Some Applications of Lawvere’s Fixpoint Theorem

The famous diagonal argument plays a prominent role in set theory as well as in the proof of undecidability results in computability theory and incompleteness results in metamathematics. Lawvere (1969) brings to light the common schema among them through a pretty neat fixpoint theorem which generalizes the diagonal argument behind Cantor’s theorem and characterizes self-reference explicitly in category theory. Not until Yanofsky (2003) rephrases Lawvere’s fixpoint theorem using sets and functions, Lawvere’s work has been overlooked by logicians. This paper will continue Yanofsky’s work, and show more applications of Lawvere’s fixpoint theorem to demonstrate the ubiquity of the theorem. For example, this paper will use it to construct uncomputable real number, unnameable real number, partial recursive but not potentially recursive function, Berry paradox, and fast growing Busy Beaver function. Many interesting lambda fixpoint combinators can also be fitted into this schema. Both Curry’s Y combinator and Turing’s Θ combinator follow from Lawvere’s theorem, as well as their call-by-value versions. At last, it can be shown that the lambda calculus version of the fixpoint lemma also fits Lawvere’s schema.     

Vagueness and Blurry Sets

This paper presents a new theory of vagueness, which is designed to retain the virtues of the fuzzy theory, while avoiding the problem of higher-order vagueness. The theory presented here accommodates the idea that for any statement S ₁ to the effect that Bob is bald is x true, for x in [0,1], there should be a further statement S ₂ which tells us how true S ₁ is, and so on – that is, it accommodates higher-order vagueness – without resorting to the claim that the metalanguage in which the semantics of vagueness is presented is itself vague, and without requiring us to abandon the idea that the logic – as opposed to the semantics – of vague discourse is classical. I model the extension of a vague predicate P as a blurry set, this being a function which assigns a degree of membership or degree function to each object o, where a degree function in turn assigns an element of [0,1] to each finite sequence of elements of [0,1]. The idea is that the assignment to the sequence 0.3,0.2, for example, represents the degree to which it is true to say that it is 0.2 true that o is P to degree 0.3. The philosophical merits of my theory are discussed in detail, and the theory is compared with other extensions and generalisations of fuzzy logic in the literature.     

Fear of falling and falls in older adults with mild cognitive impairment and Alzheimer’s disease

ABSTRACT

The purpose of this study was to explore the influence of social importance on prospective remembering in younger and older adults as a possible factor contributing to the age-prospective memory paradox. Using a between-subjects design, 40 younger and 40 older adults worked on a time-based prospective memory task in which social importance was varied. Overall, younger adults outperformed older adults in the prospective memory task. Importantly, in contrast to younger adults, older adults' prospective memory performance was significantly better in the social importance condition than in the standard condition. This interaction was not reflected in participants' time-monitoring behaviour. Findings are discussed in the context of recent prospective memory theories.     

Paradox lost: Understanding vague predicates

Abstract

The paper is concerned with the status of vague predicates. It is argued that they are for the most part ‘classifiers’, which are covertly comparatives and name not monadic properties but relations. The Sorites Paradox, it is claimed, is thus defused and a verdict theory of vague predicates is presented. Our practice in using vague words is described and it is contended that in our use of these predicates we always have a permanent possibility of independent demarcation. Wittgenstein's picture of the wall and the swamp is deployed to avoid the Transition Problem and it is argued against Fregeans that we need vague language for the advancement of knowledge and understanding.     

Chain-Arguments and the Sorites Paradox

A finite chain of valid arguments can never lead from truth to falsehood. Call this the concatenation principle, or CP. Some propose to reject CP in response to the sorites paradox. I offer a justification of a restricted version of CP based on intuitively-appealing principles, all of which are already assumed in mathematics. The restricted version excludes soritical cases, but covers intuitively correct applications of CP in mathematics and elsewhere. The upshot is that the cost of rejecting the unrestricted CP is much lower than is often assumed.     

Replies to Bacon,Eklund, and Greenough on Replacing Truth

Andrew Bacon, Matti Eklund, and Patrick Greenough have individually proposed objections to the project in my book, Replacing Truth. Briefly, the book outlines a conceptual engineering project – our defective concept of truth is replaced for certain purposes with a team of concepts that can do some of the jobs we thought truth could do. Here, I respond to their objections and develop the views expressed in Replacing Truth in various ways.     

Psychological Comparison Processes and Self-Concept in Relation to Five Distinct Frame-of-Reference Effects: Pan-Human Cross-Cultural Generalizability over 68 Countries

The concept of self is central to personhood, but personality research has largely ignored the relevance of recent advances in self-concept theory: multidimensionality of self-concept (focusing instead on self-esteem, an implicit unidimensional approach), domain specificity (generalizability of trait manifestations over different domains), and multilevel perspectives in which social-cognitive processes and contextual effects drive self-perceptions at different levels (individual, group/institution, and country) aligned to Bronfenbrenner's ecological model. Here, we provide theoretical and empirical support for psychological comparison processes that influence self-perceptions and their relation to distal outcomes. Our meta-theoretical integration of social and dimensional comparison theories synthesizes five seemingly paradoxical frame-of-reference and contextual effects in self-concept formation that occur at different levels. The effects were tested with a sample of 485,490 fifteen-year-old students (68 countries/regions, 18,292 schools). Consistent with the dimensional comparison theory, the effects on math self-concept were positive for math achievement but negative for verbal achievement. Consistent with the social comparison theory, the effects on math self-concept were negative for school-average math achievement (big-fish-little-pond effect), country-average achievement (paradoxical cross-cultural effect), and being young relative to year in school but positive for school-average verbal achievement (big-fish-little-pond effect—compensatory effect). We demonstrate cross-cultural generalizability/universality of support for predictions and discuss implications for personality research. © 2020 European Association of Personality Psychology     

Mind your meat: Religious differences in the social perception of animals

While previous work demonstrated that animals are categorised based on their edibility, little research has systematically evaluated the role of religion in the perception of animal edibility, particularly when specific animals are deemed sacred in a religion. In two studies, we explored a key psychological mechanism through which sacred animals are deemed inedible by members of a faith: mind attribution. In Study 1, non-vegetarian Hindus in Singapore (N = 70) evaluated 19 animals that differed in terms of their sacredness and edibility. Results showed that participants categorised animals into three groups: holy animals (high sacredness but low edibility), food animals (low sacredness but high edibility) and neutral animals (low sacredness and low edibility). Holy animals were deemed to possess greater mental life compared to other animal categories. In Study 2, we replicated this key finding with Hindus in India (N = 100), and further demonstrated that the observed pattern of results was specific to Hindus but not Muslims (N = 90). In both studies, mind attribution mediated the negative association between sacredness and edibility. Our findings illustrate how religious groups diverge in animal perception, thereby highlighting the role of mind attribution as a crucial link between sacredness and edibility.