The purpose of this paper is to show that the Elementary Process Theory (EPT) agrees with the knowledge of the physical world obtained from the successful predictions of Special Relativity (SR). For that matter, a recently developed method is applied: a categorical model of the EPT that incorporates SR is fully specified. Ultimate constituents of the universe of the EPT are modeled as point-particles, gamma-rays, or time-like strings, all represented by integrable hyperreal functions on Minkowski space. This proves (...) that the EPT agrees with SR. -/- Edit: this preprint differs significantly from the published chapter. (shrink)

This paper is concerned with the application of constructions from category theory to Smith and McIntyre's interpretation of Husserl's intentionality. 1 Not only did Hussefl's own ideas change in the course of his lifetime 2 but there are a number of interpretations of Husserl's work 3 so that the line of philosophical investigation that Husserl strongly influenced is still in the process of development. In this vein, Smith and McIntyre have recognized the potential for a possible worlds interpretation of intentionality (...) in Hussefl's writing 4 which has led them to extend their interpretation to give a many worlds account of intentionality. 5 Thus, while Smith and McIntyre have refuted interpretations of Husserl's noema in terms of ideal objects, they have been willing to explicate the noerna using possible objects. (shrink)

We present a categorical/denotational semantics for the Lambek Syntactic Calculus , indeed for a λlD-typed version Curry-Howard isomorphic to it. The main novelty of our approach is an abstract noncommutative construction with right and left adjoints, called sequential product. It is defined through a hierarchical structure of categories reflecting the implicit permission to sequence expressions and the inductive construction of compound expressions. We claim that Lambek's noncommutative product corresponds to a noncommutative bi-endofunctor into a category, which encloses all categories (...) of such hierarchical structure. A soundness theorem for LSC is shown with respect to this semantical framework. (shrink)

Recently, the second author, Briseid, and Safarik introduced nonstandard Dialectica, a functional interpretation capable of eliminating instances of familiar principles of nonstandard arithmetic—including overspill, underspill, and generalizations to higher types—from proofs. We show that the properties of this interpretation are mirrored by first-order logic in a constructive sheaf model of nonstandard arithmetic due to Moerdijk, later developed by Palmgren, and draw some new connections between nonstandard principles and principles that are rejected by strict constructivism. Furthermore, we introduce a variant (...) of the Diller–Nahm interpretation with two different kinds of quantifiers, similar to Hernest’s light Dialectica interpretation, and show that one can obtain nonstandard Dialectica by weakening the computational content of the existential quantifiers—a process called herbrandization. We also define a constructive sheaf model mirroring this new functional interpretation, and show that the process of herbrandization has a clear meaning in terms of these sheaf models. (shrink)

ABSTRACTThe present study describes the development and validation of a facial expression database comprising five different horizontal face angles in dynamic and static presentations. The database includes twelve expression types portrayed by eight Japanese models. This database was inspired by the dimensional and categorical model of emotions: surprise, fear, sadness, anger with open mouth, anger with closed mouth, disgust with open mouth, disgust with closed mouth, excitement, happiness, relaxation, sleepiness, and neutral. The expressions were validated using emotion classification (...) and Affect Grid rating tasks [Russell, Weiss, & Mendelsohn, 1989. Affect Grid: A single-item scale of pleasure and arousal. Journal of Personality and Social Psychology, 57, 493–502]. The results indicate that most of the expressions were recognised as the intended emotions and could systematically represent affective valence and arousal. Furthermore, face angle and facial motion information influenced emot... (shrink)

We provide here the first steps toward a Classification Theory ofElementary Classes with no maximal models, plus some mild set theoretical assumptions, when the class is categorical in some λ greater than its Löwenheim-Skolem number. We study the degree to which amalgamation may be recovered, the behaviour of non μ-splitting types. Most importantly, the existence of saturated models in a strong enough sense is proved, as a first step toward a complete solution to the o Conjecture for these classes. (...) Further results are in preparation. (shrink)

The results in this paper are in a context of abstract elementary classes identified by Shelah and Villaveces in which the amalgamation property is not assumed. The long-term goal is to solve Shelah’s Categoricity Conjecture in this context. Here we tackle a problem of Shelah and Villaveces by proving that in their context, the uniqueness of limit models follows from categoricity under the assumption that the subclass of amalgamation bases is closed under unions of bounded, -increasing chains.

An important part of the theory of algebraizable sentential logics consists of studying the algebraic semantics of these logics. As developed by Czelakowski, Blok, and Pigozzi and Font and Jansana, among others, it includes studying the properties of logical matrices serving as models of deductive systems and the properties of abstract logics serving as models of sentential logics. The present paper contributes to the development of the categorical theory by abstracting some of these model theoretic aspects and results (...) from the level of sentential logics to the level of π-institutions. (shrink)

We study versions of limit models adapted to the context of metric abstract elementary classes. Under categoricity and superstability-like assumptions, we generalize some theorems from 7, 15-17. We prove criteria for existence and uniqueness of limit models in the metric context.

We show that if T is a trivial uncountably categorical theory of Morley Rank 1 then T is model complete after naming constants for a model.

We explore the existence and the size of infinite models of categorical theories having cardinality less than the size of the associated Tarski-Lindenbaum algebra. Restricting to totally transcendental, categorical theories we show that “Every tiny model is countable” is independent of ZFC. IfT is trivial there is at most one tiny model, which must be the algebraic closure of the empty set. We give a new proof that there are no tiny models ifT is not totally (...) transcendental and is non-trivial. (shrink)

We prove Łos conjecture = Morley theorem in ZF, with the same characterization, i.e., of first order countable theories categorical in $N_\alpha $ for some (equivalently for every ordinal) α > 0. Another central result here in this context is: the number of models of a countable first order T of cardinality $N_\alpha $ is either ≥ |α| for every α or it has a small upper bound (independent of α close to Ð₂).

Frege’s project has been characterized as an attempt to formulate a complete system of logic adequate to characterize mathematical theories such as arithmetic and set theory. As such, it was seen to fail by Gödel’s incompleteness theorem of 1931. It is argued, however, that this is to impose a later interpretation on the word ‘complete’ it is clear from Dedekind’s writings that at least as good as interpretation of completeness is categoricity. Whereas few interesting first-order mathematical theories are categorical (...) or complete, there are logical extensions of these theories into second-order and by the addition of generalized quantifiers which are categorical. Frege’s project really found success through Gödel’s completeness theorem of 1930 and the subsequent development of first- and higher-order model theory. (shrink)

Due to Gӧdel’s incompleteness results, the categoricity of a sufficiently rich mathematical theory and the semantic completeness of its underlying logic are two mutually exclusive ideals. For first- and second-order logics we obtain one of them with the cost of losing the other. In addition, in both these logics the rules of deduction for their quantifiers are non-categorical. In this paper I examine two recent arguments –Warren (2020), Murzi and Topey (2021)– for the idea that the natural deduction rules (...) for the first-order universal quantifier are categorical, i.e., they uniquely determine its semantic intended meaning. Both of them make use of McGee’s open-endedness requirement and the second one uses in addition Garson’s (2013) local models for defining the validity of these rules. I argue that the success of both these arguments is relative to their semantic or infinitary assumptions, which could be easily discharged if the introduction rule for the universal quantifier is taken to be an infinitary rule, i.e. non-compact. Consequently, I reconsider the use of the ω-rule and I show that the addition of the ω-rule to the standard formalizations of first-order logic is categorical. In addition, I argue that the open-endedness requirement does not make the first-order Peano Arithmetic categorical and I advance an argument for its categoricity based on the inferential conservativity requirement. (shrink)

We build an א₁-categorical but not א₀-categorical theory whose only computably presentable model is the saturated one. As a tool, we introduce a notion related to limitwise monotonic functions.

Rosen's modelling relations constitute a conceptual schema for the understanding of the bidirectional process of correspondence between natural systems and formal symbolic systems. The notion of formal systems used in this study refers to information structures constructed as algebraic rings of observable attributes of natural systems, in which the notion of observable signifies a physical attribute that, in principle, can be measured. Due to the fact that modelling relations are bidirectional by construction, they admit a precise categorical formulation in (...) terms of the category-theoretic syntactic language of adjoint functors, representing the inverse processes of information encoding/decoding via adjunctions. As an application, we construct a topological modelling schema of complex systems. The crucial distinguishing requirement between simple and complex systems in this schema is reflected with respect to their rings of observables by the property of global commutativity. The global information structure representing the behaviour of a complex system is modelled functorially in terms of its spectrum functor. An exact modelling relation is obtained by means of a complex encoding/decoding adjunction restricted to an equivalence between the category of complex information structures and the category of sheaves over a base category of partial or local information carriers equipped with an appropriate topology. (shrink)

Alex has written an excellent thesis in the area of computable model theory. The latter is a subject that nicely combines model-theoretic ideas with delicate recursiontheoretic constructions. The results demand good knowledge of both fields. In his thesis, Alex begins by reviewing the essential model-theoretic facts, especially the Baldwin-Lachlan result about uncountably categorical theories. This he follows with a brief discussion of recursion theory, including mention of the priority method. The deepest part of the thesis concerns (...) the study of the recursive spectrum of an uncountably categorical theory, i.e. the set of n ≤ ! such that the n-th model of the theory (in the Baldwin-Lachlan sense) has a computable presentation. This is a deep and very active area of conteporary research in computable model theory which Alex discusses in considerable detail. The exposition is very good and therefore the thesis makes a nice introduction to the subject for a wide community of logicians. (Sy-David Friedman, Professor of Mathematical Logic, Director of the Kurt Godel Research Center for Mathematical Logic, University of Vienna). (shrink)

In the paper “Categoricity in abstract elementary classes with no maximal models”, we address gaps in Saharon Shelah and Andrés Villavecesʼ proof in [4] of the uniqueness of limit models of cardinality μ in λ-categorical abstract elementary classes with no maximal models, where λ is some cardinal larger than μ. Both [4] and [5] employ set theoretic assumptions, namely GCH and Φμ+μ+).Recently, Tapani Hyttinen pointed out a problem in an early draft of [3] to Villaveces. This problem stems from (...) the proof in Shelah and Villavecesʼ [4] that reduced towers are continuous. Residues of this problem also infect the proof of Proposition II.7.2 in VanDieren [5]. We respond to the issues in Shelah and Villaveces [4] and VanDieren [5] with alternative proofs under the strengthened assumption that the abstract elementary class is categorical in μ+. (shrink)

According to the Lockean thesis, a proposition is believed just in case it is highly probable. While this thesis enjoys strong intuitive support, it is known to conflict with seemingly plausible logical constraints on our beliefs. One way out of this conflict is to make probability 1 a requirement for belief, but most have rejected this option for entailing what they see as an untenable skepticism. Recently, two new solutions to the conflict have been proposed that are alleged to be (...) non-skeptical. We compare these proposals with each other and with the Lockean thesis, in particular with regard to the question of how much we gain by adopting any one of them instead of the probability 1 requirement, that is, of how likely it is that one believes more than the things one is fully certain of. (shrink)

David Danks. Psychological Theories of Categorizations as Probabilistic Models.

We investigate a notion called uniqueness in power κ that is akin to categoricity in power κ, but is based on the cardinality of the generating sets of models instead of on the cardinality of their universes. The notion is quite useful for formulating categoricity-like questions regarding powers below the cardinality of a theory. We prove, for universal theories T, that if T is κ-unique for one uncountable κ, then it is κ-unique for every uncountable κ; in particular, it is (...) categorical in powers greater than the cardinality of T. (shrink)

A system of logic usually comprises a language for which a model-theory and a proof-theory are defined. The model-theory defines the semantic notion of model-theoretic logical consequence (⊨), while the proof-theory defines the proof- theoretic notion of logical consequence (or logical derivability, ⊢). If the system in question is sound and complete, then the two notions of logical consequence are extensionally equivalent. The concept of full formalization is a more restrictive one and requires in addition the preservation (...) of the standard meanings of the logical terms in all the admissible interpretations of the logical calculus, as it is proof-theoretically defined. Although classical first-order logic is sound and complete, its standard formalizations fall short to be full formalizations since they allow non-intended interpretations. This fact poses a challenge for the logical inferentialism program, whose main tenet is that the meanings of the logical terms are uniquely determined by the formal axioms or rules of inference that govern their use in a logical calculus, i.e., logical inferentialism requires a categorical calculus. This paper is the first part of a more elaborated study which will analyze the categoricity problem from its beginning until the most recent approaches. I will first start by describing the problem of a full formalization in the general framework in which Carnap (1934/1937, 1943) formulated it for classical logic. Then, in sections IV and V, I shall discuss the way in which the mathematicians B.A. Bernstein (1932) and E.V. Huntington (1933) have previously formulated and analyzed it in algebraic terms for propositional logic and, finally, I shall discuss some critical reactions Nagel (1943), Hempel (1943), Fitch (1944), and Church (1944) formulated to these approaches. (shrink)

On a widespread naturalist view, the meanings of mathematical terms are determined, and can only be determined, by the way we use mathematical language—in particular, by the basic mathematical principles we’re disposed to accept. But it’s mysterious how this can be so, since, as is well known, minimally strong first-order theories are non-categorical and so are compatible with countless non-isomorphic interpretations. As for second-order theories: though they typically enjoy categoricity results—for instance, Dedekind’s categoricity theorem for second-order PA and Zermelo’s (...) quasi-categoricity theorem for second-order ZFC—these results require full second-order logic. So appealing to these results seems only to push the problem back, since the principles of second-order logic are themselves non-categorical: those principles are compatible with restricted interpretations of the second-order quantifiers on which Dedekind’s and Zermelo’s results are no longer available. In this paper, we provide a naturalist-friendly, non-revisionary solution to an analogous but seemingly more basic problem—Carnap’s Categoricity Problem for propositional and first-order logic—and show that our solution generalizes, giving us full second-order logic and thereby securing the categoricity or quasi-categoricity of second-order mathematical theories. Briefly, the first-order quantifiers have their intended interpretation, we claim, because we’re disposed to follow the quantifier rules in an open-ended way. As we show, given this open-endedness, the interpretation of the quantifiers must be permutation-invariant and so, by a theorem recently proved by Bonnay and Westerståhl, must be the standard interpretation. Analogously for the second-order case: we prove, by generalizing Bonnay and Westerståhl’s theorem, that the permutation invariance of the interpretation of the second-order quantifiers, guaranteed once again by the open-endedness of our inferential dispositions, suffices to yield full second-order logic. (shrink)

We show that the categoricity of second-order Peano axioms can be proved from the comprehension axioms. We also show that the categoricity of second-order Zermelo–Fraenkel axioms, given the order type of the ordinals, can be proved from the comprehension axioms. Thus these well-known categoricity results do not need the so-called “full” second-order logic, the Henkin second-order logic is enough. We also address the question of “consistency” of these axiom systems in the second-order sense, that is, the question of existence of (...) models for these systems. In both cases we give a consistency proof, but naturally we have to assume more than the mere comprehension axioms. (shrink)

We prove that from categoricity in λ+ we can get categoricity in all cardinals ≥ λ+ in a χ-tame abstract elementary classe [Formula: see text] which has arbitrarily large models and satisfies the amalgamation and joint embedding properties, provided [Formula: see text] and λ ≥ χ. For the missing case when [Formula: see text], we prove that [Formula: see text] is totally categorical provided that [Formula: see text] is categorical in [Formula: see text] and [Formula: see text].

Categorical perception (CP) refers to how similar things look different depending on whether they are classified as the same category. Many studies demonstrate that adult humans show CP for human emotional faces. It is widely debated whether the effect can be accounted for solely by perceptual differences (structural differences among emotional faces) or whether additional perceiver-based conceptual knowledge is required. In this review, I discuss the phenomenon of CP and key studies showing CP for emotional faces. I then discuss (...) a new model of emotion which highlights how perceptual and conceptual knowledge interact to explain how people see discrete emotions in others’ faces. In doing so, I discuss how language (emotion words included in the paradigm) contribute to CP. (shrink)

Notoriously, the dispositional view of natural properties is thought to face a number of regress problems, one of which points to an epistemological worry. In this paper, I argue that the rival categorical view is also susceptible to the same kind of regress problem. This problem can be overcome, most plausibly, with the development of a structuralist epistemology. After identifying problems faced by alternative solutions, I sketch the main features of this structuralist epistemological approach, referring to graph-theoretic modelling in (...) the process. Given that both the categoricalists and dispositionalists are under pressure to adopt this same epistemological approach in light of the regress problem, this suggests that the categoricalist versus dispositionalist debate is best fought on metaphysical rather than epistemological grounds. (shrink)

We consider two systems of constructive modal logic which are computationally motivated. Their modalities admit several computational interpretations and are used to capture intensional features such as notions of computation, constraints, concurrency, etc. Both systems have so far been studied mainly from type-theoretic and category-theoretic perspectives, but Kripke models for similar systems were studied independently. Here we bring these threads together and prove duality results which show how to relate Kripke models to algebraic models and these in turn to the (...) appropriate categorical models for these logics. (shrink)

This article proposes to explicate theoretical equivalence by supplementing formal equivalence criteria with preservation conditions concerning interpretation. I argue that both the internal structure of models and choices of morphisms are aspects of formalisms that are relevant when it comes to their interpretation. Hence, a formal criterion suitable for being supplemented with preservation conditions concerning interpretation should take these two aspects into account. The two currently most important criteria—gener-alized definitional equivalence (Morita equivalence) and categorical equivalence—are not optimal in this (...) respect. I put forward a criterion that takes both aspects into account: the criterion of definable categorical equivalence. (shrink)

In the paper there are introduced and discussed the concepts of an indexed category with quantifications and a higher level indexed category to present an algebraic characterization of some version of Martin-Löf Type Theory. This characterization is given by specifying an additional equational structure of those indexed categories which are models of Martin-Löf Type Theory. One can consider the presented characterization as an essentially algebraic theory of categorical models of Martin-Löf Type Theory. The paper contains a construction of an (...) indexed category with quantifications from terms and types of the language of Martin-Löf Type Theory given in the manner of Troelstra [11]. The paper contains also an inductive definition of a valuation of these terms and types in an indexed category with quantifications. (shrink)

Two models of second-order ZFC need not be isomorphic to each other, but at least one is isomorphic to an initial segment of the other. The situation is subtler for impure set theory, but Vann McGee has recently proved a categoricity result for second-order ZFCU plus the axiom that the urelements form a set. Two models of this theory with the same universe of discourse need not be isomorphic to each other, but the pure sets of one are isomorphic to (...) the pure sets of the other. This paper argues that similar results obtain for considerably weaker second-order axiomatizations of impure set theory that are in line with two different conceptions of set, the iterative conception and the limitation of size doctrine. (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.

This paper provides a historically sensitive discussion of Carnaps theory will be assessed with respect to two interpretive issues. The first concerns his mathematical sources, that is, the mathematical axioms on which his extremal axioms were based. The second concerns Carnapcompleteness of the modelss different attempts to explicate the extremal properties of a theory and puts his results in context with related metamathematical research at the time.