Mathematics and the definitions of religion

In 2014, I published a proposal for a definition of “religion”. My goal was to offer a definition of this contentious term that would include Buddhism, Daoism, and other non-theistic forms of life widely considered religions in the contemporary world. That proposal suggested necessary and sufficient conditions for treating a form of life as a religious one. It was critiqued as too broad, however, on the grounds that it would include the study of math as a religion. How can one include forms of life based on non-theistic realities without including math? In this paper, I show the flaw in the previous definition and the weaknesses of two attempts to evade that flaw, before recommending a shift, first, to a Wittgenstein-inspired polythetic definition of “religion” and, second, to a certain kind of polythetic definition that I call “anchored”.     

Attributive adjectives and the predicate calculus

Conclusion I have attempted to show that many attributive adjectives can be dealt with within the framework of first-order predicate calculus by the method suggested in this paper. I've also supplied independent reasons for the claim that attributive adjectives that are not responsive to this method require a formal treatment different from the one that the adjectives successfully dealt with by that method require. Thus, if the method I've argued for is sound, then the scope of first-order predicate calculus was shown to be wider than assumed by several logicians. This I take to be of interest from a logical point of view.     

Interpreting HOL in the calculus of constructions

The purpose of this paper is to consider a representation of the HOL theorem-prover in the calculus of constructions with the property that consistency results from the calculus of constructions imply such results in HOL. This kind of representation is impossible using the propositions-as-types representation of logic and equality, but it is possible if a different representation is used.     

On the calculus ratiocinator

The project, entertained by Leibniz and others, of creating an ideal language to facilitate ratiocination, is investigated in detail. Six possible relations between the ideal language (IL) and the natural language (NL) it replaces are studied. (1) IL says exactly what NL says, but says it much more clearly. (2) IL says exactly what NL says, but does so more economically. (3) IL says exactly what NL says, but does so more succinctly. (4) IL says part of what NL says, and says it more perspicuously. (5) IL says part of what NL says, and says it more perspicuously; moreover, there is an effective procedure for going from NL to IL. (6) IL says everything that NL says, plus some things that NL cannot say.     

Attributions of trust and the calculus of reciprocity

This research investigated trust and reciprocity in two experiments using the Trust Game. In the Trust Game, Player 1 can “trust” an unknown Player 2 by sending some portion of a monetary endowment. The amount sent triples on its way to Player 2, who can then “reciprocate” by returning as much as he or she wishes to Player 1. Initial endowments were either $10 or $20 and were known to recipients; amounts sent were experimentally manipulated and varied from $2 to the entire endowment. Although many trusted parties returned enough money to equalize outcomes, trustors only benefited, on average, when they sent all or almost all of their endowments. Results suggested that recipients viewed sending less than everything as a lack of trust and that felt obligations mediated choices to reciprocate. These and other results contrast markedly with traditional, incremental models of the trust process, which suggest that initial trustors should take small risks and build trust gradually.     

Construct representation and definitions in psychopathology: the case of delusion

Background  

Delusion is one of the most intriguing psychopathological phenomena and its conceptualization remains the subject of genuine debate. Claims that it is ill-defined, however, are typically grounded on essentialist expectations that a given definition should capture the core of every instance acknowledged as delusion in the clinical setting.     

Teachers' and pupils' definitions of bullying

Background. Comparison of teachers' and pupils' definitions of bullying is important for considering the implications for reports of its incidence in schools, for the study of developmental trends in children's and adolescents' perceptions of the phenomenon and for evaluating the effectiveness of interventions designed to combat bullying. Aims. To investigate the effects of gender, teacher/pupil status and, for pupils, bullied/non‐bullied (target/non‐target) status and age on the definition of bullying. Samples. Teachers (N = 225: 158 women, 67 men) and pupils (N = 1,820: 466 boys, 460 girls were 11–12 years old, year 7, and 415 boys, 479 girls were 13–14 years, year 9) in 51 UK secondary schools participated in a questionnaire survey. A total of 557 of the pupils (117 girls and 117 boys aged 11–12 years, and 197 girls and 126 boys aged 13–14 years) reported that they had been bullied at some time in their present school. Methods. Written questionnaire responses to the question, ‘Say what you think bullying is’ have been content analysed to derive two sets of categories, one of bullying behaviour and the other of effects of bullying on the target. Results. Regarding both bullying behaviour and the effects of bullying on the target, teachers – by comparison with pupils – have been found to express more comprehensive ideas in their definitions. Specifically, pupils compared with teachers are more likely to restrict their definitions to direct bullying (verbal and/or physical abuse) and are less likely to refer to social exclusion, a power imbalance in the bully's favour and the bully's intention to cause the target hurt or harm and to feel threatened. Analysis of definitions on the bases of sex, pupil age and target/non‐target status show that: targets are more likely than non‐targets are to refer to the bully's physically and verbally abusive behaviour, and for Year 7 compared with Year 9 pupils, to suggest that bullies socially exclude targets; girls are more likely than boys are to mention verbal abuse and the effects on the target of ‘Feels hurt/harm’, but boys are more likely than girls are to construe bullying as involving repetition; older pupils are more likely than younger ones are to refer to a power imbalance in the bully's favour but, for bully targets, younger ones compared with older ones are more likely to invoke the idea of social exclusion in their definitions. Conclusions. The most important implication of the findings of this study that there are important differences between teachers' and pupils' definitions of bullying is that teachers need to listen carefully to what pupils have to say about bullying and work with and help them to develop their conceptions of the phenomenon. Some teachers, too, need to develop their conceptions of bullying.     

On a three-valued logical calculus and its application to the analysis of the paradoxes of the classical extended functional calculus

A three-valued propositional logic is presented, within which the three values are read as ‘true’, ‘false’ and ‘nonsense’. A three-valued extended functional calculus, unrestricted by the theory of types, is then developed. Within the latter system, Bochvar analyzes the Russell paradox and the Grelling-Weyl paradox, formally demonstrating the meaninglessness of both.