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)

Immanuel Kant's Critique of Judgment is widely recognized as a founding document of modern aesthetics, but its legacy has fallen into disrepute. In this book Katalin Makkai calls for the rediscovery of Kant's aesthetics, showing that its centerpiece, his investigation of the judgment of taste, paints a compelling portrait of our relationships with works of art that we love. At its heart is a scene of aesthetic encounter in which one feels oneself to be 'animated' - brought to life - (...) by an object, finding there to be something in one's experience of it, beyond what there is to know about it, that one wants to explore and articulate. Tracing Kant's insight that to judge is to reveal one's sense of what bears judging, and hence of what matters, Makkai situates Kant's aesthetics within his larger study, begun in the first Critique, of judgment's fundamental role in the life of the mind. (shrink)

The dissertation offers a study of Kant's aesthetic theory as it is developed in the "Critique of Aesthetic Judgment", the first half of his Critique of Judgment, which is widely acknowledged to be the founding text of modern philosophical aesthetics. I aim to show that this work elaborates an important and deeply interesting study of the nature and conditions of aesthetic judgment which---despite the recent resurgence of commentary and of secondary literature---has not yet been inherited. I give an account focused (...) on understanding Kant's idea that aesthetic judgment makes a claim to "subjective universality" and "exemplary necessity", the "twofold peculiarity" of aesthetic judgment that Kant himself presents as the main thematic guiding the course of the Critique. I argue for an interpretation of Kant's notion of aesthetic judgment which takes seriously his idea that its demand for the agreement of others takes the form of a presumption, in which the judge offers her judgment as exemplary of a "universal voice", apart from any independently or transcendentally assured authorization. (shrink)

Let C be a small Barr-exact category, Reg the category of all regular functors from C to the category of small sets. A form of M. Barr's full embedding theorem states that the evaluation functor e : C →[Reg, Set ] is full and faithful. We prove that the essential image of e consists of the functors that preserve all small products and filtered colimits. The concept of κ-Barr-exact category is introduced, for κ any infinite regular cardinal, and the natural (...) generalization to κ-Barr-exact categories of the above result is proved. The treatment combines methods of model theory and category theory. Some applications to module categories are given. (shrink)

Released in 1958, Vertigo is widely regarded as Alfred Hitchcock’s masterpiece and one of the greatest films of all time. This is the first book devoted to exploring the philosophical aspects of Vertigo . Following an introduction by the editor that places the film in context, each chapter reflects upon Hitchcock’s film from a philosophical perspective. Topics discussed include: memory, loss, memorialisation, and creativity mimetic or representational art and art as magic the nature of romantic love gender, sexual objectification, and (...) identity looking, "the gaze", and voyeurism film and psychoanalysis fantasy, illusion, and reality the phenomenology of colour. Including annotated further reading at the end of each chapter, this collection is essential reading for anyone interested in Vertigo , and an ideal resource for students of film and philosophy. (shrink)

Using the framework of categorical logic, this paper analyzes and streamlines Gabbay's semantical proof of the Craig interpolation theorem for intuitionistic predicate logic. In the process, an apparently new and interesting fact about the relation of coherent and intuitionistic logic is found.

Abstract: Kant declares the judgment of beauty to be neither ‘objective’ nor ‘merely subjective’. This essay takes up the question of what this might mean and whether it can be taken seriously. It is often supposed that Kant's denials of ‘objectivity’ to the judgment of beauty express a rejection of realism about beauty. I suggest that Kant's thought is not to be understood in these terms—that it does not properly belong in the arena of debates about the constituents of ‘reality’—motivating (...) the suggestion by first considering a pair of opposing views on the question of whether Kant can be understood to develop a real alternative to realism about beauty at all. (shrink)

We use a fundamental theorem of Vaught, called the covering theorem in [V] (cf. theorem 0.1 below) as well as a generalization of it (cf. Theorem $0.1^\ast$ below) to derive several known and a few new results related to the logic $L_{\omega_1\omega}$. Among others, we prove that if every countable model in a $PC_{\omega_1\omega}$ class has only countably many automorphisms, then the class has either $\leq\aleph_0$ or exactly $2^{\aleph_0}$ nonisomorphic countable members (cf. Theorem $4.3^\ast$) and that the class of countable (...) saturated structures of a sufficiently large countable similarity type is not $PC_{\omega_1\omega}$ among countable structures (cf. Theorem 5.2). We also give a simple proof of the Lachlan-Sacks theorem on bounds of Morley ranks ($\s 7$). (shrink)

A listing is given of the published writings of the logician and philosopher Hao Wang , which includes all items known to the authors, including writings in Chinese and translations into other languages.

A listing is given of the published writings of the logician and philosopher Hao Wang (1921—1995), which includes all items known to the authors, including writings in Chinese and translations into other languages.

Averroës, 1126-1198; views on Sufism; philosophy and islam; comparati.

A review of Katalin Makkai, _Kant’s Critique of Taste: The Feeling of Life Cambridge_: Cambridge University Press, 2021.

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Kant’s Critique of Taste: The Feeling of Life by Katalin MakkaiYoon ChoiKatalin Makkai. Kant’s Critique of Taste: The Feeling of Life. Cambridge: Cambridge University Press, 2021. Pp. viii + 209. Hardback, $99.99. Paperback, $29.99.This monograph offers a bold and original interpretation of Kant’s theory of reflective judgment, focusing on judgments of taste (hereafter “aesthetic judgments”) and the special problem that Kant takes such judgments to raise. In Makkai’s (...) preferred terms, the problem is that aesthetic judgments are normative—they demand universal agreement—but also subjective (14). How are such judgments possible? Though the problem may seem clear enough, Makkai suggests that a closer look is in order; indeed, her “main focus is on working out a proper understanding of the problem” (15). Makkai undertakes this task over the course of five relatively short chapters, introducing several interpretive theses that come together to transform the problem of aesthetic judgment in illuminating, sometimes surprising, ways. [End Page 509]Chapter 1 introduces Kant’s theory of judgment, arguing that it is primarily about what Kant calls the “power of judgment” (44). This is the power exercised in the activity of “adopting a stance,” an activity that is not only identificatory but also evaluative, that reveals something about the judging subject as well as the object judged (63). Though aesthetic judgment is one way in which the power of judgment is exercised, there is something special about it, as Kant himself flags (KU AA 5:193, quoted on 37). For aesthetic judgment “models” an “essential aspect” of the power of judgment in a way that no other kind of judgment does (38). One of Makkai’s interpretive aims is to make sense of this claim; she accordingly returns to it in her conclusion.Chapter 2, however, turns to aesthetic judgment, focusing on the Second Moment of the Analytic (KU AA 5:211–19). In this section, Kant introduces the much-discussed claim that aesthetic judgment involves the “free play” of imagination and understanding (KU AA 5:217). According to Makkai, it is significant that Kant introduces the idea of “universal communicability” along with the idea of “free play” (KU AA 5:217). Communicability, in Makkai’s view, is underwritten by the activity of “making [one’s thoughts and feelings] communicable,” an activity that is involved not just in communicating one’s thoughts and feelings, but in knowing what one is thinking and feeling in the first place, and indeed in making aesthetic judgment possible, as Makkai goes on to argue (81). For when I have an “aesthetic encounter,” I come across a beautiful object, and though I “cognize it,... there is ‘something else’ there in my experience of it that I find I want to articulate” (90). In the activity of trying to “make communicable” what this “something else” is, the imagination and understanding are brought into free play, a state “characterized,” according to Makkai, by the “exercising” of my “capacity for rendering (something) communicable” (89; emphasis in original). The profound pleasure that accompanies this state, Makkai adds, arises from being “animated” by the beautiful object to this heightened activity in which my faculties are in “attunement” with each other and with the object; it is the feeling of being “brought to life” that Makkai refers to in her title (100–101).It is a central part of this view that to experience an object as beautiful is to respond to the way the object really is. As Makkai puts it, “a condition of the possibility of performing a judgment of taste is a prior fact of the object being beautiful” (125). But in what sense, then, is aesthetic judgment subjective? Makkai addresses this question in chapter 3, arguing that whereas cognitive judgment asserts, aesthetic judgment “acknowledges” (136). And to acknowledge a beautiful object, it is not enough to identify a beautiful object correctly. It is necessary to experience the object as beautiful: to encounter it and take pleasure in it (136). Aesthetic judgment thus has an “essentially first-personal character” (127). As Makkai puts it later, in such a judgment, I represent “my experience of the object, my pleasure in it”—that is... (shrink)

The perennial challenge for any Kant scholar is to offer close reading of a dense and often exceedingly abstract text that is faithful both to its driving philo.

European Journal of Philosophy, EarlyView.

The author of this paper presents the main interpretative orientations regarding the concept on the minority being of the reformed Transylvanian bishop Makkai Sándor who lived in the inter-war period. The author tries to point out the philosophical, moral, and existential sides of this problem which has become deep-rooted and permanent in the consciousness of the Hungarian intellectuals from Transylvania, and which has been known as the problem of the minority existential paradox. To accomplish this, the author relies on the (...) theoretical and methodological possibilities provided by the philosophical hermeneutics. (shrink)

Makkai [10] produced an arithmetical structure of Scott rank $\omega _{1}^{\mathit{CK}}$. In [9]. Makkai's example is made computable. Here we show that there are computable trees of Scott rank $\omega _{1}^{\mathit{CK}}$. We introduce a notion of "rank homogeneity". In rank homogeneous trees, orbits of tuples can be understood relatively easily. By using these trees, we avoid the need to pass to the more complicated "group trees" of [10] and [9]. Using the same kind of trees, we obtain one of rank (...) $\omega _{1}^{\mathit{CK}}$ that is "strongly computably approximable". We also develop some technology that may yield further results of this kind. (shrink)

Given a regular cardinal $\kappa $ such that $\kappa ^{<\kappa }=\kappa $, we study a class of toposes with enough points, the $\kappa $ -separable toposes. These are equivalent to sheaf toposes over a site with $\kappa $ -small limits that has at most $\kappa $ many objects and morphisms, the topology being generated by at most $\kappa $ many covering families, and that satisfy a further exactness property T. We prove that these toposes have enough $\kappa $ -points, that (...) is, points whose inverse image preserve all $\kappa $ -small limits. This generalizes the separable toposes of Makkai and Reyes, that are a particular case when $\kappa =\omega $, when property T is trivially satisfied. This result is essentially a completeness theorem for a certain infinitary logic that we call $\kappa $ -geometric, where conjunctions of less than $\kappa $ formulas and existential quantification on less than $\kappa $ many variables is allowed. We prove that $\kappa $ -geometric theories have a $\kappa $ -classifying topos having property T, the universal property being that models of the theory in a Grothendieck topos with property T correspond to $\kappa $ -geometric morphisms into that topos. Moreover, we prove that $\kappa $ -separable toposes occur as the $\kappa $ -classifying toposes of $\kappa $ -geometric theories of at most $\kappa $ many axioms in canonical form, and that every such $\kappa $ -classifying topos is $\kappa $ -separable. Finally, we consider the case when $\kappa $ is weakly compact and study the $\kappa $ -classifying topos of a $\kappa $ -coherent theory, that is, a theory where only disjunction of less than $\kappa $ formulas are allowed, obtaining a version of Deligne’s theorem for $\kappa $ -coherent toposes from which we can derive, among other things, Karp’s completeness theorem for infinitary classical logic. (shrink)

For countable structure, "Scott rank" provides a measure of internal, model-theoretic complexity. For a computable structure, the Scott rank is at most [Formula: see text]. There are familiar examples of computable structures of various computable ranks, and there is an old example of rank [Formula: see text]. In the present paper, we show that there is a computable structure of Scott rank [Formula: see text]. We give two different constructions. The first starts with an arithmetical example due to Makkai, and (...) codes it into a computable structure. The second re-works Makkai's construction, incorporating an idea of Sacks. (shrink)

Canonical extension has proven to be a powerful tool in algebraic study of propositional logics. In this paper we describe a generalisation of the theory of canonical extension to the setting of first order logic. We define a notion of canonical extension for coherent categories. These are the categorical analogues of distributive lattices and they provide categorical semantics for coherent logic, the fragment of first order logic in the connectives ∧, ∨, 0, 1 and ∃. We describe a universal property (...) of our construction and show that it generalises the existing notion of canonical extension for distributive lattices. Our new construction for coherent categories has led us to an alternative description of the topos of types, introduced by Makkai in [22]. This allows us to give new and transparent proofs of some properties of the action of the topos of types construction on morphisms. Furthermore, we prove a new result relating, for a coherent category C, its topos of types to its category of models. (shrink)

There exists a countable structure of Scott rank where and where the -theory of is not ω-categorical. The Scott rank of a model is the least ordinal β where the model is prime in its -theory. Most well-known models with unbounded atoms below also realize a non-principal -type; such a model that preserves the Σ1-admissibility of will have Scott rank . Makkai [M. Makkai, An example concerning Scott heights, J. Symbolic Logic 46 301–318. [4]] produces a hyperarithmetical model of Scott (...) rank whose -theory is ω-categorical. A computable variant of Makkai’s example is produced in [W. Calvert, S.S. Goncharov, J.F. Knight, J. Millar, Categoricity of computable infinitary theories, Arch. Math. Logic . [1]; J. Knight, J. Millar, Computable structures of rank J. Math. Logic . [2]]. (shrink)

In this paper we deal with infinitary universal Horn logic both with and without equality. First, we obtain a relative Lyndon-style interpolation theorem. Using this result, we prove a non-standard preservation theorem which contains, as a particular case, a Lyndon-style theorem on surjective homomorphisms in its Makkai-style formulation. Another consequence of the preservation theorem is a theorem on bimorphisms, which, in particular, provides a tool for immediate obtaining characterizations of infinitary universal Horn classes without equality from those with equality. From (...) the theorem on surjective homomorphisms we also derive a non-standard Beth-style preservation theorem that yields a non-standard Beth-style definability theorem, according to which implicit definability of a relation symbol in an infinitary universal Horn theory implies its explicit definability by a conjunction of atomic formulas. We also apply our theorem on surjective homomorphisms, theorem on bimorphisms and definability theorem to algebraic logic for general propositional logic. (shrink)