Are There Enough Injective Sets?

The axiom of choice ensures precisely that, in ZFC, every set is projective: that is, a projective object in the category of sets. In constructive ZF (CZF) the existence of enough projective sets has been discussed as an additional axiom taken from the interpretation of CZF in Martin-Löf’s intuitionistic type theory. On the other hand, every non-empty set is injective in classical ZF, which argument fails to work in CZF. The aim of this paper is to shed some light on the problem whether there are (enough) injective sets in CZF. We show that no two element set is injective unless the law of excluded middle is admitted for negated formulas, and that the axiom of power set is required for proving that “there are strongly enough injective sets”. The latter notion is abstracted from the singleton embedding into the power set, which ensures enough injectives both in every topos and in IZF. We further show that it is consistent with CZF to assume that the only injective sets are the singletons. In particular, assuming the consistency of CZF one cannot prove in CZF that there are enough injective sets. As a complement we revisit the duality between injective and projective sets from the point of view of intuitionistic type theory.     

Complexity Results for Modal Dependence Logic

Modal dependence logic was introduced recently by Väänänen. It enhances the basic modal language by an operator = (). For propositional variables p ₁, . . . , p _n , = (p ₁, . . . , p _n-1, p _n ) intuitively states that the value of p _n is determined by those of p ₁, . . . , p _n-1. Sevenster (J. Logic and Computation, 2009) showed that satisfiability for modal dependence logic is complete for nondeterministic exponential time. In this paper we consider fragments of modal dependence logic obtained by restricting the set of allowed propositional connectives. We show that satisfiability for poor man’s dependence logic, the language consisting of formulas built from literals and dependence atoms using ${\wedge, \square, \lozenge}$ (i. e., disallowing disjunction), remains NEXPTIME-complete. If we only allow monotone formulas (without negation, but with disjunction), the complexity drops to PSPACE-completeness. We also extend Väänänen’s language by allowing classical disjunction besides dependence disjunction and show that the satisfiability problem remains NEXPTIME-complete. If we then disallow both negation and dependence disjunction, satisfiability is complete for the second level of the polynomial hierarchy. Additionally we consider the restriction of modal dependence logic where the length of each single dependence atom is bounded by a number that is fixed for the whole logic. We show that the satisfiability problem for this bounded arity dependence logic is PSPACE-complete and that the complexity drops to the third level of the polynomial hierarchy if we then disallow disjunction. In this way we completely classify the computational complexity of the satisfiability problem for all restrictions of propositional and dependence operators considered by Väänänen and Sevenster.     

The procedural epistemic value of deliberation

Collective deliberation is fuelled by disagreements and its epistemic value depends, inter alia, on how the participants respond to each other in disagreements. I use this accountability thesis to argue that deliberation may be valued not just instrumentally but also for its procedural features. The instrumental epistemic value of deliberation depends on whether it leads to more or less accurate beliefs among the participants. The procedural epistemic value of deliberation hinges on the relationships of mutual accountability that characterize appropriately conducted deliberation. I will argue that it only comes into view from the second-person standpoint. I shall explain what the second-person standpoint in the epistemic context entails and how it compares to Stephen Darwall’s interpretation of the second-person standpoint in ethics.     

Retribution and forgiveness: The healing effects of punishing for just deserts

Although punishment and forgiveness frequently are considered to be opposites, in the present paper we propose that victims who punish their offender are subsequently more likely to forgive. Notably, punishment means that victims get justice (i.e. just deserts), which facilitates forgiveness. Study 1 reveals that participants were more likely to forgive a friend's negligence after being primed with punishment than after being primed with inability to punish. In Study 2, participants were more forgiving towards a criminal offender if the offender was punished by a judge than if the offender escaped punishment, a finding that was mediated by the just deserts motive. Study 3 was in the context of actual recalled ongoing interpersonal relations and revealed that punishment predicted forgiveness indirectly via just deserts, not via victims' vengeful motivations. It is concluded that punishment facilitates forgiveness because of its capacity to restore a sense of justice. Copyright © 2013 John Wiley & Sons, Ltd.     

Personal values and prosocial behaviour in strategic interactions: Distinguishing value‐expressive from value‐ambivalent behaviours

Relying on the framework provided by Schwartz's theory of personal values, we investigated whether values can help explain prosocial behaviour. We first distinguished value‐expressive behaviours from value‐ambivalent behaviours. The former are compatible with primarily one value or with congruent values, the latter with mutually conflicting values. In Study 1, an analysis over all 41 (39 unpublished) samples in which we measured personal values and prosocial behaviour in monetarily incentivized strategic interactions (N = 1289; data collected between 2007 and 2010 in China, Finland, Germany, Israel, and the West Bank) supported our idea that personal values, universalism in particular, predict value‐expressive (prisoner's dilemma cooperation and trust game back‐transfers) but not value‐ambivalent behaviours (trust game transfers and ultimatum game proposals and responses). Study 2 (N = 56) focused on dictator game behaviours, which we expected and found to be strongly value‐expressive. The findings contribute to the ongoing discussion on whether and under which circumstances values shape behaviour. Copyright © 2013 John Wiley & Sons, Ltd.     

The Doomsday Argument and the Simulation Argument

The Doomsday Argument and the Simulation Argument share certain structural features, and hence are often discussed together (Bostrom 2003, Are you living in a computer simulation, Philosophical Quarterly, 53:243–255; Aranyosi 2004, The Doomsday Simulation Argument. Or why isn’t the end nigh, and you’re not living in a simulation, http://philsci-archive.pitt.edu/190/; Richmond 2008, Doomsday, Bishop Ussher and simulated worlds, Ratio, 21:201–217; Bostrom and Kulczycki 2011 A patch for the Simulation Argument, Analysis, 71:54–61). Both are cases where reflecting on one’s location among a set of possibilities yields a counter-intuitive conclusion—in the first case that the end of humankind is closer than you initially thought, and in the second case that it is more likely than you initially thought that you are living in a computer simulation. Indeed, the two arguments do have some structural similarities. But there are also significant disanalogies between the two arguments, and I argue that these disanalogies mean that the Simulation Argument succeeds and the Doomsday Argument fails.     

Is history important for a historical argument in religious studies?

This review article addresses the historical argument of Norenzayan's Big Gods. It questions the claim that there is enough historical evidence to support the thesis that gods, who are observing human behaviour from high, enable societies to grow beyond the stage of face-to-face groups.