Appeared in 2012. It was presented in conference form in the concluding session of the 2011 Leibniz-Kongress. Complete concepts, a key notion of Leibniz’s philosophy, are analysed in their metaphysical genesis in Leibniz’s theory of creation. Both forms they are supposed to have (collections of predicates, individual histories) are discussed in the framework of Leibniz’s metaphysics of individual essences.

A theoretically rigorous approach to the key problems of Molinism leads to a clear distinction between semantic and metaphysical problems. Answers to semantic problems do not provide answers to metaphysical problems that arise from the theory of middle knowledge. The so-called ‘grounding objection’ to Molinism raises a metaphysical problem. The most promising solution to it is a revised form of the traditional ‘essence solution’. Inspired by Leibniz’s idea of a ‘notio completa’ (complete concept), we propose a mathematical model of (...) ‘possibilistic’ (Molinist) complete concepts. They ground middle knowledge within the very being of the agents themselves. Molinist Complete Concepts can thus serve to reject consequence-style arguments against Molinism. They also allow the Molinist to safeguard a robustly libertarian notion of the ability to do otherwise. (shrink)

Theoretically rigorous approach to the key problems of Molinism.

This paper develops a valid reconstruction in first-order predicate logic of Leibniz’s argument for his complete concept definition of substance in §8 of the Discours de Métaphysique. Following G. Rodriguez-Pereyra, it construes the argument as resting on two substantial premises, the “merely verbal” Aristotelian definition and Leibniz’s concept containment theory of truth, and it understands the resulting “real” definition as saying not that an entity is a substance iff its complete concept contains every predicate of that entity, but (...) iff its complete concept contains every predicate of any subject to which that concept is truly attributable. An account is suggested of why Leibniz criticises the Aristotelian definition as merely nominal and how he takes his own definition to overcome this shortcoming: while on the Aristotelian basis the predication relation could generate endless chains, so that substances as endpoints of predication would be impossible, Leibniz’s definition reveals lowest species as such endpoints, which he therefore identifies with individual substances. Since duplicate lowest species make no sense, the Identity of Indiscernibles for substances follows. The reading suggests a Platonist interpretation according to which substances do not so much have but are individual essences, natures or forms. (shrink)

A formula is (materially) valid iff all its instances are true sentences; and an axiomatic system is called (materially) sound and complete iff it proves all and only valid formulas. These are 'natural' concepts of validity and completeness, which were, however, in the course of the history of modern logic, stealthily replaced by their formal descendants: formal validity and completeness. A formula is formally valid iff it is true under all interpretations in all universes; and an axiomatic system (...) is called formally sound and complete iff it proves all and only formulas valid in this sense. Though the step from material to formal validity and completeness may seem to be merely an unproblematic case of explication, I argue that it is not; and that mistaking the latter concepts for the former ones may lead to serious conceptual confusions. (shrink)

In this paper, two concepts of completing an infinite number of tasks are considered. After discussing supertasks, equisupertasks are introduced. I suggest that equisupertasks are logically possible.

Extended Abstract This book introduces and develops a new, universal method of the scientific comprehension of reality providing the objective, ...

We apply some tools developed in categorical logic to give an abstract description of constructions used to formalize constructive mathematics in foundations based on intensional type theory. The key concept we employ is that of a Lawvere hyperdoctrine for which we describe a notion of quotient completion. That notion includes the exact completion on a category with weak finite limits as an instance as well as examples from type theory that fall apart from this.

The type of completeness Whitehead and Russell aimed for in their Principia Mathematica was what I call descriptive completeness. This is completeness with respect to the propositions that have been proved in traditional mathematics. The notion of completeness addressed by Gödel in his famous work of 1930 and 1931 was completeness with respect to the truths expressible in a given language. What are the relative significances of these different conceptions of completeness for traditional mathematics? What, if any, effects does incompleteness (...) of the type Gödel had in mind have for the evaluation of the Principia Mathematica project and other projects like it? (shrink)

In previous articles, it has been shown that the deductive system developed by Aristotle in his "second logic" is a natural deduction system and not an axiomatic system as previously had been thought. It was also stated that Aristotle's logic is self-sufficient in two senses: First, that it presupposed no other logical concepts, not even those of propositional logic; second, that it is (strongly) complete in the sense that every valid argument expressible in the language of the system (...) is deducible by means of a formal deduction in the system. Review of the system makes the first point obvious. The purpose of the present article is to prove the second. Strong completeness is demonstrated for the Aristotelian system. (shrink)

Machery proposes that the construct of detracts from research progress. However, ignoring development also detracts from research progress. Developmental research has advanced our understanding of how concepts are acquired and thus is essential to a complete theory. We propose a framework that both accounts for development and holds great promise as a new direction for thinking about concepts.

Concepts and Cases in Nursing Ethics maps the ethical landscape of contemporary nursing. The book is the product of a collaboration between philosopher-ethicist Michael Yeo, nurse-ethicist Anne Moorhouse, and six representatives of various areas of professional nursing. It thus combines philosophical and ethical analysis with nursing knowledge and experience in a manner that is both understandable and relevant. The book is organized around six main concepts in nursing ethics: beneficence, autonomy, confidentiality, truth-telling, justice, and integrity. A chapter is (...) devoted to the elucidation of each of these concepts. In each chapter, historical background and conceptual analysis are supplemented by case studies that exemplify issues and show how the concept applies in nursing practice. In this new edition, the materials in each chapter have been updated to reflect recent developments in nursing and more generally in health care. In addition, a totally new chapter on ethical theory has been added. Complete with bibliographies and study questions for further analysis of cases, this book is ideally suited for textbook use. It will help both practitioners and students to deal better with the clinical problems and issues that are encountered in the field. However, it's simple prose and clear exposition of complex issues will make Concepts and Cases in Nursing Ethics attractive to anyone concerned about health care. (shrink)

오늘날 이성 개념은 여러 방면에서 비판의 대상이 되고 있다. 전통적으로 이성은 대상을 필연성과 합리성의 관점에서 해명하는 능력으로서 해방의 기획의 주체로 간주된다. 그런데 이성의 정점으로 평가되는 계몽의 이성 혹은 과학주의적 이성은 오히려 새로운 억압체계를 만들어 냈다. 이런 해방의 역설을 해결하고자 수많은 비합리주의적 운동이 일어나지만, 헤겔은 좀 더 포괄적 이성으로 이를 극복하고자 한다. 헤겔은 계몽의 이성을 오성으로 평가하면서 그 일면성을 비판할 뿐 아니라 이를 극복하고자 하는 낭만주의 역시 일면적인 것으로 비판한다. 그는 이런 분열을 화해하는 이성, 혹은 이성의 통합적 능력으로 극복하고자 한다. 그에게서 (...) 존재하는 것은 이성이며, 이성은 분열을 극복하는 힘으로 간주되기 때문이다. 헤겔의 이러한 노력은 근거와 이유로부터 대상을 파악하고자 하는 전통적인 이성관을 근본으로부터 성찰하고 있는 것으로 평가할 수 있다. (shrink)

The Concept of Law is one of the most influential texts in English-language jurisprudence. 50 years after its first publication its relevance has not diminished and in this third edition, Leslie Green adds an introduction that places the book in a contemporary context, highlighting key questions about Hart's arguments and outlining the main debates it has prompted in the field. The complete text of the second edition is replicated here, including Hart's Postscript, with fully updated notes to include modern (...) references and further reading. (shrink)

Since the publication of The verb “be” in Ancient Greek by Charles Kahn, people have put a lot of emphasis and invested too much labor in all kinds of historico-philological analyses in order to resolve philosophical questions regarding the concept of existence in Greek thought. Useful as these analyses might be, they cannot provide us with conclusive answers to the specific philosophical questions under scrutiny, and, perhaps, it is time for us to abandon the overwhelming optimist motivating the pioneers behind (...) these attempts. I use the Sophist as a case study. I conclude that whether the Form of Being represents existence or predication cannot be settled on speculation regarding the syntax of the verb εἶναι; absolute εἶναι need not be existential in meaning, and complemented εἶναι may be existential in meaning. (shrink)

The concept of space has always been a fundamental theme and issue since the beginning of philosophy and abstract thinking in ancient Greece, and has been fundamentally change due to cultural-historical changes of spatiality throughout the history of knowledge. At the beginning of philosophy, there was a metaphysical question about the beginning or the first cause of all things, to which the concept of space, as a fundamental concept, is the answer. The main lines of philosophical discourse in ancient Greece, (...) which flowed in the philosophy of Plato and Aristotle, were conceived in the context of Euclid's geometry and Ptolemy's worldview. In the modern era, Descartes, Leibniz and Kant tried to conceptually influence space with Copernican rotation and Newtonian physics. Finally, spatial imagery is challenged by the application of physiological problems through non-Euclidean geometry, and especially through phenomenology. Hence, the opening of phenomenological concepts in philosophy and other disciplines that focus on the concept of space could be of interest to researchers in the foundations of phenomenology. Therefore, the aspect of effort in this article is the descriptive analytical development of the opinions of three expert phenomenologists on space. These three philosophers base their theories on a critique of the way of speaking on space. This approach taken by Descartes and Newton has been completed in recent times. In this article, we discuss the philosophical foundations of these philosophers. We discuss the space with each of these three philosophers in terms of a key concept in their philosophical system. Therefore, for Cassirer, we discuss the concept of symbolic spaces, for Heidegger, the concept of being-in-the-world, and for Schmitz, the concept of surfaceless space and felt-bodily space. (shrink)

We systematically study the completion of choice problems in the Weihrauch lattice. Choice problems play a pivotal rôle in Weihrauch complexity. For one, they can be used as landmarks that characterize important equivalences classes in the Weihrauch lattice. On the other hand, choice problems also characterize several natural classes of computable problems, such as finite mind change computable problems, non-deterministically computable problems, Las Vegas computable problems and effectively Borel measurable functions. The closure operator of completion generates the concept of total (...) Weihrauch reducibility, which is a variant of Weihrauch reducibility with total realizers. Logically speaking, the completion of a problem is a version of the problem that is independent of its premise. Hence, studying the completion of choice problems allows us to study simultaneously choice problems in the total Weihrauch lattice, as well as the question which choice problems can be made independent of their premises in the usual Weihrauch lattice. The outcome shows that many important choice problems that are related to compact spaces are complete, whereas choice problems for unbounded spaces or closed sets of positive measure are typically not complete. (shrink)

There are several forms of the Ontological Argument, but it is more or less fair to say that all hang on the contention that the notion of a perfect being entails the existence of that being, since existence is involved in perfection. My first interest is in the word "perfect." The word, I think, is usually vague but it seems to me that, in the context of the proof, it has a meaning which turns out to be much more pedestrian (...) than we would suspect. The argument, if it is valid, must assert that any perfect being would have to have every possible characteristic. It alleges that existence is a characteristic and it is, of course, over this that most debate has centered. It seems clear, however, that "perfect," as employed in the argument, merely means what we would call "complete" in less august transactions. In some forms of the proof, indeed, this is made somewhat clearer since the expression "greatest possible" being is substituted for "perfect." Ordinarily, no one would be tempted to pass lightly from the notion of "that which is complete" to the notion of "the deity." Both "perfect" and "greatest possible" appear to have moral connotations and "complete" usually does not. The slide, however, is made easier by the recurrent notion that evil is not a positive quality but merely the absence of another positive quality, usually goodness. If this view is held, it certainly does follow that whatever is "complete" must be "perfect" in the moral sense as well as in the other usual senses. It is not, here, my concern to pass judgment on this notion. I am, at the moment, merely trying to trace possible sources of confusion in the argument. (shrink)

Jusqu’à récemment, les deux textes écrits par Georges Canguilhem sur l’écologie restaient peu connus. Aujourd’hui, ils sont réunis dans le tome V des Œuvres complètes. Chez Can-guilhem, le sujet de l’écologie est intrinsèquement lié aux concepts de technique et de milieu, dont la généalogie théorique remonte à ses premiers ouvrages des années 1930. Ce qui éclaire ces questions : l’être humain doit-il renoncer aux techniques pour restaurer un équilibre supposé avec la nature, ne serait-il pas plutôt nécessaire de rendre (...) à ses besoins leur priorité sur les intérêts économiques? (shrink)

Versions and extensions of intuitionistic and modal logic involving biHeyting and bimodal operators, the axiom of constant domains and Barcan's formula, are formulated as structured categories. Representation theorems for the resulting concepts are proved. Essentially stronger versions, requiring new methods of proof, of known completeness theorems are consequences. A new type of completeness result, with a topos theoretic character, is given for theories satisfying a condition considered by Lawvere . The completeness theorems are used to conclude results asserting that (...) certain logics are conservatively interpretable in others. (shrink)

This paper examines the concept of isēgoria. It looks especially at Herodotus, comparing his use of the term to that of other authors. The term does not primarily refer to ‘the equal right to speak in the assembly’. Rather, it is a ‘language ideology’ that characterizes the bearing of the free, full citizen. Isēgoria was a negative concept, defined by what it was not more than what it was: not flattery; not fearful; not indirect. Isēgoria could only exist in a (...) community of complete equality, and was threatened when someone became too powerful or too important. It can best be translated as ‘the speech of equals before equals’. (shrink)

Self-concept differentiation is a sign of fragmentation of the self rather than specialization of role identities for its robust relationship with psychological adjustment. However, little is known about the mechanisms underlying the relationship between SCD and psychological adjustment. The aim of this study was to examine the mediating role of self-consistency and congruence in the association between SCD and psychological adjustment, and the moderating role of age in the relationship between SCD and SCC. This moderated mediation model was examined among (...) 158 Chinese retirees, who completed measurements regarding SCD, SCC, psychological well-being, anxiety, and depression. The results showed that SCC partially mediated the links between SCD and the indices of psychological adjustment. Furthermore, age moderated this mediation effect, which was found in mean and high-age participants, but not in low-age ones. Our findings indicate that, at different age stages, the internal mechanisms of SCD affecting psychological adaptation are not the same, and a low differentiated or highly integrated self can serve as an adaptive resource to maintain high subjective well-being of the elderly and protect them from anxiety and depression. (shrink)

This paper focuses on the evolution of the notion of completeness in contemporary logic. We discuss the differences between the notions of completeness of a theory, the completeness of a calculus, and the completeness of a logic in the light of Gödel's and Tarski's crucial contributions.We place special emphasis on understanding the differences in how these concepts were used then and now, as well as on the role they play in logic. Nevertheless, we can still observe a certain ambiguity (...) in the use of the close notions of completeness of a calculus and completeness of a logic. We analyze the state of the art under which Gödel's proof of completeness was developed, particularly when dealing with the decision problem for first-order logic. We believe that Gödel had to face the following dilemma: either semantics is decidable, in which case the completeness of the logic is trivial or, completeness is a critical property but in this case it cannot be obtained as a corollary of a previous decidability result. As far as first-order logic is concerned, our thesis is that the contemporary understanding of completeness of a calculus was born as a generalization of the concept of completeness of a theory. The last part of this study is devoted to Henkin's work concerning the generalization of his completeness proof to any logic from his initial work in type theory. (shrink)

We constructively prove completeness for intuitionistic first-order logic, iFOL, showing that a formula is provable in iFOL if and only if it is uniformly valid in intuitionistic evidence semantics as defined in intuitionistic type theory extended with an intersection operator.Our completeness proof provides an effective procedure that converts any uniform evidence into a formal iFOL proof. Uniform evidence can involve arbitrary concepts from type theory such as ordinals, topological structures, algebras and so forth. We have implemented that procedure in (...) the Nuprl proof assistant.Our result demonstrates the value of uniform validity as a semantic notion for studying logical theories, and it provides new techniques for showing that formulas are not intuitionistically provable. Here we demonstrate its value for minimal and intuitionistic first-order logic. (shrink)

Concepts are highly theoretical entities. One cannot study them empirically without committing oneself to substantial preliminary assumptions. Among the competing theories of concepts and categorization developed by psychologists in the last thirty years, the implicit theoretical assumption that what falls under a concept is determined by description () has never been seriously challenged. I present a nondescriptionist theory of our most basic concepts, which include (1) stuffs (gold, milk), (2) real kinds (cat, chair), and (3) individuals (Mama, (...) Bill Clinton, the Empire State Building). On the basis of something important that all three have in common, our earliest and most basic concepts of substances are identical in structure. The membership of the category like that of is a natural unit in nature, to which the concept does something like pointing, and continues to point despite large changes in the properties the thinker represents the unit as having. For example, large changes can occur in the way a child identifies cats and the things it is willing to call without affecting the extension of its word The difficulty is to cash in the metaphor of in this context. Having substance concepts need not depend on knowing words, but language interacts with substance concepts, completely transforming the conceptual repertoire. I will discuss how public language plays a crucial role in both the acquisition of substance concepts and their completed structure. (shrink)

We propose a new approach to positive modal logics, hereby called anodic modal logics. Our treatment is completely positive since the language has neither negation nor any falsum or minimal particle. The elimination of the minimal particle of the language requires introducing the new concept of factual sets and factual deductions which permit us to talk about deductions in the actual world. We start from a positive fragment of the standard system K, denoted by K⊃, ∧, ◊, which is a (...) bimodal system with □ and ◊ as primitive. This system is then extended to a class of fragments of the Lemmon and Scott systems (cf. (Lemmon et al., 1977)), denoted by K⊃, ∧, ◊ + Gk;l;m;n + Gm;n;k;l. It is shown that such classes of systems are characterized with respect to the usual Kripke-style semantics. The proof is by way of a Henkin-style construction, with “possible worlds” being taken to be prime theories as introduced in the modal context by J. M. Dunn in (Dunn, 1995). We also obtain a surprising limiting result showing that the incompleteness phenomenon in modal logic is independent of negation. (shrink)

The concept of fitness has generated a lot of discussion in philosophy of biology. There is, however, relative agreement about the need to distinguish at least two uses of the term: ecological fitness on the one hand, and population genetics fitness on the other. The goal of this paper is to give an explication of the concept of ecological fitness by providing a reconstruction of the theory of natural selection in which this concept was framed, that is, based on the (...) way the theory was put to use in Darwin’s main texts. I will contend that this reconstruction enables us to account for the current use of the theory of natural selection. The framework presupposed in the analysis will be that of metatheoretical structuralism. This framework will provide both a better understanding of the nature of ecological fitness and a more complete reconstruction of the theory. In particular, it will provide what I think is a better way of understanding how the concept of fitness is applied through heterogeneous cases. One of the major advantages of my way of thinking about natural selection theory is that it would not have the peculiar metatheoretical status that it has in other available views. I will argue that in order to achieve these goals it is necessary to make several concepts explicit, concepts that are frequently omitted in usual reconstructions. (shrink)

In Pietroski ( 2018 ) a simple representation language called SMPL is introduced, construed as a hypothesis about core conceptual structure. The present work is a study of this system from a logical perspective. In addition to establishing a completeness result and a complexity characterization for reasoning in the system, we also pinpoint its expressive limits, in particular showing that the fourth corner in the square of opposition (“ Some_not ”) eludes expression. We then study a seemingly small extension, called (...) SMPL +, which allows for a minimal predicate-binding operator. Perhaps surprisingly, the resulting system is shown to encode precisely the concepts expressible in first-order logic. However, unlike the latter class, the class of SMPL + expressions admits a simple procedural (context-free) characterization. Our contribution brings together research strands in logic—including natural logic, modal logic, description logic, and hybrid logic—with recent advances in semantics and philosophy of language. (shrink)

We provide a sound and complete proof system for an extension of Kleene’s ternary logic to predicates. The concept of theory is extended with, for each function symbol, a formula that specifies when the function is defined. The notion of “is defined” is extended to terms and formulas via a straightforward recursive algorithm. The “is defined” formulas are constructed so that they themselves are always defined. The completeness proof relies on the Henkin construction. For each formula, precisely one of (...) the formula, its negation, and the negation of its “is defined” formula is true on the constructed model. Many other ternary logics in the literature can be reduced to ours. Partial functions are ubiquitous in computer science and even in (in)equation solving at schools. Our work was motivated by an attempt to precisely explain, in terms of logic, typical informal methods of reasoning in such applications. (shrink)

In the first year of the twentieth century, in Gottingen, Husserl delivered two talks dealing with a problem that proved central in his philosophical development, that of imaginary elements in mathematics. In order to solve this problem Husserl introduced a logical notion, called “definiteness”, and variants of it, that are somehow related, he claimed, to Hilbert’s notions of completeness. Many different interpretations of what precisely Husserl meant by this notion, and its relations with Hilbert’s ones, have been proposed, but no (...) consensus has been reached. In this paper I approach this question afresh and thoroughly, taking into consideration not only the relevant texts and context, as others have also done before, but, more importantly, Husserl’s philosophy, his intuition-based epistemology in particular. Based on a system of clearly defined concepts that I here present, I reinforce an interpretation—definiteness as a form of syntactic completeness—that has, I believe, some advantages vis-à-vis alternative interpretations. It is in conformity with the available texts; it makes clear that Husserl’s notion of definiteness is indeed close to Hilbert’s notions of completeness; it solves the important problem of imaginaries for which it was created; and last, but not least, it fits naturally into Husserl’s system of concepts and ideas. (shrink)

In the first year of the twentieth century, in Gottingen, Husserl delivered two talks dealing with a problem that proved central in his philosophical development, that of imaginary elements in mathematics. In order to solve this problem Husserl introduced a logical notion, called “definiteness”, and variants of it, that are somehow related, he claimed, to Hilbert’s notions of completeness. Many different interpretations of what precisely Husserl meant by this notion, and its relations with Hilbert’s ones, have been proposed, but no (...) consensus has been reached. In this paper I approach this question afresh and thoroughly, taking into consideration not only the relevant texts and context, as others have also done before, but, more importantly, Husserl’s philosophy, his intuition-based epistemology in particular. Based on a system of clearly defined concepts that I here present, I reinforce an interpretation—definiteness as a form of syntactic completeness—that has, I believe, some advantages vis-à-vis alternative interpretations. It is in conformity with the available texts; it makes clear that Husserl’s notion of definiteness is indeed close to Hilbert’s notions of completeness; it solves the important problem of imaginaries for which it was created; and last, but not least, it fits naturally into Husserl’s system of concepts and ideas. (shrink)

This article analyzes how five leading Hungarian postdissident liberal thinkers conceptually constructed their view of liberalism in the early years of postcommunism. Studying Beszélő, the most signi cant liberal journal during the early years of representative democracy, it shows how they did so through references to political “threats” and the idea of a “liberal minimum” (János Kis), local liberal and democratic traditions and “progressive patriotism” (Miklós Szabó), the ongoing “liberal-conservative revolution” and the creation of a “new political community” (Gáspár Miklós (...) Tamás), antipolitics and “expertise” (György Konrád), and the “complete catalog of human rights” and the agenda of “modernization” (István Eörsi), respectively. Next to its conceptual analysis of heavily influential individual thinkers, the article discusses the ambition of postdissident Hungarian liberals to harmonize liberal and democratic tenets. Last but not least, it elaborates on the left-wing origins of many of their central concepts that, as suggested here, ultimately hindered liberalism's assumption of a central position in the new political system. (shrink)

It is a celebrated result of McKinsey and Tarski [28] thatS4is the logic of the closure algebraΧ+over any dense-in-itself separable metrizable space. In particular,S4is the logic of the closure algebra over the realsR, the rationalsQ, or the Cantor spaceC. By [5], each logic aboveS4that has the finite model property is the logic of a subalgebra ofQ+, as well as the logic of a subalgebra ofC+. This is no longer true forR, and the main result of [5] states that each connected (...) logic aboveS4with the finite model property is the logic of a subalgebra of the closure algebraR+.In this paper we extend these results to all logics aboveS4. Namely, for a normal modal logicL, we prove that the following conditions are equivalent: (i)Lis aboveS4, (ii)Lis the logic of a subalgebra ofQ+, (iii)Lis the logic of a subalgebra ofC+. We introduce the concept of a well-connected logic aboveS4and prove that the following conditions are equivalent: (i)Lis a well-connected logic, (ii)Lis the logic of a subalgebra of the closure algebra$\xi _2^ + $over the infinite binary tree, (iii)Lis the logic of a subalgebra of the closure algebra${\bf{L}}_2^ + $over the infinite binary tree with limits equipped with the Scott topology. Finally, we prove that a logicLaboveS4is connected iffLis the logic of a subalgebra ofR+, and transfer our results to the setting of intermediate logics.Proving these general completeness results requires new tools. We introduce the countable general frame property (CGFP) and prove that each normal modal logic has the CGFP. We introduce general topological semantics forS4, which generalizes topological semantics the same way general frame semantics generalizes Kripke semantics. We prove that the categories of descriptive frames forS4and descriptive spaces are isomorphic. It follows that every logic aboveS4is complete with respect to the corresponding class of descriptive spaces. We provide several ways of realizing the infinite binary tree with limits, and prove that when equipped with the Scott topology, it is an interior image of bothCandR. Finally, we introduce gluing of general spaces and prove that the space obtained by appropriate gluing involving certain quotients ofL2is an interior image ofR. (shrink)