Observational equality: now for good

Building on the recent extension of dependent type theory with a universe of definitionally proof-irrelevant types, we introduce TTobs, a new type theory based on the setoidal interpretation of dependent type theory. TTobs equips every type with an identity relation that satisfies function extensionality, propositional extensionality, and definitional uniqueness of identity proofs (UIP). Compared to other existing proposals to enrich dependent type theory with these principles, our theory features a notion of reduction that is normalizing and provides an algorithmic canonicity result, which we formally prove in Agda using the logical relation framework of Abel et al. Our paper thoroughly develops the meta-theoretical properties of TTobs, such as the decidability of the conversion and of the type checking, as well as consistency. We also explain how to extend our theory with quotient types, and we introduce a setoidal version of Swan's Id types that turn it into a proper extension of MLTT with inductive equality.

On the Overlap Between Everything and Nothing

Graham Priest has recently proposed a solution to the problem of the One and the Many which involves inconsistent objects and a non-transitive identity relation. We show that his solution entails either that the object everything is identical with the object nothing or that they are mutual parts; depending on whether Priest goes for an extensional or a non-extensional mereology.

Epilogue

In the Epilogue, various possible implications of Survival Nihilism are considered. It is argued, first, that Rational Egoism must be rejected if Survival Nihilism is true. Second, the question of whether Survival Nihilism excludes both the possibilities of compensation and of moral responsibility is examined. The conclusion is that it does but only if compensation and moral responsible require that there some relation matters in survival, but it is uncertain one way or the other that this is a requirement of either compensation or moral responsibility. Finally, the possibility of a purely pragmatic justification for having a practice of prudential concern—organized around identity or some other relation—that cannot be defeated by metaphysical considerations is assessed. It is suggested that, in fact, given the metaphysical reflections of this work, a pragmatic justification for adopting a practice of prudential concern, so organized, would not mean that identity or an alternative non-identity relation would give you a non-derivative or derivative reason for prudential concern.

Ehrenfeucht-Fraïssé games without identity

This note defines Ehrenfeucht-Fraïssé games where identity is not present in the basic language. The formulation is applied to show that there is no elementary theory in the language of one binary relation that exactly characterizes models in which the relation is the identity relation.

Consequences

The consequences of our actions seem to matter. But what is the nature of the consequence relation that a particular act bears to, well, its consequences? This essay considers a number of traditional approaches to understanding the consequence relation. While many traditional approaches treat the consequence relation as built upon a causal relation, I hold that there are good reasons to doubt that the consequence relation should be understood in terms of causal relations, even if supplemented with the identity relation. Instead, I argue for a contrastive approach that, while not entirely free of problems, does a better job than standard accounts at capturing the relationship between an act and its consequences.

Intensional Composition as Identity

Composition as Identity claims that a composite object is identical to its parts taken collectively. This is often understood as reducing the identity of composite objects to the identity of their parts. The author argues that Composition as Identity is not such a reduction. His central claim is that an intensional notion of composition, which is sensitive to the arrangement of the composing objects, avoids criticisms based on an extensional understanding of composition. The key is to understand composition as an intensional kind of identity relation, many-one identity. Eventually, the author suggests an arrangement condition for many-one identity that allows him to distinguish between composite objects, even if they have the same parts.

Thinking, Experiencing and Rethinking Mereological Interdependence

Summary The present article is a partly ontological, partly Gestalt-psychological discussion of the thinkability of structures in which parts and whole are interdependent (MI). In the first section, I show that in the framework of E. Husserl’s formal part–whole ontology, the conceptualization of such an interdependence leads to (mereo)logical problems. The second section turns to and affirms the experience of this interplay between parts and whole, exemplified with B. Pinna’s recent research on meaningful Gestalt perception. In the final section, I take the experienceability of MI as a justification to suggest a way of rethinking it. This entails an implementation of the process of foregrounding and backgrounding displayed by reversible figures and originally described by E. Rubin. This can avoid both an identity relation between parts and whole and their mutual exclusion as well as hierarchization due to their apparent differences. It would also guarantee the inherent dynamics of interdependence.

Green’s Relations on a Semigroup of Transformations with Restricted Range that Preserves an Equivalence Relation and a Cross-Section

Let T(X,Y) be the semigroup consisting of all total transformations from X into a fixed nonempty subset Y of X. For an equivalence relation ρ on X, let ρ^ be the restriction of ρ on Y, R a cross-section of Y/ρ^ and define T(X,Y,ρ,R) to be the set of all total transformations α from X into Y such that α preserves both ρ (if (a,b)∈ρ, then (aα,bα)∈ρ) and R (if r∈R, then rα∈R). T(X,Y,ρ,R) is then a subsemigroup of T(X,Y). In this paper, we give descriptions of Green’s relations on T(X,Y,ρ,R), and these results extend the results on T(X,Y) and T(X,ρ,R) when taking ρ to be the identity relation and Y=X, respectively.