The conjunction fallacy and the many meanings of and

According to the conjunction rule, the probability of A and B cannot exceed the probability of either single event. This rule reads and in terms of the logical operator wedge, interpreting A and B as an intersection of two events. As linguists have long argued, in natural language "and" can convey a wide range of relationships between conjuncts such as temporal order ("I went to the store and bought some whisky"), causal relationships ("Smile and the world smiles with you"), and can indicate a collection of sets rather than their intersection (as in "He invited friends and colleagues to the party"). When "and" is used in word problems researching the conjunction fallacy, the conjunction rule, which assumes the logical operator wedge, therefore cannot be mechanically invoked as a norm. Across several studies, we used different methods of probing people's understanding of and-conjunctions, and found evidence that many of those respondents who violated the conjunction rule in their probability or frequency judgments inferred a meaning of and that differs from the logical operator wedge. We argue that these findings have implications for whether judgments involving ambiguous and-conjunctions that violate the conjunction rule should be considered manifestations of fallacious reasoning or of reasonable pragmatic and semantic inferences.     

Is ZF a hack?: Comparing the complexity of some (formalist interpretations of) foundational systems for mathematics

This paper presents Automath encodings (which are also valid in LF/λP) of various kinds of foundations of mathematics. Then it compares these encodings according to their size, to find out which foundation is the simplest.

The systems analyzed in this way are two kinds of set theory (ZFC and NF), two systems based on Church's higher order logic (Isabelle/Pure and HOL), three kinds of type theory (the calculus of constructions, Luo's extended calculus of constructions, and Martin-Löf's predicative type theory) and one foundation based on category theory.

The conclusions of this paper are that the simplest system is type theory (the calculus of constructions), but that type theories that know about serious mathematics are not simple at all. In that case the set theories are the simplest. If one looks at the number of concepts needed to explain such a system, then higher order logic is the simplest, with twenty-five concepts. On the other side of the scale, category theory is relatively complex, as is Martin-Löf's type theory.

(The full Automath sources of the contexts described in this paper are one the web at http://www.cs.ru.nl/~freek/zfc-etc/.)     

Carnap’s logical empiricism,values, and American pragmatism

Value judgments are meaningless. This thesis was one of the notorious tenets of Carnap’s mature logical empiricism. Less well known is the fact that in the Aufbau values were considered as philosophically respectable entities that could be constituted from value experiences. About 1930, however, values and value judgments were banished to the realm of meaningless metaphysics, and Carnap came to endorse a strict emotivism. The aim of this paper is to shed light on the question why Carnap abandoned his originally positive attitude concerning values. It is argued that his non-cognitivist attitude was the symptom of a deep-rooted and never properly dissolved tension between conflicting inclinations towards Neokantianism and Lebensphilosophie. In America Carnap’s non-cognitivism became a major obstacle for a closer collaboration between logical empiricists and American pragmatists. Carnap’s persisting adherence to the dualism of practical life and theoretical science was the ultimate reason why he could not accept Morris’s and Kaplan’s pragmatist theses that cognitivism might well be compatible with a logical and empiricist scientific philosophy.     

现实预期和意愿预期在文本阅读中的激活强度

文本阅读中,读者往往对事件的后续发展进行预期推理。预期推理有两种倾向,要么是倾向于根据客观现实条件进行的现实预期,要么是倾向于根据主观个人意愿进行的意愿预期。两个实验探讨了文本阅读中读者产生的现实预期和意愿预期的激活强度。结果发现,现实预期和意愿预期都可以在阅读中即时产生,意愿预期强于现实预期;把读者分为"情感导向型"和"非情感导向型",发现"情感导向型"的读者,阅读过程中产生的意愿预期强于现实预期,而"非情感导向型"的读者则不是。     

Counting distinctions: on the conceptual foundations of Shannon’s information theory

Categorical logic has shown that modern logic is essentially the logic of subsets (or “subobjects”). In “subset logic,” predicates are modeled as subsets of a universe and a predicate applies to an individual if the individual is in the subset. Partitions are dual to subsets so there is a dual logic of partitions where a “distinction” [an ordered pair of distinct elements (u, u′) from the universe U] is dual to an “element”. A predicate modeled by a partition π on U would apply to a distinction if the pair of elements was distinguished by the partition π, i.e., if u and u′ were in different blocks of π. Subset logic leads to finite probability theory by taking the (Laplacian) probability as the normalized size of each subset-event of a finite universe. The analogous step in the logic of partitions is to assign to a partition the number of distinctions made by a partition normalized by the total number of ordered |U|² pairs from the finite universe. That yields a notion of “logical entropy” for partitions and a “logical information theory.” The logical theory directly counts the (normalized) number of distinctions in a partition while Shannon’s theory gives the average number of binary partitions needed to make those same distinctions. Thus the logical theory is seen as providing a conceptual underpinning for Shannon’s theory based on the logical notion of “distinctions.” This paper is dedicated to the memory of Gian-Carlo Rota—mathematician, philosopher, mentor, and friend.     

A plea for logical objects

An account of validity that makes what is invalid conditional on how many individuals there are is what I call a conditional account of validity. Here I defend conditional accounts against a criticism derived from Etchemendy’s well-known criticism of the model-theoretic analysis of validity. The criticism is essentially that knowledge of the size of the universe is non-logical and so by making knowledge of the extension of validity depend on knowledge of how many individuals there are, conditional accounts fail to reflect that the former knowledge is basic, i.e., independent of knowledge derived from other sciences. Appealing to Russell’s pre-Principia logic, I defend conditional accounts against this criticism by sketching a rationale for thinking that there are infinitely many logical objects.     

The development of category learning strategies: What makes the difference?

Category learning can be achieved by identifying common features among category members, distinctive features among non-members, or both. These processes are psychologically and computationally distinct, and may have implications for the acquisition of categories at different hierarchical levels. The present study examines an account of children’s difficulty in acquiring categories at the subordinate level grounded on these distinct comparison processes. Adults and children performed category learning tasks in which they were exposed either to pairs of objects from the same novel category or pairs of objects from different categories. The objects were designed so that for each category learning task, two features determined category membership whereas two other features were task irrelevant. In the learning stage participants compared pairs of objects noted to be either from the same category or from different categories. Object pairs were chosen so that the objective amount of information provided to the participants was identical in the two learning conditions. We found that when presented only with object pairs noted to be from the same category, young children (6 ? YO ? 9.5) learned the novel categories just as well as older children (10 ? YO ? 14) and adults. However, when presented only with object pairs known to be from different categories, unlike older children and adults, young children failed to learn the novel categories. We discuss cognitive and computational factors that may give rise to this comparison bias, as well as its expected outcomes.     

Using logical relevance for question answering

We propose a novel method of determining the appropriateness of an answer to a question through a proof of logical relevance rather than a logical proof of truth. We define logical relevance as the idea that answers should not be considered as absolutely true or false in relation to a question, but should be considered true more flexibly in a sliding scale of aptness. This enables us to reason rigorously about the appropriateness of an answer even in cases where the sources we are getting answers from are incomplete or inconsistent or contain errors. We show how logical relevance can be implemented through the use of measured simplification, a form of constraint relaxation, in order to seek a logical proof than an answer is in fact an answer to a particular question. We then give an example of such an implementation providing a set of specific rules for this purpose.     

Intra‐sentential context effects on the interpretation of logical metonymy⋆

Verbs such as enjoy in the student enjoyed the book exhibit logical metonymy: enjoy is interpreted as enjoy reading. Theoreticalwork [Computational Linguistics 17 (4) (1991) 409; The Generative Lexicon, MIT Press, Cambridge, MA, 1995] predicts that this interpretation can be influenced by intra‐sentential context, e.g., by the subject of enjoy. In this article, we test this prediction using a completion experiment and find that the interpretation of a metonymic verb is influenced by the semantic role of its subject. We present a Bayesian model that accounts for the interpretation of logical metonymy and achieves a good fit on our experimental data. We show that the parameters of the model can be estimated from completion data or from corpus data.     

The development of spontaneous trait inferences about the actor and spontaneous trait transferences about the informant: Evidence from children aged 8–13 years

In social‐cognitive research, little attention has been paid to the developmental course of spontaneous trait inferences about the actor (STIs about the actor) and spontaneous trait transferences about the informant (STTs about the informant). Using a false recognition paradigm, Study 1 investigated the developmental course of STIs and Study 2 investigated the developmental course of STTs, comparing 8‐, 9‐, 10‐, 11‐, 12‐ and 13‐year olds. The results of Study 1 showed that 8‐year olds could make STIs about the actor, and the magnitude of STIs increased from ages 8 to 10 years, stabilised at the age of 10, 11, 12 years, and decreased from ages 12 through 13 years. The results of Study 2 showed that 8‐year olds could make STTs about the informant, and the magnitude of STTs did not vary with age. In all age groups, the magnitude of STIs about the actor was greater than that of STTs about the informant.