In this paper, we develop a mathematical model of awareness based on the idea of plurality. Instead of positing a singular principle, telos, or essence as noumenon, we model it as plurality accessible through multiple forms of awareness (“n-awareness”). In contrast to many other approaches, our model is committed to pluralist thinking. The noumenon is plural, and reality is neither reducible nor irreducible. Nothing dies out in meaning making. We begin by mathematizing the concept of awareness by appealing to the (...) mathematical formalism of higher category theory. The beauty of higher category theory lies in its universality. Pluralism is categorical. In particular, we model awareness using the theories of derived categories and (infinity, 1)-topoi which will give rise to our meta-language. We then posit a “grammar” (“n-declension”) which could express n-awareness, accompanied by a new temporal ontology (“n-time”). Our framework allows us to revisit old problems in the philosophy of time: how is change possible and what do we mean by simultaneity and coincidence? Another question which could be re-conceptualized in our model is one of soteriology related to this pluralism: what is a self in this context? A new model of “personal identity over time” is thus introduced. (shrink)

Instead of the half-century old foundational feud between set theory and category theory, this paper argues that they are theories about two different complementary types of universals. The set-theoretic antinomies forced naïve set theory to be reformulated using some iterative notion of a set so that a set would always have higher type or rank than its members. Then the universal u_{F}={x|F(x)} for a property F() could never be self-predicative in the sense of u_{F}∈u_{F}. But (...) the mathematical theory of categories, dating from the mid-twentieth century, includes a theory of always-self-predicative universals--which can be seen as forming the "other bookend" to the never-self-predicative universals of set theory. The self-predicative universals of category theory show that the problem in the antinomies was not self-predication per se, but negated self-predication. They also provide a model (in the Platonic Heaven of mathematics) for the self-predicative strand of Plato's Theory of Forms as well as for the idea of a "concrete universal" in Hegel and similar ideas of paradigmatic exemplars in ordinary thought. (shrink)

Starting from a generalization of the standard axioms for a monoid we present a stepwise development of various, mutually equivalent foundational axiom systems for category theory. Our axiom sets have been formalized in the Isabelle/HOL interactive proof assistant, and this formalization utilizes a semantically correct embedding of free logic in classical higher-order logic. The modeling and formal analysis of our axiom sets has been significantly supported by series of experiments with automated reasoning tools integrated with Isabelle/HOL. We (...) also address the relation of our axiom systems to alternative proposals from the literature, including an axiom set proposed by Freyd and Scedrov for which we reveal a technical issue (when encoded in free logic where free variables range over defined and undefined objects): either all operations, e.g. morphism composition, are total or their axiom system is inconsistent. The repair for this problem is quite straightforward, however. (shrink)

Talk of higher categories (ranks) like Genus and Family is ubiquitous in biology. Yet there is widespread skepticism about these categories. We can locate the source of this skepticism in the lack of “robust concepts” for these categories, robust theories of what it is to be in a certain category. A common defense of category talk is that its virtues are “just pragmatic and not theoretic”. But this strains credulity. We should suppose rather that talk of the (...) higher categories does theoretical work. What work? The paper proposes a “minimal concept” for a category, according to which the category is at least explanatory in marking out, in a rough and ready way, a level in the hierarchy of explanatory taxa. The skepticism is exaggerated. (shrink)

This volume explores the many different meanings of the notion of the axiomatic method, offering an insightful historical and philosophical discussion about how these notions changed over the millennia. The author, a well-known philosopher and historian of mathematics, first examines Euclid, who is considered the father of the axiomatic method, before moving onto Hilbert and Lawvere. He then presents a deep textual analysis of each writer and describes how their ideas are different and even how their ideas progressed over time. (...) Next, the book explores category theory and details how it has revolutionized the notion of the axiomatic method. It considers the question of identity/equality in mathematics as well as examines the received theories of mathematical structuralism. In the end, Rodin presents a hypothetical New Axiomatic Method, which establishes closer relationships between mathematics and physics. Lawvere's axiomatization of topos theory and Voevodsky's axiomatization of higher homotopy theory exemplify a new way of axiomatic theory building, which goes beyond the classical Hilbert-style Axiomatic Method. The new notion of Axiomatic Method that emerges in categorical logic opens new possibilities for using this method in physics and other natural sciences. This volume offers readers a coherent look at the past, present and anticipated future of the Axiomatic Method. (shrink)

Higher types can readily be added to set theory, Bernays-Morse set theory being an example. A type for each ordinal is added in [2]. Adding higher types to set theory provides a neat solution to the problem of how to handle higher type categories. We give the basic definitions, and prove cocompleteness of some higher type categories. MSC: 14A15.

Science does not exist in vacuum; it arises and works in context. Ground-breaking achievements transforming the scientific landscape often stem from philosophical thought, just as symbolic logic and computer science were born from the early analytic philosophy, and for the very reason they impact our global worldview as a coherent whole as well as local knowledge production in different specialised domains. Here we take first steps in elucidating rich philosophical contexts in which Samson Abramsky’s far-reaching work centring around categorical science (...) as a new kind of science may be placed, explicated, and articulated. We argue, inter alia, that Abramsky’s work, as a whole, may be construed as demonstrating the categorical unity of science, or rather the sciences, in a mathematically rigorous, down-to-earth manner, which has been a salient feature of his work. At the same time we trace his intellectual history, leading from duality, to intensionality, and to contextuality, and place it in a broader context of philosophy beyond the analytic-continental divide, namely towards the reintegration of them as in the post-analytic tradition. Besides, we address issues in philosophy of category theory, such as the foundational autonomy of category theory and the (presumably two) dogmas of set-theoretical foundationalism, which Abramsky actually touch upon in one of his few philosophically inclined works. As to philosophy of category theory, we also address categorical structuralism as higher-order structuralism, categorical epistemology as elucidating higher-order meta-laws, and categorical ontology as allowing for reduction of ontological commitment via structural realism, the structuralist resolution of Benaceraff’s dilemma, and the pluralistic multiverse view of science as opposed to the set-theoretical reductionist ‘universe’ view. We conclude by speculating about the existence of the Oxford School of (Pluralistic) Unified Science as opposed to the Vienna Circle of (Monistic) Unified Science and to the Stanford School of (Pluralistic) Disunified Science; Categorical Unified Science may potentially allow us to reconcile the two camps on the unity and disunity of science whilst doing justice to both of them. Categorical unity arguably allows for unification via epistemological and ontological networking, and via knowledge transfer thus enabled, rather than unification via the reduction of everything and every truth to a single foundationalist framework, whilst taking at face value disunity, plurality, and diversity, and their significance in science and in human civilisation as a whole. (shrink)

This Element is an exposition of second- and higher-order logic and type theory. It begins with a presentation of the syntax and semantics of classical second-order logic, pointing up the contrasts with first-order logic. This leads to a discussion of higher-order logic based on the concept of a type. The second Section contains an account of the origins and nature of type theory, and its relationship to set theory. Section 3 introduces Local Set Theory, (...) an important form of type theory based on intuitionistic logic. In Section 4 number of contemporary forms of type theory are described, all of which are based on the so-called 'doctrine of propositions as types'. We conclude with an Appendix in which the semantics for Local Set Theory - based on category theory - is outlined. (shrink)

We consider formal theories synonymous with various free categories . Their Lindenbaum algebras may be described as the lattices of subobjects of a terminator. These theories have intuitionistic logic. We show that the Lindenbaum algebras of second order and higher order arithmetic , and set theory are not isomorphic to the Lindenbaum algebras of first order theories such as arithmetic . We also show that there are only five kernels of representations of the free Heyting algebra on one (...) generator in these Lindenbaum algebras. (shrink)

Dual-process and dual-system theories in both cognitive and social psychology have been subjected to a number of recently published criticisms. However, they have been attacked as a category, incorrectly assuming there is a generic version that applies to all. We identify and respond to 5 main lines of argument made by such critics. We agree that some of these arguments have force against some of the theories in the literature but believe them to be overstated. We argue that the (...) dual-processing distinction is supported by much recent evidence in cognitive science. Our preferred theoretical approach is one in which rapid autonomous processes are assumed to yield default responses unless intervened on by distinctive higher order reasoning processes. What defines the difference is that Type 2 processing supports hypothetical thinking and load heavily on working memory. (shrink)

Let T be the theory of dense cyclically ordered sets with at least two elements. We determine the classifying space of $\mathsf {Mod}(T)$ to be homotopically equivalent to $\mathbb {CP}^\infty $. In particular, $\pi _2(\lvert \mathsf {Mod}(T)\rvert )=\mathbb {Z}$, which answers a question in our previous work. The computation is based on Connes’ cycle category $\Lambda $.

We define a class of higher inductive types that can be constructed in the category of sets under the assumptions of Zermelo‐Fraenkel set theory without the axiom of choice or the existence of uncountable regular cardinals. This class includes the example of unordered trees of any arity.

In this paper, we propose a mathematical model of subjective experience in terms of classes of hierarchical geometries of representations (“n-awareness”). We first outline a general framework by recalling concepts from higher category theory, homotopy theory, and the theory of (infinity,1)-topoi. We then state three conjectures that enrich this framework. We first propose that the (infinity,1)-category of a geometric structure known as perfectoid diamond is an (infinity,1)-topos. In order to construct a topology on the (...) (infinity,1)-category of diamonds we then propose that topological localization, in the sense of Grothendieck-Rezk-Lurie (infinity,1)-topoi, extends to the (infinity,1)-category of diamonds. We provide a small-scale model using triangulated categories. Finally, our meta-model takes the form of Efimov K-theory of the (infinity,1)-category of perfectoid diamonds, which illustrates structural equivalences between the category of diamonds and subjective experience (i.e.its privacy, self-containedness, and self-reflexivity). Based on this, we investigate implications of the model. We posit a grammar (“n-declension”) for a novel language to express n-awareness, accompanied by a new temporal scheme (“n-time”). Our framework allows us to revisit old problems in the philosophy of time: how is change possible and what do we mean by simultaneity and coincidence? We also examine the notion of “self” within our framework. A new model of personal identity is introduced which resembles a categorical version of the “bundle theory”: selves are not substances in which properties inhere but (weakly) persistent moduli spaces in the K-theory of perfectoid diamonds. (shrink)

This paper studies categories and functors in the context of reverse and computable mathematics. In ordinary reverse mathematics, we only focuses on categories whose objects and morphisms can be represented by natural numbers. We first consider morphism sets of categories and prove several associated theorems equivalent to $$\mathrm ACA_{0}$$ over the base system $$\mathrm RCA_{0}$$. The Yoneda Lemma is a basic result in category theory and homological algebra. We then develop an effective version of the Yoneda Lemma in (...) $$\mathrm RCA_{0}$$ ; as an application, we formalize an effective version of the Yoneda Embedding in $$\mathrm RCA_{0}$$. Products and coproducts are basic notions for defining special categories like semi-additive categories and additive categories. We study properties of products and coproducts of a sequence of objects of categories and provide effective characterizations of semi-additive categories and additive categories in terms of products and coproducts. Finally, we further consider the strength of theorems of category theory that are studied in this paper by methods of higher-order reverse mathematics. (shrink)

From the perspective of an African ethic, analytically interpreted as a philosophical principle of right action, what are the proper final ends of a publicly funded university and how should they be ranked? To answer this question, I first provide a brief but inclusive review of the literature on Africanising higher education from the past 50 years, and contend that the prominent final ends suggested in it can be reduced to five major categories. Then, I spell out an intuitively (...) attractive African moral theory and apply it to these five final ends, arguing that three of them are appropriate but that two of them are not. After that, I maintain that the African moral theory prescribes two additional final ends for a public university that are not salient in the literature. Next, I argue that employing the African moral theory as I do enables one to rebut several criticisms of Africanising higher education that have recently been made from a liberal perspective. I conclude by posing questions suitable for future research. (shrink)

The purpose of this informal survey article is to introduce the reader to a new and surprising connection between Logic, Geometry, and Algebra which has recently come to light in the form of an interpretation of the constructive type theory of Per Martin-Löf into homotopy theory and higher-dimensional category theory.

A categorical, higher dimensional algebra and generalized topos framework for Łukasiewicz–Moisil Algebraic–Logic models of non-linear dynamics in complex functional genomes and cell interactomes is proposed. Łukasiewicz–Moisil Algebraic–Logic models of neural, genetic and neoplastic cell networks, as well as signaling pathways in cells are formulated in terms of non-linear dynamic systems with n-state components that allow for the generalization of previous logical models of both genetic activities and neural networks. An algebraic formulation of variable ‘next-state functions’ is extended to a (...) Łukasiewicz–Moisil Topos with an n-valued Łukasiewicz–Moisil Algebraic Logic subobject classifier description that represents non-random and non-linear network activities as well as their transformations in developmental processes and carcinogenesis. The unification of the theories of organismic sets, molecular sets and Robert Rosen’s (M,R)-systems is also considered here in terms of natural transformations of organismal structures which generate higher dimensional algebras based on consistent axioms, thus avoiding well known logical paradoxes occurring with sets. Quantum bionetworks, such as quantum neural nets and quantum genetic networks, are also discussed and their underlying, non-commutative quantum logics are considered in the context of an emerging Quantum Relational Biology. (shrink)

Martin-Löf's constructive type theory forms the basis of this paper. His central notions of category and set, and their relations with Russell's type theories, are discussed. It is shown that addition of an axiom - treating the category of propositions as a set and thereby enabling higher order quantification - leads to inconsistency. This theorem is a variant of Girard's paradox, which is a translation into type theory of Mirimanoff's paradox (concerning the set of all (...) well-founded sets). The occurrence of the contradiction is explained in set theoretical terms. Crucial here is the way a proof-object of an existential proposition is understood. It is shown that also Russell's paradox can be translated into type theory. The type theory extended with the axiom mentioned above contains constructive higher order logic, but even if one only adds constructive second order logic to type theory the contradictions arise. (shrink)

A theory of higher-order vagueness for partially-defined, context-sensitive predicates like is blue is offered. According to the theory, the predicate is determinately blue means roughly is an object o such that the claim that o is blue is a necessary consequence of the rules of the language plus the underlying non-linguistic facts in the world. Because the question of which rules count as rules of the language is itself vague, the predicate is determinately blue is both vague (...) and partial in a sense analogous (though not completely identical) to the sense in which is blue is vague and partial. However, higher-order vagueness stops there. Although the anti-extension of is determinately blue differs from that of is blue, further iterations of determinately have no semantic effect. The paper closes with a discussion of the need to posit sharp and precise lines dividing the range of potential application of a vague predicate into semantically distinct categories. It is argued that this is less of a problem for the present theory than for many others, because the categories into which the range is divided by the present theory -- like the category of counting as blue because the rules of the language offer speakers no discretion, as opposed to the category of counting as blue because speakers have invoked an absolute minimum of discretion in adjusting the contextually sensitive boundaries of the predicate -- are not those that the linguistic competence of ordinary speakers equips them to make highly accurate and reliable judgments about. (shrink)

Theories of relational concept acquisition based on structured intersection discovery predict that relational concepts with a probabilistic structure ought to be extremely difficult to learn. We report four experiments testing this prediction by investigating conditions hypothesized to facilitate the learning of such categories. Experiment 1 showed that changing the task from a category-learning task to choosing the “winning” object in each stimulus greatly facilitated participants' ability to learn probabilistic relational categories. Experiments 2 and 3 further investigated the mechanisms underlying (...) this “who's winning” effect. Experiment 4 replicated and generalized the “who's winning” effect with more natural stimuli. Together, our findings suggest that people learn relational concepts by a process of intersection discovery akin to schema induction, and that any task that encourages people to discover a higher order relation that remains invariant over members of a category will facilitate the learning of putatively probabilistic relational concepts. (shrink)

Higher order unification is a way of combining information (or equivalently, solving equations) expressed as terms of a typed higher order logic. A suitably restricted form of the notion has been used as a simple and perspicuous basis for the resolution of the meaning of elliptical expressions and for the interpretation of some non-compositional types of comparative construction also involving ellipsis. This paper explores another area of application for this concept in the interpretation of sentences containing intonationally marked (...) focus, or various semantic constructs which are sensitive to focus.Similarities and differences between this approach, and theories using alternative semantics, structured meanings, or flexible categorial grammars, are described. The paper argues that the higher order unification approach offers descriptive advantages over these alternatives, as well as the practical advantage of being capable of fairly direct computational implementation. (shrink)

In this book the authors reconcile two different viewpoints of the foundations of mathematics, namely mathematical logic and category theory. In Part I, they show that typed lambda-calculi, a formulation of higher order logic, and cartesian closed categories are essentially the same. In Part II, it is demonstrated that another formulation of higher order logic is closely related to topos theory. Part III is devoted to recursive functions. Numerous applications of the close relationship between traditional (...) logic and the algebraic language of category theory are given. The authors have included an introduction to category theory and develop the necessary logic as required, making the book essentially self-contained. Detailed historical references are provided throughout, and each section concludes with a set of exercises. Thus it is well-suited for graduate courses and research in mathematics and logic. Researchers in theoretical computer science, artificial intelligence and mathematical linguistics will also find this an accessible introduction to a subject of increasing application to these disciplines. (shrink)

Most descriptions of higher-order vagueness in terms of traditional modal logic generate so-called higher-order vagueness paradoxes. The one that doesn't is problematic otherwise. Consequently, the present trend is toward more complex, non-standard theories. However, there is no need for this.In this paper I introduce a theory of higher-order vagueness that is paradox-free and can be expressed in the first-order extension of a normal modal system that is complete with respect to single-domain Kripke-frame semantics. This is the (...) system QS4M+BF+FIN. It corresponds to the class of transitive, reflexive and final frames. With borderlineness defined logically as usual, it then follows that something is borderline precisely when it is higher-order borderline, and that a predicate is vague precisely when it is higher-order vague.Like Williamson's, the theory proposed here has no clear borderline cases in Sorites sequences. I argue that objections that there must be clear borderline cases ensue from the confusion of two notions of borderlineness—one associated with genuine higher-order vagueness, the other employed to sort objects into categories—and that the higher-order vagueness paradoxes result from superimposing the second notion onto the first. Lastly, I address some further potential objections. (shrink)

A theory of higher-order vagueness for partially-defined, context-sensitive predicates like is blue is offered. According to the theory, the predicate is determinately blue means roughly is an object o such that the claim that o is blue is a necessary consequence of the rules of the language plus the underlying non-linguistic facts in the world. Because the question of which rules count as rules of the language is itself vague, the predicate is determinately blue is both vague (...) and partial in a sense analogous (though not completely identical) to the sense in which is blue is vague and partial. However, higher-order vagueness stops there. Although the anti-extension of is determinately blue differs from that of is blue, further iterations of determinately have no semantic effect. -/- The paper closes with a discussion of the need to posit sharp and precise lines dividing the range of potential application of a vague predicate into semantically distinct categories. It is argued that this is less of a problem for the present theory than for many others, because the categories into which the range is divided by the present theory -- like the category of counting as blue because the rules of the language offer speakers no discretion, as opposed to the category of counting as blue because speakers have invoked an absolute minimum of discretion in adjusting the contextually sensitive boundaries of the predicate -- are not those that the linguistic competence of ordinary speakers equips them to make highly accurate and reliable judgments about. (shrink)

A natural extension of Natural Deduction was defined by Schroder-Heister where not only formulas but also rules could be used as hypotheses and hence discharged. It was shown that this extension allows the definition of higher-level introduction and elimination schemes and that the set $\{ \vee, \wedge, \rightarrow, \bot \}$ of intuitionist sentential operators forms a {\it complete} set of operators modulo the higher level introduction and elimination schemes, i.e., that any operator whose introduction and elimination rules are (...) instances of the higher-level schemes can be defined in terms of $\{ \vee, \wedge, \rightarrow, \bot \}$.The aim of the present work is to extend the well-known connections between Proof Theory and Category Theory to higher-level Natural Deduction. To be precise, we will show how an adjointness between cartesian closed categories with finite coproducts can be associated, in a systematic way, with any operator $\phi$ defined by the higher-level schemes. The objects in the categories will be rules instead of formulas. (shrink)

Journal of Mathematical Logic, Volume 22, Issue 02, August 2022. We introduce bounded category forcing axioms for well-behaved classes [math]. These are strong forms of bounded forcing axioms which completely decide the theory of some initial segment of the universe [math] modulo forcing in [math], for some cardinal [math] naturally associated to [math]. These axioms naturally extend projective absoluteness for arbitrary set-forcing — in this situation [math] — to classes [math] with [math]. Unlike projective absoluteness, these higher (...) bounded category forcing axioms do not follow from large cardinal axioms but can be forced under mild large cardinal assumptions on [math]. We also show the existence of many classes [math] with [math] giving rise to pairwise incompatible theories for [math]. (shrink)

Many argue that higher-cognitive emotions such as pride arose de novo in humans, and thus fall outside of the scope of the kinds of evolutionary explanations offered for ?basic emotions,? like fear. This approach fractures the general category of ?emotion? into two deeply distinct kinds of emotion. However, an increasing number of emotion researchers are converging on the conclusion that higher-cognitive emotions are evolutionarily rooted in simpler emotional responses found in primates. I argue that pride fits this (...) pattern, and then consider some of the possible mechanisms by which the basic forms of these emotions have been transformed, arguing that we can see this transformation in part as the progressive internalization of originally externalized processes associated with norms, selfhood, and social structure. This connects the emotion literature with the literatures on the Social/Machiavellian Intelligence Hypothesis and situated cognition, offering the possibility of a more unified theory of human emotions. (shrink)

Higher-order logic, with its type-theoretic apparatus known as the simple theory of types (STT), has increasingly come to be employed in theorizing about properties, relations, and states of affairs—or ‘intensional entities’ for short. This paper argues against this employment of STT and offers an alternative: ordinal type theory (OTT). Very roughly, STT and OTT can be regarded as complementary simplifications of the ‘ramified theory of types’ outlined in the Introduction to Principia Mathematica (on a realist reading). (...) While STT, understood as a theory of intensional entities, retains the Fregean division of properties and relations into a multiplicity of categories according to their adicities and ‘input types’ and discards the division of intensional entities into different ‘orders’, OTT takes the opposite approach: it retains the hierarchy of orders (though with some modifications) and discards the categorisation of properties and relations according to their adicities and input types. In contrast to STT, this latter approach avoids intensional counterparts of the Epimenides and related paradoxes. Fundamental intensional entities lie at the base of the proposed hierarchy and are also given a prominent part to play in the individuation of non-fundamental intensional entities. (shrink)

This paper is just a comment to the impressive work by A. C. Ehresmann and J.-P. Vanbremeersch on the theory of Memory Evolutive Systems (MES). MES are truly higher order systems. Hyperstructures represent a new concept which I introduced in order to capture the essence of what a higher order structure is—encompassing hierarchies and emergence. Hyperstructures are motivated by cobordism theory in topology and higher category theory. The morphism concept is replaced by the (...) concept of a bond. In the paper I briefly introduce hyperstructures motivated geometrically and suggest further developments of the MESs along these lines, which could widen up new areas of applications. (shrink)

It is unreasonable to assume that our pre-scientific emotion vocabulary embodies all and only those distinctions required for a scientific psychology of emotion. The psychoevolutionary approach to emotion yields an alternative classification of certain emotion phenomena. The new categories are based on a set of evolved adaptive responses, or affect-programs, which are found in all cultures. The triggering of these responses involves a modular system of stimulus appraisal, whose evoluations may conflict with those of higher-level cognitive processes. Whilst the (...) structure of the adaptive responses is innate, the contents of the system which triggers them are largely learnt. The circuits subserving the adaptive responses are probably located in the limbic system. This theory of emotion is directly applicable only to a small sub-domain of the traditional realm of emotion. It can be used, however, to explain the grouping of various other phenomena under the heading of emotion, and to explain various characteristic failings of the pre-scientific conception of emotion. (shrink)

Cartesian closed categories (CCCs) have played and continue to play an important role in the study of the semantics of programming languages. An axiomatization of the isomorphisms which hold in all Cartesian closed categories discovered independently by Soloviev and Bruce, Di Cosmo and Longo leads to seven equalities. We show that the unification problem for this theory is undecidable, thus settling an open question. We also show that an important subcase, namely unification modulo the linear isomorphisms, is NP-complete. Furthermore, (...) the problem of matching in CCCs is NP-complete when the subject term is irreducible. CCC-matching and unification form the basis for an elegant and practical solution to the problem of retrieving functions from a library indexed by types investigated by Rittri. It also has potential applications to the problem of polymorphic type inference and polymorphic higher-order unification, which in turn is relevant to theorem proving and logic programming. (shrink)

In the last 10 years, several authors including Griffiths and Matthen have employed classificatory principles from biology to argue for a radical revision in the way that we individuate psychological traits. Arguing that the fundamental basis for classification of traits in biology is that of ‘homology’ (similarity due to common descent) rather than ‘analogy’, or ‘shared function’, and that psychological traits are a special case of biological traits, they maintain that psychological categories should be individuated primarily by relations of homology (...) rather than in terms of shared function. This poses a direct challenge to the dominant philosophical view of how to define psychological categories, viz., ‘functionalism’. Although the implications of this position extend to all psychological traits, the debate has centered around ‘emotion’ as an example of a psychological category ripe for reinterpretation within this new framework of classification. I address arguments by Griffiths that emotions should be divided into at least two distinct classes, basic emotions and higher cognitive emotions, and that these two classes require radically different theories to explain them. Griffiths argues that while basic emotions in humans are homologous to the corresponding states in other animals, higher cognitive emotions are dependent on mental capacities unique to humans, and are therefore not homologous to basic emotions. Using the example of shame, I argue that (a) many emotions that are commonly classified as being higher cognitive emotions actually correspond to certain basic emotions, and that (b) the “higher cognitive forms” of these emotions are best seen as being homologous to their basic forms. (shrink)

It is unreasonable to assume that our pre-scientific emotion vocabulary embodies all and only those distinctions required for a scientific psychology of emotion. The psychoevolutionary approach to emotion yields an alternative classification of certain emotion phenomena. The new categories are based on a set of evolved adaptive responses, or affect-programs, which are found in all cultures. The triggering of these responses involves a modular system of stimulus appraisal, whose evoluations may conflict with those of higher-level cognitive processes. Whilst the (...) structure of the adaptive responses is innate, the contents of the system which triggers them are largely learnt. The circuits subserving the adaptive responses are probably located in the limbic system. This theory of emotion is directly applicable only to a small sub-domain of the traditional realm of emotion. It can be used, however, to explain the grouping of various other phenomena under the heading of emotion, and to explain various characteristic failings of the pre-scientific conception of emotion. (shrink)

An appropriate framework is put forward for the construction of $$\lambda $$ -models with $$\infty $$ -groupoid structure, which we call homotopic $$\lambda $$ -models, through the use of an $$\infty $$ -category with cartesian closure and enough points. With this, we establish the start of a project of generalization of Domain Theory and $$\lambda $$ -calculus, in the sense that the concept of proof (path) of equality of $$\lambda $$ -terms is raised to higher proof (homotopy).

Researchers have evaluated how broad categories of emotion influence judgments of learning relative to neutral items. Specifically, JOLs are typically higher for emotional relative to neutral items. The novel goal of the present research was to evaluate JOLs for fine-grained categories of emotion. Participants studied faces with afraid, angry, sad, or neutral expressions and with afraid, angry, or sad expressions. Participants identified the expressed emotion, made a JOL for each, and completed a recognition test. JOLs were higher for (...) the emotional relative to neutral expressions. However, JOLs were insensitive to the categories of negative emotion. Using a survey design in Experiment 3, participants demonstrated idiosyncratic beliefs about emotion. Some people believed the fine-grained emotions were equally memorable, whereas others believed a specific emotion was most memorable. Thus, beliefs about emotion are nuanced, which has important implications for JOL theory. (shrink)

We present a class of first-order modal logics, called transformational logics, which are designed for working with sentences that hold up to a certain type of transformation. An inference system is given, and com- pleteness for the basic transformational logic HOS is proved. In order to capture ‘up to isomorphism’, we express a very weak version of higher category theory in terms of first-order models, which makes tranforma- tional logics applicable to category theory. A (...) class='Hi'>category-theoretical concept of isomorphism is used to arrive at a modal operator nisoφ expressing ‘up to isomorphism, φ’, which is such that category equivalence comes out as literally isomorphism up to isomorphism. In the final part of the paper, we explore the possibility of using trans- formational logics to define weak higher categories. We end with two informal comparisons: one between HOS and counterpart semantics, and one between isomorphism logic, as a transformational logic, and Homo- topy Type Theory. (shrink)

The eta rule for a set A says that an arbitrary element of A is judgementally identical to an element of constructor form. Eta rules are not part of what may be called canonical Martin-Löf type theory. They are, however, justified by the meaning explanations, and a higher-order eta rule is part of that type theory. The main aim of this paper is to clarify this somewhat puzzling situation. It will be argued that lower-order eta rules do (...) not, whereas the higher-order eta rule does, accord with the understanding of judgemental identity as definitional identity. A subsidiary aim is to clarify precisely what an eta rule is. This will involve showing how such rules relate to various other notions of type theory, proof theory, and category theory. (shrink)

Les algèbres de dimensions supérieures libèrent les mathématiques de la restriction d'une notation purement linéaire. Elles aident ainsi à la modélisation de la géométrie et procurent une meilleure compréhension et plus de possibilités pour les calculs. Elles nous donnent de nouveaux outils pour l'étude de problèmes non-commutatifs, de dimension supérieure qui assurent le passage du local au global, en utilisant la notion d' «inverse algébrique de subdivision». Nous allons exposer comment ces idées sont venues aux auteurs en prolongeant initialement la (...) notion classique de groupe abstrait à celle de groupoïde abstrait, dont la composition n'est que partiellement définie, et qui ajoute une composante spatiale à la théorie habituelle des groupes. La théorie des noeuds nous fournit un exemple en indiquant comment une telle algèbre peut être utilisée pour décrire la structure d'un espace. Le prolongement à la dimension 2 utilise des compositions de carrés dans deux directions et la richesse de l'algèbre qui en résulte est montrée par certains calculs de dimension 2. La difficulté de la transition de la dimension 1 à la dimension 2 est également illustrée par la comparaison de la notion de carré commutatif à celle de cube commutatif - le traitement de cette dernière nécessitant de nouvelles notions. L'importance de la théorie des catégories est expliquée, de même que les possibilités de l'application d'algèbres de dimensions supérieures. (shrink)

Hefty and serious—that is how this book feels when you pick it up. That was my subjective aesthetic experience anyway. Aesthetic judgment is, after all, one key to assessing our thoughts and perceptions. More on that soon, as you might expect.Hefty and serious also describes the questions with which the volume grapples: Is there, or can there be, a clear American Aesthetics, not merely aesthetics practiced by Americans? What would that look like? How would such a process affect the minds (...) and actions of Americans?The answers? It's complicated.The twenty essays in this volume trace the possibility of an American Aesthetics from foundational ideas of the past to the contemporary landscape. The goal is not only articulating an aesthetics that is distinctively American—aesthetics as a method for examining the arts and the artful—but also the possibility of aesthetics as a dynamic tool for pursuing a flourishing life within our hyper-capitalist, consumerist culture.Constant amongst the scholars gathered here is the assertion that aesthetics is not an abstract academic pursuit interesting only to the privileged and interested few but rather that aesthetics is the key element in the human pursuit of meaning and purpose in our lives. Aristotle would approve!Speaking of ancestors, the scholars here trace the work of grandparents such as Ralph Waldo Emerson, Charles Pierce, and William James, and parents such as the pragmatist John Dewey and the process philosopher Alfred North Whitehead. (Whitehead is from the UK but “American” in that he died in the US after a long career at Harvard and wrote his magnum opus, Process and Reality, in the US.) From these ancestors, the scholars gathered in this volume see a path forward: Art as conscious experience, as Dewey posited, and/or art as committed immersion in the process, as Whitehead claimed.Whitehead's process approach—process aesthetics—reminds us that each moment of our experience in this at once mundane and amazing world is an occasion of beauty and a telos—each moment is in the larger process of reality quite literally our last.William James attempted to map how we might free ourselves from the everyday and from everyday-ness: “We are stuffed with abstract conceptions, and glib verbalities and verbosities and in the culture of these higher functions the peculiar sources of joy connected with our simpler functions often dry up” (109). Yet, according to James, there is hope due to the fact that we human beings seek art and do art first and foremost because we enjoy it.Enjoyment was not the angle from which John Dewey approached the subject. According to Dewey our aesthetic reactions and interactions create the experience of art (and life) that re-accesses “an underlying harmony... that we vaguely remember” (112). For pragmatists, noticing a “thing” is entering a relationship with that thing, what David Strong terms in his essay, “correlational coexistence” (371).From these building blocks the scholars assembled propose ways to change US culture from one prone to ask, “What does it do?” into one in which it is increasingly possible to ask, “What does it mean?” This would be a fundamental shift in the dominant paradigms in US culture.How to get there? Randall E. Auxier offers a concise definition of “bad”—bad is something that is “lifeless and too much indebted to the actual.” My attempts at singing fall into that category, as do many American experiences conceived in the mode of what they do, not what they mean.That works of art escape or repel descriptions of their constitutive elements has proven to be a persistent philosophical conundrum through the centuries, as has the question of whether human feelings create an object or an object creates feelings, or both/and—Dewey's interaction of “organism and environment.” From whichever philosophical direction we approach this crux, aesthetic thinking can lead us to recognize and experience the inwardness of what we view as things—objects—and this examination can realign our personal axiology—our personal valuation of reality—the “ten thousand things” of ancient Chinese philosophy.Several authors cite Suzanne Langer's theory of symbol, broadening the scope of “art” well beyond Art with a capital A into the aesthetic and the ritualistic.The scholars gathered here are optimistic that aesthetic examination can create more of those rare moments (moments art can reliably create) in which our very being adjusts to the reality of existence, and we feel exhilarated and a part of the whole process of existence. We call this “nirvana” or “enlightenment.”That is another thing that feels hefty about this volume: the persistent optimism among the scholars that psychic wholeness and human flourishing are possible for more Americans. Robert S. Corrington terms this the “selving process,” roughly what Carl Jung termed “individuation.” This collection adds up to much more than the sum of its parts, as one idea inflected in a particular way by one author sparks off the same idea approached from a different perspective in another. This collection contains scholars on the cutting edge of process theology, pragmatism, aesthetics, and religion, many of whom are adept at crossing the traditional fences between academic fields. The US needs an American Aesthetics. As American poet William Carlos Williams wrote,It is difficultto get the news from poems yet men die miserably every day for lackof what is found there. (shrink)

A novel conceptual framework is introduced for the Complexity Levels Theory in a Categorical Ontology of Space and Time. This conceptual and formal construction is intended for ontological studies of Emergent Biosystems, Super-complex Dynamics, Evolution and Human Consciousness. A claim is defended concerning the universal representation of an item’s essence in categorical terms. As an essential example, relational structures of living organisms are well represented by applying the important categorical concept of natural transformations to biomolecular reactions and relational structures (...) that emerge from the latter in living systems. Thus, several relational theories of living systems can be represented by natural transformations of organismic, relational structures. The ascent of man and other living organisms through adaptation, is viewed in novel categorical terms, such as variable biogroupoid representations of evolving species. Such precise but flexible evolutionary concepts will allow the further development of the unifying theme of local-to-global approaches to highly complex systems in order to represent novel patterns of relations that emerge in super- and ultra-complex systems in terms of compositions of local procedures. Solutions to such local-to-global problems in highly complex systems with ‘broken symmetry’ might be possible to be reached with the help of higher homotopy theorems in algebraic topology such as the generalized van Kampen theorems (HHvKT). Categories of many-valued, Łukasiewicz-Moisil (LM) logic algebras provide useful concepts for representing the intrinsic dynamic ‘asymmetry’ of genetic networks in organismic development and evolution, as well as to derive novel results for (non-commutative) Quantum Logics. Furthermore, as recently pointed out by Baianu and Poli (Theory and applications of ontology, vol 1. Springer, Berlin, in press), LM-logic algebras may also provide the appropriate framework for future developments of the ontological theory of levels with its complex/entangled/intertwined ramifications in psychology, sociology and ecology. As shown in the preceding two papers in this issue, a paradigm shift towards non-commutative, or non-Abelian, theories of highly complex dynamics—which is presently unfolding in physics, mathematics, life and cognitive sciences—may be implemented through realizations of higher dimensional algebras in neurosciences and psychology, as well as in human genomics, bioinformatics and interactomics. (shrink)

Structuralism, the view that mathematics is the science of structures, can be characterized as a philosophical response to a general structural turn in modern mathematics. Structuralists aim to understand the ontological, epistemological, and semantical implications of this structural approach in mathematics. Theories of structuralism began to develop following the publication of Paul Benacerraf’s paper ‘What numbers could not be’ in 1965. These theories include non-eliminative approaches, formulated in a background ontology of sui generis structures, such as Stewart Shapiro’s ante rem (...) structuralism and Michael Resnik’s pattern structuralism. In contrast, there are also eliminativist accounts of structuralism, such as Geoffrey Hellman’s modal structuralism, which avoids sui generis structures. These research projects have guided a more systematic focus on philosophical topics related to mathematical structuralism, including the identity criteria for objects in structures, dependence relations between objects and structures, and also, more recently, structural abstraction principles. Parallel to these developments are approaches that describe mathematical structure in category-theoretic terms. Category-theoretic approaches have been further developed using tools from homotopy type theory. Here we find a strong relationship between mathematical structuralism and the univalent foundations project, an approach to the foundations of mathematics based on higher category theory. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Hegel’s Theory of Self-Conscious Life by Guido SeddoneWill DesmondSEDDONE, Guido. Hegel’s Theory of Self-Conscious Life. Leiden: Brill, 2023. 155 pp. Cloth, $138.00Guido Seddone’s monograph explores an ensemble of issues centering on what he terms Hegelian “naturalism.” He argues that “Hegel’s philosophy represents a novel version of naturalism since it stresses the mutual dependence between nature and spirit, rather than just conceiving of spirit as a substance (...) emerging and separating from bodily natural requisites.” Seddone avoids scholarly exegeses to address instead “theoretical and systematic aspects of the Hegelian philosophy.” He focusses on “life” and “self-consciousness” as the conceptual keys to those parts of Hegel that remain compelling for relevant contemporary debates. “This book is, therefore, based on the dialogue between Hegel [End Page 361] and contemporary thinkers and strives to enhance our understanding of theoretical and practical issues and to widen the grammar of the philosophical thought.” The book’s chapters proceed from the more abstract to the more concrete—essentially from issues of logic, epistemology, and psychology to morality, society, and history.Chapter 1 (“Science of Logic: The Logical Premises of Hegel’s Naturalism”) argues that the categories of “Teleology” and “Life” in Hegel’s Science of Logic mark the critical transition in his naturalism from the sciences of physics (“Mechanism”) and chemistry (“Chemism”) to biology, as well as from external nature to inner self-consciousness and cognition. Seddone helpfully points to Hegel’s revisions of Kant’s subordination of teleological language (merely heuristic) to mechanical explanation. Most promising is the comparison of the “self-referring negativity” of Hegelian “life” to discussions by Evan Thompson and Francisco Varela of the cell—whose permeable membrane enables it to be more than its merely chemical environment.Chapter 2 (“Self-Consciousness”) takes a rather bird’s-eye approach to the Phenomenology of Spirit, offering a basic reading of Hegelian self-consciousness and Geist in relation to the Cartesian cogito and Kantian unity of apperception. Scattered remarks about animal desire and the master–slave relation could be clarified to better explore a naturalistic basis of the Hegelian conception of mind. One is struck by the dearth of specific texts from the section “Self-Consciousness”: Seddone seems content with some well-known quotations (for example, Geist as an “I that is We and We that is I”) punctuating large generalizations.Chapter 3 does not quite live up to its title (“The Hegelian Theory about the (Human) Biological Organism”) in that it focuses primarily on Hegel’s “treatise about habits” in the Encyclopaedia of the Philosophical Sciences. Here habit is construed as a second nature, and hence a further step toward self-conscious, relational mind: Habits are “universal ways of the soul, i.e. ways in which individual soul [sic] incorporates feelings by evolving general rules and practices.”The last section of chapter 4 (“Extended and Embodied Mind”) reverts to habits as moments of the feeling soul but dissociates Hegel’s notion of Verleiblichung from contemporary discussions of embodied mind. By contrast, Seddone argues for retranslating Hegelian Geist as “common mind” and juxtaposes this with the related concept of mind (in, for example, Philip Pettit) as “extended” through social entities, with the result that Hegel’s approach is praised as “the state-of-the-art of this topic.”Chapter 5 (“Natural and Self-conscious Agency”) is perhaps the clearest chapter as it presents the dialectic of the abstractly universal will in the initial sections of Philosophy of Right, links this and the Hegelian conception of free agency to his logical category of “actuality,” and proceeds to an illuminating contrast of the Hobbesian and Hegelian visions of society and state. [End Page 362]In tackling attempts to reconcile “Normativity and Freedom,” chapter 6 spends over six pages rehearsing the basics of Kantian autonomy before turning to Hegel’s Phenomenology of Spirit, specifically the master–slave dialectic (again) as the critical moment in the evolution of social norms. “My thesis is that self-conscious life needs recognition in order to achieve the higher form of independence created by social acknowledgement and justification, what [sic] fosters the condition for an authoritative reason that can be socially... (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Advances in Peircean Mathematics: The Colombian School ed. by Fernando ZalameaGianluca CaterinaFernando Zalamea (Ed.) Advances in Peircean Mathematics: The Colombian School Berlin, Boston: De Gruyter, 2022. 212 pp. (incl. index).The volume Advances in Peircean Mathematics is an important, very much needed contribution towards a deeper understanding of the impact of Peirce's work especially in the fields of mathematics, logic, and semiotic. It fills a gap in the current (...) literature by recasting some of [End Page 373] Peirce's profound mathematical insights into a formal framework. This is a finely edited editorial project, which summarizes the work of more than two decades done by a group of scholars—the Colombian School—lead by Fernando Zalamea, a pioneer of Peirce studies, and pragmatism in general, in Latin America.The structure of the volume unfolds in three dense chapters, each corresponding to a specific mathematical context connected to various aspects of Peirce's work. Broadly speaking, in the first chapter, Gustavo Arengas proposes the theory of higher category theory as a natural framework to represent the main traits of Peirce's semiotic; in the second chapter, Francisco Vargas develops a thorough set-theoretical model of Peirce's continuum; in the third chapter, Arnold Oostra, in a truly Peircean spirit, formalizes the notion of Intuitionistic Existential Graphs, emphasizing the idea that intuitionistic logic is the most natural context to represent and understand Peirce's diagrammatic logic. The fourth and last chapter, penned by Zalamea, provides the reader with a solid summary of the main results presented in the volume along with a robust road map to navigate through the technical details of the text.Mathematics, as implied by the title of the volume, is at the core of this project, and the authors do not shy away from the complexity, details, and depth of it in order to shed light on some of the most fundamental and groundbreaking nature of Peirce's ideas. That level of rigor and formal soundness is certainly one of the strengths of the volume, although readers who may not be acquainted with at least some basic notions of category theory or with the mathematics of Peirce's work, may find some of the text a bit terse—this is not a casual reading and it does require a certain level of commitment in order to appreciate the depth of the provided insights. On the other hand, all the background material is clearly laid out, making the volume a self-contained work which can be thoroughly enjoyed by a general audience.The real upshot of recasting some of Peirce's intuitions within the fabric of formal mathematics is not only that of providing the instruments to clarify those ideas, but also that of highlighting the role that Peirce's work has been playing in the development of modern mathematics in many relevant aspects, from the very genesis of category theory—which today is widely regarded as a necessary and unifying language in several fields of the mathematical and physical sciences—to the development of non-standard analysis, just to mention a few of them.An elegant instance of such perspective is offered by Gustavo Arengas in the first chapter, where it is argued that ∞-categories can be used to model the natural order of hierarchies implicit in Peirce's conception of the basic triadic relation between sign, object, and interpreter. What, at first glance, may seem just a simple analogy, unfolds onto a methodical and enlightening analysis of Peirce's semiotic, as the author builds [End Page 374] the connections with the relevant categorical constructions in a precise manner. Most of the rest of the chapter focuses on Peirce's notions of the pragmatic and pragmaticist maxim, and the use of basic structures such as monads, limits, presheaves and sheaves as the natural environment to better convey their meaning, both from a mathematical and a philosophical perspective. The reader will find it interesting to realize that, by delving deeper into the understanding of the relational nature of category theory, a great chunk of Peirce's work can be seen as fundamentally intertwined with the apparatus and the language of modern mathematics... (shrink)

We analyse the incentives of individuals to misrepresent their truthful judgments when engaged in collective decision-making. Our focus is on scenarios in which individuals reason about the incentives of others before choosing which judgments to report themselves. To this end, we introduce a formal model of strategic behaviour in logic-based judgment aggregation that accounts for such higher-level reasoning as well as the fact that individuals may only have partial information about the truthful judgments and preferences of their peers. We (...) find that every aggregation rule must belong to exactly one of three possible categories: it is either (i) immune to strategic manipulation for every level of reasoning, or (ii) manipulable for every level of reasoning, or (iii) immune to manipulation only for everykth level of reasoning, for some natural number kgreater than 1. (shrink)

This book concerns the foundations of epistemic modality and hyperintensionality and their applications to the philosophy of mathematics. David Elohim examines the nature of epistemic modality, when the modal operator is interpreted as concerning both apriority and conceivability, as well as states of knowledge and belief. The book demonstrates how epistemic modality and hyperintensionality relate to the computational theory of mind; metaphysical modality and hyperintensionality; the types of mathematical modality and hyperintensionality; to the epistemic status of large cardinal axioms, (...) undecidable propositions, and abstraction principles in the philosophy of mathematics; to the modal and hyperintensional profiles of the logic of rational intuition; and to the types of intention, when the latter is interpreted as a hyperintensional mental state. Chapter \textbf{2} argues for a novel type of expressivism based on the duality between the categories of coalgebras and algebras, and argues that the duality permits of the reconciliation between modal and hyperintensional cognitivism and modal and hyperintensional expressivism. Elohim develops a novel, topic-sensitive truthmaker semantics for dynamic epistemic logic, and develops a novel, dynamic two-dimensional semantics grounded in two-dimensional hyperintensional Turing machines. Chapter \textbf{3} provides an abstraction principle for two-dimensional (hyper-)intensions. Chapter \textbf{4} advances a topic-sensitive two-dimensional truthmaker semantics, and provides three novel interpretations of the framework along with the epistemic and metasemantic. Chapter \textbf{5} applies the fixed points of the modal $\mu$-calculus in order to account for the iteration of epistemic states in a single agent, by contrast to availing of modal axiom 4 (i.e. the KK principle). The fixed point operators in the modal $\mu$-calculus are rendered hyperintensional, which yields the first hyperintensional construal of the modal $\mu$-calculus in the literature and the first application of the calculus to the iteration of epistemic states in a single agent instead of the common knowledge of a group of agents. Chapter \textbf{6} advances a solution to the Julius Caesar problem based on Fine's `criterial' identity conditions which incorporate conditions on essentiality and grounding. Chapter \textbf{7} provides a ground-theoretic regimentation of the proposals in the metaphysics of consciousness and examines its bearing on the two-dimensional conceivability argument against physicalism. The topic-sensitive epistemic two-dimensional truthmaker semantics developed in chapters \textbf{2} and \textbf{4} is availed of in order for epistemic states to be a guide to metaphysical states in the hyperintensional setting. -/- Chapters \textbf{8-12} provide cases demonstrating how the two-dimensional hyperintensions of hyperintensional, i.e. topic-sensitive epistemic two-dimensional truthmaker, semantics, solve the access problem in the epistemology of mathematics. Chapter \textbf{8} examines the interaction between Elohim's hyperintensional semantics and the axioms of epistemic set theory, large cardinal axioms, the Epistemic Church-Turing Thesis, the modal axioms governing the modal profile of $\Omega$-logic, Orey sentences such as the Generalized Continuum Hypothesis, and absolute decidability. These results yield inter alia the first hyperintensional Epistemic Church-Turing Thesis and hyperintensional epistemic set theories in the literature. Chapter \textbf{9} examines the modal and hyperintensional commitments of abstractionism, in particular necessitism, and epistemic hyperintensionality, epistemic utility theory, and the epistemology of abstraction. Elohim countenances a hyperintensional semantics for novel epistemic abstractionist modalities. Elohim suggests, too, that higher observational type theory can be applied to first-order abstraction principles in order to make first-order abstraction principles recursively enumerable, i.e. Turing machine computable, and that the truth of the first-order abstraction principle for two-dimensional hyperintensions is grounded in its being possibly recursively enumerable and the machine being physically implementable. Chapter \textbf{10} examines the philosophical significance of hyperintensional $\Omega$-logic in set theory and discusses the hyperintensionality of metamathematics. Chapter \textbf{11} provides a modal logic for rational intuition and provides a hyperintensional semantics. Chapter \textbf{12} avails of modal coalgebras to interpret the defining properties of indefinite extensibility, and avails of hyperintensional epistemic two-dimensional semantics in order to account for the interaction between interpretational and objective modalities and the truthmakers thereof. This yields the first hyperintensional category theory in the literature. Elohim invents a new mathematical trick in which first-order structures are treated as categories, and Vopenka's principle can be satisfied because of the elementary embeddings between the categories and generate Vopenka cardinals in the category of Set in category theory. Chapter \textbf{13} examines modal responses to the alethic paradoxes. Elohim provides a counter-example to epistemic closure for logical deduction. Chapter \textbf{14} examines, finally, the modal and hyperintensional semantics for the different types of intention and the relation of the latter to evidential decision theory. (shrink)

In this paper we present a general theory of normal forms, based on a categorial result for the free monoid construction. We shall use the theory mainly for proposictional modal logic, although it seems to have a wider range of applications. We shall formally represent normal forms as combinatorial objects, basically labelled trees and forests. This geometric conceptualization is implicit in and our approach will extend it to other cases and make it more direct: operations of a purely (...) geometric and combinatorial nature will be introduced in order to give a mathematical description of the basic logical/algebraic constructions .We begin by recalling the above-mentioned categorial construction: we need a careful inspection of it because in the various examples considered later we plan to deduce from it in a uniform way the normal forms and the description of finitely generated free algebras. This method always works whenever we can describe the category of algebras corresponding to the logic under consideration as a T-objects category. When this simple description seems not to be available, still the general theory might be of some interest, because a description of the category of algebras as a T-objects category plus equation is possible .The central part of the paper is more advanced and specific: we show how the general approach presented here can provide some insights even in the basic case of the modal system K. Section 4 contains a contribution to the theory of normal forms, namely the description of the uniform substitution. This result will enable us to give a duality theorem for the category of finitely generated free modal algebras and in Section 5 to provide a characterization of the collections of normal forms which happen to be normal forms for a logic, thus giving a description of the lattice of modal logics.Section 6 deals with some applications: we shall show how to use normal forms in order to prove for the modal system K the definability of higher-order propositional quantifiers and of the tense operator F .As to the prerequisites, the paper is almost self-contained. The reader is only assumed to have familiarity with standard techniques in algebraic logic ). Knowledge of the basic facts about adjoint functors is required too, see e.g. McLane or the appendix. (shrink)

A categorical structure suitable for interpreting polymorphic lambda calculus (PLC) is defined, providing an algebraic semantics for PLC which is sound and complete. In fact, there is an equivalence between the theories and the categories. Also presented is a definitional extension of PLC including "subtypes", for example, equality subtypes, together with a construction providing models of the extended language, and a context for Girard's extension of the Dialectica interpretation.

People categorize objects slowly when visual input is highly impoverished instead of optimal. While bottom-up models may explain a decision with optimal input, perceptual hypothesis testing (PHT) theories implicate top-down processes with impoverished input. Brain mechanisms and the time course of PHT are largely unknown. This event-related potential study used a neuroimaging paradigm that implicated prefrontal cortex in top-down modulation of occipitotemporal cortex. Subjects categorized more impoverished and less impoverished real and pseudo objects. PHT theories predict larger impoverishment effects for (...) real than pseudo objects because top-down processes modulate knowledge only for real objects, but different PHT variants predict different timing. Consistent with parietal-prefrontal PHT variants, around 250 ms, the earliest impoverished real object interaction started on an N3 complex, which reflects interactive cortical activity for object cognition. N3 impoverishment effects localized to both prefrontal and occipitotemporal cortex for real objects only. The N3 also showed knowledge effects by 230 ms that localized to occipitotemporal cortex. Later effects reflected (a) word meaning in temporal cortex during the N400, (b) internal evaluation of prior decision and memory processes and secondary higher-order memory involving anterotemporal parts of a default mode network during posterior positivity (P600), and (c) response related activity in posterior cingulate during an anterior slow wave (SW) after 700 ms. Finally, response activity in supplementary motor area during a posterior SW after 900 ms showed impoverishment effects that correlated with RTs. Convergent evidence from studies of vision, memory, and mental imagery which reflects purely top-down inputs, indicates that the N3 reflects the critical top-down processes of PHT. A hybrid multiple-state interactive, PHT and decision theory best explains the visual constancy of object cognition. (shrink)

The objective of the article is to present Hans Kelsen’s basic norm concept that allows the combination of the two relevant dimensions in relation to juridical science, namely the positivity and validity of law. The role of the concept of basic norm is presented by the author of the Reine Rechtslehre with reference to Kant as a concept enabling formulation of an answer to the question “To what extent is it possible to interpret certain facts as objectively valid legal norms?” (...) The epistemological problem of the object of cognition of juridical science is connected with the issue of normativity. According to Kelsen, only the assumption of a certain nonpositive hypothetical norm regulating the legislation of norms of a given system enables normative interpretation of certain facts. The basic norm authorizes the way of issuing norms, yet not their content. The structure of the legal order creates a hierarchical system in which the higher category norms delegate the law-making power to create the lower category norms. The legal system creates a dynamic system of norms. (shrink)

Robert Rosen has proposed several characteristics to distinguish “simple” physical systems (or “mechanisms”) from “complex” systems, such as living systems, which he calls “organisms”. The Memory Evolutive Systems (MES) introduced by the authors in preceding papers are shown to provide a mathematical model, based on category theory, which satisfies his characteristics of organisms, in particular the merger of the Aristotelian causes. Moreover they identify the condition for the emergence of objects and systems of increasing complexity. As an application, (...) the cognitive system of an animal is modeled by the “MES of cat-neurons” obtained by successive complexifications of his neural system, in which the emergence of higher order cognitive processes gives support to Mario Bunge’s “emergentist monism.”. (shrink)