Entailment and Bivalence

My purpose in this paper is to argue that the classical notion of entailment is not suitable for non-bivalent logics, to propose an appropriate alternative and to suggest a generalized entailment notion suitable to bivalent and non-bivalent logics alike. In classical two valued logic, one can not infer a false statement from one that is not false, any more than one can infer from a true statement a statement that is not true. In classical logic in fact preserving truth and preserving non-falsity are one and the same thing. They are not the same in non-bivalent logics however and I will argue that the classical notion of entailment that preserves only truth is not strong enough for such a logic. I will show that if we retain the classical notion of entailment in a logic that has three values, true, false and a third value in between, an inconsistency can be derived that can be resolved only by measures that seriously disable the logic. I will show this for a logic designed to allow for semantic presuppositions, then I will show that we get the same result in any three valued logic with the same value ordering. I will finally suggest how the notion of entailment should be generalized so that this problem may be avoided. The strengthened notion of entailment I am proposing is a conservative extension of the classical notion that preserves not only truth but the order of all values in a logic, so that the value of an entailed statement must alway be at least as great as the value of the sequence of statements entailing it. A notion of entailment this strong or stronger will, I believe, be found to be applicable to non-classical logics generally. In the opinion of Dana Scott, no really workable three valued logic has yet been developed. It is hard to disagree with this. A workable three valued logic however could perhaps be developed however, if we had a notion of entailment suitable to non-bivalent logics.     

On the Standard and Rational Completeness of some Axiomatic Extensions of the Monoidal T-norm Logic

The monoidal t-norm based logic MTL is obtained from Hájek's Basic Fuzzy logic BL by dropping the divisibility condition for the strong (or monoidal) conjunction. Recently, Jenei and Montgana have shown MTL to be standard complete, i.e. complete with respect to the class of residuated lattices in the real unit interval [0,1] defined by left-continuous t-norms and their residua. Its corresponding algebraic semantics is given by pre-linear residuated lattices. In this paper we address the issue of standard and rational completeness (rational completeness meaning completeness with respect to a class of algebras in the rational unit interval [0,1]) of some important axiomatic extensions of MTL corresponding to well-known parallel extensions of BL. Moreover, we investigate varieties of MTL algebras whose linearly ordered countable algebras embed into algebras whose lattice reduct is the real and/or the rational interval [0,1]. These embedding properties are used to investigate finite strong standard and/or rational completeness of the corresponding logics.     

Putnam, Peano, and the Malin Génie: could we possibly bewrong about elementary number-theory?

This article examines Hilary Putnam's work in the philosophy of mathematics and - more specifically - his arguments against mathematical realism or objectivism. These include a wide range of considerations, from Gödel's incompleteness-theorem and the limits of axiomatic set-theory as formalised in the Löwenheim-Skolem proof to Wittgenstein's sceptical thoughts about rule-following (along with Saul Kripke's ‘scepticalsolution’), Michael Dummett's anti-realist philosophy of mathematics, and certain problems – as Putnam sees them – with the conceptual foundations of Peano arithmetic. He also adopts a thought-experimental approach – a variant of Descartes' dream scenario – in order to establish the in-principle possibility that we might be deceived by the apparent self-evidence of basic arithmetical truths or that it might be ‘rational’ to doubt them under some conceivable (even if imaginary) set of circumstances. Thus Putnam assumes that mathematical realism involves a self-contradictory ‘Platonist’ idea of our somehow having quasi-perceptual epistemic ‘contact’ with truths that in their very nature transcend the utmost reach of human cognitive grasp. On this account, quite simply, ‘nothing works’ in philosophy of mathematics since wecan either cling to that unworkable notion of objective (recognition-transcendent) truth or abandon mathematical realism in favour of a verificationist approach that restricts the range of admissible statements to those for which we happen to possess some means of proof or ascertainment. My essay puts the case, conversely, that these hyperbolic doubts are not forced upon us but result from a false understanding of mathematical realism – a curious mixture of idealist and empiricist themes – which effectively skews the debate toward a preordained sceptical conclusion. I then go on to mount a defence of mathematical realism with reference to recent work in this field and also to indicate some problems – as I seethem – with Putnam's thought-experimental approach as well ashis use of anti-realist arguments from Dummett, Kripke, Wittgenstein, and others.     

Effing the Ineffable: The Logocentric Fallacy in Argumentation

Words, just because they are words, are not inherently clear. The message they contain becomes clear to those who speak the language and are familiar with the issues and contexts. If the message lacks linguistic clarity the recipient of the message will typically make a query that will bring forth further information intended to clarify. The result might be more words, but it might also involve pointing or drawing, or words that utilize other modes such as references to context, history, and so on. If the ambiguity derives from an inconsistency between, say, words and behaviour, one may look to either mode for clarity. Communication, we must accept, actually occurs in messages, and our ability to transmit information may be limited by any number of factors. When we focus entirely on discursive aspects of communication we limit both the ways in which we receive and ways in which we transmit information. The logocentric fallacy is committed when language, especially in it's most logical guise, is seen to be the only form of rational communication.     

关于归因定向与行为预测的实验研究

不同的归因方式将导致人们对行为的不同预测倾向。本文正是基于这样的假设,来考察在信息匮乏的情况下,性情归因和情景归因对他人积极行为和消极行为的预测倾向的影响。结果发现,两种归因方式都导致被试对积极行为的预测多于对消极行为的预测,而情景归因下的被试对积极行为的预测又显著多于性情归因下的被试。在此基础上,在引人价值判断的因素后,发现不同的归因方式启动了不同的价值判断,进而影响到对行为的预测。     

Teaching Genesis: A Present-Day Approach Inspired by the Prophet Nathan

The prophets Nathan (2 Samuel 12:1–15) and John the Baptist (Mark 6:16–28) had comparable tasks before them: to convince their respective kings about the wrongs of taking somebody else's wife and marrying her. Nathan succeeded, while John failed and furthermore lost his life. What made the difference? One possible explanation is that Nathan proceeded in two steps: (1) Tell an interesting, nonthreatening story that nevertheless makes the point at issue; (2) transfer that message to the case at hand. In contrast, John used a direct approach, which raised apprehension, even fear (on the part of Herodias, the woman involved), and led to failure. That lesson has wider applications, as illustrated here for teaching the biblical Genesis narration. The other ingredient in this teaching is relational and contextual reasoning (RCR), the use of which is also indicated for other issues besides teaching Genesis.     

Smart and easy: Co-occurring activation of spontaneous trait inferences and spontaneous situational inferences

Social perceivers have been shown to draw spontaneous trait inferences (STI’s) about the behavior of an actor as well as spontaneous situational inferences (SSI’s) about the situation the actor is in. In two studies, we examined inferences about behaviors that allow for both an STI and an SSI. In Experiment 1, using a probe recognition paradigm, we found activation of both STI’s and SSI’s. In Experiment 2, using a relearning paradigm, we again found activation of both STI’s and SSI’s, regardless of temporarily activated processing goals. Results are discussed in light of three-stage models of the process of social inference.     

Logic Games are Complete for Game Logics

Game logics describe general games through powers of players for forcing outcomes. In particular, they encode an algebra of sequential game operations such as choice, dual and composition. Logic games are special games for specific purposes such as proof or semantical evaluation for first-order or modal languages. We show that the general algebra of game operations coincides with that over just logical evaluation games, whence the latter are quite general after all. The main tool in proving this is a representation of arbitrary games as modal or first-order evaluation games. We probe how far our analysis extends to product operations on games. We also discuss some more general consequences of this new perspective for standard logic.