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.
Certain axiomatic systems involve more than one category of fundamental objects; for example, points, lines, and planes in geometry; individuals, classes of individuals, etc. in the theory of types or in predicate calculi of orders higher than one. It is natural to use variables of different kinds with their ranges respectively restricted to different categories of objects, and to assume as substructure the usual quantification theory (the restricted predicate calculus) for each of the various kinds of variables together with the usual theory of truth functions for the formulas of the system. An axiomatic theory set up in this manner will be called many-sorted. We shall refer to the theory of truth functions and quantifiers in it as its (many-sorted) elementary logic, and call the primitive symbols and axioms (including axiom schemata) the proper primitive symbols and proper axioms of the system. Our purpose in this paper is to investigate the many-sorted systems and their elementary logics.Among the proper primitive symbols of a many-sorted system Tn (n = 2, …, ω) there may be included symbols of some or all of the following kinds: (1) predicates denoting the properties and relations treated in the system; (2) functors denoting the functions treated in the system; (3) constant names for certain objects of the system. We may either take as primitive or define a predicate denoting the identity relation in Tn.
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.
AbstractAn electronic identity relation, relating capacitively coupled plasma sources to corresponding inductively coupled plasma sources, has been derived, starting from the Maxwell relations for matter and the characteristics of a capacitor and of an inductor. Furthermore, the breakdown conditions for both capacitively coupled plasmas and for inductively coupled plasmas as well as their optimal operation frequency ranges are discussed.
“The confusion of a logical with a real predicate,” according to the Critique of Pure Reason, “is almost beyond correction” (A598/B626). Kant did not assert that existence is no predicate, but that it is only a “logical” one, and not a “real” one. Much the same thing has been said about identity, although Kant himself thought it is real and not logical. We have long lacked a rigorous criterion to distinguish real from logical predicates, and hence have not been able to say why the difference matters. This paper has two objects. First it provides a demarcation between real and logical predicates that confirms Kant's dictum that existence is only “logical.” Secondly it states the theory of a “logical” (but not “real”) relation of identity. Perhaps this is not the only identity relation. I show only that once it has been precisely defined in the right setting, there are definite answers to a number of disputed questions about identity. Maybe there are other concepts of identity for which different answers are to be given, but I shall not discuss that disagreeable prospect here. A third application concerns the ontological argument.
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.
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.
This article examines the role of the mediaeval theory of the propositio in re, as proposed by Walter Burley and others, which bears a striking resemblance to the theory of the “proposition” advocated by G. E. Moore and B. Russell. Burley’s proposition composed of real things has the function of an ultimate significate for every sentence of natural language. The main problems of such a theory are on the one hand absurdities like a bird flying between the subject and predicate of a sentence, on the other hand Burley’s assumption that a relation of identity holds between subject and predicate, which might render propositiones in re tautological. Moreover, the particular nature of this relation is left unexplained. But these difficulties can be solved: The former by applying objective being, being as being cognized, to the terms of a propositio, as did Scotus and Franciscus de Prato, the latter by specifying multiple forms of real predication as being or being-in-something apart from a mere identity-relation (e. g. William Milverley).
Green’s Relations on a Semigroup of Transformations with Restricted Range that Preserves an Equivalence Relation and a Cross-Section