we shall show that this defect does not occur in S4, nor in S5, nor in any “quasi-normal” extension of S5; we give an example, on the other hand, of an unreasonable system which lies between S4 and S5.When we speak, in what follows, of a system of modal logic, we shall mean a system having the same class of well-formed formulas as have the various Lewis calculi. Thus the well-formed formulas of a system of modal logic, when written in unabbreviated form, are just those formulas which can be built up from sentential variables by use of the binary connective ‘·’ (conjunction sign), and the two unary connectives ‘˜’ (negation sign) and ‘◇’ (possibility sign). We shall, however, also make use of some of the defined signs of Lewis.