Mathematical concepts for analyzing ambiguous situations

Abstract

We examine some mathematical tools for dealing with ambiguous situations. The main tool is the use of non-standard logic with truth-values in what is called a locale. This approach is related to fuzzy set theory, which we briefly discuss. We also consider probabilistic concepts. We include specific examples and describe the way a researcher can set up a suitable locale to analyse a concrete situation.     

Probabilities on Sentences in an Expressive Logic

Automated reasoning about uncertain knowledge has many applications. One difficulty when developing such systems is the lack of a completely satisfactory integration of logic and probability. We address this problem directly. Expressive languages like higher-order logic are ideally suited for representing and reasoning about structured knowledge. Uncertain knowledge can be modeled by using graded probabilities rather than binary truth values. The main technical problem studied in this paper is the following: Given a set of sentences, each having some probability of being true, what probability should be ascribed to other (query) sentences? A natural wish-list, among others, is that the probability distribution (i) is consistent with the knowledge base, (ii) allows for a consistent inference procedure and in particular (iii) reduces to deductive logic in the limit of probabilities being 0 and 1, (iv) allows (Bayesian) inductive reasoning and (v) learning in the limit and in particular (vi) allows confirmation of universally quantified hypotheses/sentences. We translate this wish-list into technical requirements for a prior probability and show that probabilities satisfying all our criteria exist. We also give explicit constructions and several general characterizations of probabilities that satisfy some or all of the criteria and various (counter)examples. We also derive necessary and sufficient conditions for extending beliefs about finitely many sentences to suitable probabilities over all sentences, and in particular least dogmatic or least biased ones. We conclude with a brief outlook on how the developed theory might be used and approximated in autonomous reasoning agents. Our theory is a step towards a globally consistent and empirically satisfactory unification of probability and logic.     

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.     

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.     

The Undecidability of Grisin's Set Theory

We investigate a contractionless naive set theory, due to Grisin [11]. We prove that the theory is undecidable.     

大学生的认知风格与逻辑推理的关系

以大学生为被试,探讨认知风格与逻辑推理的关系。结果表明：（1）在三段论推理中,场依存型学生和场独立型学生推理成绩不存在显著差异,学过和未学过逻辑学的学生推理成绩也不存在显著差异,但男生和女生推理成绩差异显著,女生比男生更易受“气氛”的影响。（2）在命题推理中,对抽象材料构成的命题,场独立型学生和场依存型学生的推理存在显著差异,场独立型学生表现出更强的证伪倾向,场依存型学生表现出更强的换位倾向;对具体材料构成的命题,场独立型学生和场依存型学生的推理不存在显著差异。