Even though the strong relationship between proof-theoretic and model-theoretic notions in one’s logical theory can be shown by soundness and completeness proofs, whether we can define the model-theoretic notions by means of the inferences in a proof system is not at all trivial. For instance, provable inferences in a proof system of classical logic in the logical framework do not determine its intended models as shown by Carnap (Formalization of logic, Harvard University Press, Cambridge, 1943), i.e., there are non-Boolean models (...) that satisfy its provable inferences. In the literature, this is known as the Categoricity problem or Carnap’s problem. In this paper, we will discuss the Categoricity problem (or Carnap’s problem) for three-valued logics K3 and LP. We will provide three different restrictions on admissible models that will deliver us categoricity results, some of which draw from the solutions provided for the Categoricity problem for classical logic in Belnap and Massey (Stud Log 49(1):67–82, 1990) and Bonnay and Westerståhl (Erkenntis 81(4):721–739, 2016). We will then argue that two of those solutions are philosophically well-motivated: (1) restricting the admissible models where negation is interpreted as a Strong Kleene truth-function, and (2) restricting the admissible models where a complex formula is assigned the third value when its immediate subformulas are assigned the third value. (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)

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)

This article considers categorical perception (CP) as a crucial process involved in all sort of communication throughout the biological hierarchy, i.e. in all of biosemiosis. Until now, there has been consideration of CP exclusively within the functional cycle of perception–cognition–action and it has not been considered the possibility to extend this kind of phenomena to the mere physiological level. To generalise the notion of CP in this sense, I have proposed to distinguish between categorical perception (CP) and (...) class='Hi'>categorical sensing (CS) in order to extend the CP framework to all communication processes in living systems, including intracellular, intercellular, metabolic, physiological, cognitive and ecological levels. The main idea is to provide an account that considers the heterarchical embeddedness of many instances of CP and CS. This will take me to relate the hierarchical nature of categorical sensing and perception with the equally hierarchical issues of the “binding problem”, “triadic causality”, the “emergent interpretant” and the increasing semiotic freedom observed in biological and cognitive systems. (shrink)

There are many entry points into the problem of categorization. Two particularly important ones are the so-called top-down and bottom-up approaches. Top-down approaches such as artificial intelligence begin with the symbolic names and descriptions for some categories already given; computer programs are written to manipulate the symbols. Cognitive modeling involves the further assumption that such symbol-interactions resemble the way our brains do categorization. An explicit expectation of the top-down approach is that it will eventually join with the bottom-up approach, which (...) tries to model how the hardware of the brain works: sensory systems, motor systems and neural activity in general. The assumption is that the symbolic cognitive functions will be implemented in brain function and linked to the sense organs and the organs of movement in roughly the way a program is implemented in a computer, with its links to peripheral devices such as transducers and effectors. (shrink)

Categorical Monism (that is, the view that all fundamental natural properties are purely categorical) has recently been challenged by a number of philosophers. In this paper, I examine a challenge which can be based on Gabriele Contessa’s [10] defence of the view that only powers can confer dispositions. In his paper Contessa argues against what he calls the Nomic Theory of Disposition Conferral (NTDC). According to NTDC, in each world in which they exist, (categorical) properties confer specific (...) dispositions on their bearers; yet, which disposition a (categorical) property confers on its bearers depends on what the (contingent) laws of nature happen to be. Contessa, inter alia, rests his case on an intuitive analogy between cases of mimicking (in which objects do not actually possess the dispositions associated with their displayed behaviour) and cases of disposition conferral through the action of laws. In this paper, I criticize various aspects of Contessa’s argumentation and show that the conclusion he arrives at (that is, only powers can confer dispositions) is controversial. (shrink)

An ω-categorical supersimple group is finite-by-abelian-by-finite, and has finite SU-rank. Every definable subgroup is commensurable with an acl( $\emptyset$ )-definable subgroup. Every finitely based regular type in a CM-trivial ω-categorical simple theory is non-orthogonal to a type of SU-rank 1. In particular, a supersimple ω-categorical CM-trivial theory has finite SU-rank.

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)

Many of its students are first drawn to Kant's practical philosophy because it seems to promise a theory of morality both objective and practicable. Moral theory, as Kant conceives of it, must abst...

Many recent writers in the philosophy of mathematics have put great weight on the relative categoricity of the traditional axiomatizations of our foundational theories of arithmetic and set theory. Another great enterprise in contemporary philosophy of mathematics has been Wright's and Hale's project of founding mathematics on abstraction principles. In earlier work, it was noted that one traditional abstraction principle, namely Hume's Principle, had a certain relative categoricity property, which here we term natural relative categoricity. In this paper, we show (...) that most other abstraction principles are not naturally relatively categorical, so that there is in fact a large amount of incompatibility between these two recent trends in contemporary philosophy of mathematics. To better understand the precise demands of relative categoricity in the context of abstraction principles, we compare and contrast these constraints to stability-like acceptability criteria on abstraction principles, the Tarski-Sher logicality requirements on abstraction principles studied by Antonelli and Fine, and supervaluational ideas coming out of Hodes' work. (shrink)

Many of its students are first drawn to Kant's practical philosophy because it seems to promise a theory of morality both objective and practicable. Moral theory, as Kant conceives of it, must abst...

This study investigated whether and how a person's varied series of lexical categories corresponding to different discriminatory characteristics of the same colors affect his or her perception of colors. In three experiments, Chinese participants were primed to categorize four graduated colors—specifically dark green, light green, light blue, and dark blue—into green and blue; light color and dark color; and dark green, light green, light blue, and dark blue. The participants were then required to complete a visual search task. Reaction times (...) in the visual search task indicated that different lateralized categorical perceptions of color corresponded to the various priming situations. These results suggest that all of the lexical categories corresponding to different discriminatory characteristics of the same colors can influence people's perceptions of colors and that color perceptions can be influenced differently by distinct types of lexical categories depending on the context. (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)

After a short preface, the first of the three sections of this paper is devoted to historical and philosophic aspects of categoricity. The second section is a self-contained exposition, including detailed definitions, of a proof that every mathematical system whose domain is the closure of its set of distinguished individuals under its distinguished functions is categorically characterized by its induction principle together with its true atoms (atomic sentences and negations of atomic sentences). The third section deals with applications especially those (...) involving the distinction between characterizing a system and axiomatizing the truths of a system. (shrink)

We investigate effective categoricity for polymodal algebras. We prove that the class of polymodal algebras is complete with respect to degree spectra of nontrivial structures, effective dimensions, expansion by constants, and degree spectra of relations. In particular, this implies that every categoricity spectrum is the categoricity spectrum of a polymodal algebra.

Why do we draw the boundaries between “blue” and “green”, where we do? One proposed answer to this question is that we categorize color the way we do because we perceive color categorically. Starting in the 1950’s, the phenomenon of “categorical perception” (CP) encouraged such a response. CP refers to the fact that adjacent color patches are more easily discriminated when they straddle a category boundary than when they belong to the same category. In this paper, I make three (...) related claims. (1) Although what seems to guide discrimination performances seems to indeed be categorical information, the evidence in favor of the fact that categorical perception infl uences the way we perceive color is not convincing. (2) That CP offers a useful account of categorization is not obvious.While aiming at accounting for categorization, CP itself requires an account of categories. This being said, CP remains an interesting phenomenon. Why and how is our discrimination behavior linked to our categories? It is suggested that linguistic labels determine CP through a naming strategy to which participants resort while discriminating colors. This paper’s fi nal point is (3) that the naming strategy account is not enough. Beyond category labels, what seems to guide discrimination performance is category structure. (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.

Deprivation views of the badness of death are almost universally accepted among those who hold that death can be bad for the person who dies. In their most common form, deprivation views hold that death is bad because (and to the extent that) it deprives people of goods they would have gained had they not died at the time they did. Contrast this with categorical desire views, which hold that death is bad because (and to the extent that) it (...) thwarts people’s categorical desires. Categorical desires are desires that are not conditional upon one being alive; yet provide reason for the agent to continue living to ensure that those very desires are satisfied. I argue that categorical desire views are subject to two serious problems that deprivation views are not. First, categorical desire views entail that it is not bad for someone to not be resuscitated after dying a bad death. Second, categorical desire views cannot account for cases in which it is good to prevent people from coming into existence or cases in which it is good to prevent them from continuing to exist. After considering, and rejecting, various replies on behalf of categorical desire proponents, I conclude that we have good reason to reject categorical desire views in favor of deprivation views. (shrink)

We argue that, if taken seriously, Kripke's view that a language for science can dispense with a negation operator is to be rejected. Part of the argument is a proof that positive logic, i.e., classical propositional logic without negation, is not categorical.

A well-known difficulty that affects all accounts of laws of nature according to which the latter are higher-order facts involving relations between universals (the so-called DTA accounts, from Dretske in Philosophy of Science 44:248–268, 1977; Tooley in Canadian Journal of Philosophy 7:667–698, 1977 and Armstrong (What is a Law of Nature?, Cambridge University Press, Cambridge, 1983)) is the Inference Problem: how can laws construed in that way determine the first-order regularities that we find in the actual world? Bird (Analysis 65:147–55, (...) 2005) has argued that there is no solution to the Inference Problem which is consistent with both categorical monism (that is, the view that all natural properties are categorical) and basic tenets of Armstrong’s account of the laws of nature. This paper shows that, given Armstrong’s view about laws as first-order structural universals whose instantiation ‘produce’ nomic regularities and under specific plausible metaphysical assumptions concerning nomic relations which are consistent with a broadly construed DTA approach to laws, there is no extra difficulty regarding the Inference Problem in a categorical monistic context besides the ones that beset structural universals in general. (shrink)

Discussions about dispositional and categorical properties have become commonplace in metaphysics. Unfortunately, dispositionality and categoricity are disputed notions: usual characterizations are piecemeal and not widely applicable, thus threatening to make agreements and disagreements on the matter merely verbal—and also making it arduous to map a logical space of positions about dispositional and categorical properties in which all parties can comfortably fit. This paper offers a prescription for this important difficulty, or at least an inkling thereof. This will be (...) achieved by comparing pairs of positions and exploring their background metaphysics to discover where alleged agreements and disagreements concerning dispositionality and categoricity really lie; more specifically, the Pure Powers view and the Powerful Qualities view will be under scrutiny. Over this background, the prescription functions by isolating a successful identity-based characterization of categoricity, while abandoning the correspondent identity-based characterization of dispositionality. On the contrary, according to this prescription a property is dispositional if and only if it is solely in virtue of possessing that property that its bearer is assigned a certain dispositional profile. A crucial consequence of this prescription is that, while supporters of the Pure Powers view often characterize their position as an essentialist one, the dispositionality of properties needn’t always be a matter of essence. (shrink)

We prove that each ω-categorical, generically stable group is solvable-by-finite.

For a computable structure $\mathcal {M}$, the categoricity spectrum is the set of all Turing degrees capable of computing isomorphisms among arbitrary computable copies of $\mathcal {M}$. If the spectrum has a least degree, this degree is called the degree of categoricity of $\mathcal {M}$. In this paper we investigate spectra of categoricity for computable rigid structures. In particular, we give examples of rigid structures without degrees of categoricity.

During public health crises including the COVID-19 pandemic, resource scarcity and contagion risks may require health systems to shift—to some degree—from a usual clinical ethic, focused on the well-being of individual patients, to a public health ethic, focused on population health. Many triage policies exist that fall under the legal protections afforded by “crisis standards of care,” but they have key differences. We critically appraise one of the most fundamental differences among policies, namely the use of criteria to categorically exclude (...) certain patients from eligibility for otherwise standard medical services. We examine these categorical exclusion criteria from ethical, legal, disability, and implementation perspectives. Focusing our analysis on the most common type of exclusion criteria, which are disease-specific, we conclude that optimal policies for critical care resource allocation and the use of cardiopulmonary resuscitation (CPR) should not use categorical exclusions. We argue that the avoidance of categorical exclusions is often practically feasible, consistent with public health norms, and mitigates discrimination against persons with disabilities. (shrink)

The concept of “life” certainly is of some use to distinguish birds and beavers from water and stones. This pragmatic usefulness has led to its construal as a categorical predicate that can sift out living entities from non-living ones depending on their possessing specific properties—reproduction, metabolism, evolvability etc. In this paper, we argue against this binary construal of life. Using text-mining methods across over 30,000 scientific articles, we defend instead a degrees-of-life view and show how these methods can contribute (...) to experimental philosophy of science and concept explication. We apply topic-modeling algorithms to identify which specific properties are attributed to a target set of entities (bacteria, archaea, viruses, prions, plasmids, phages and the molecule of adenine). Eight major clusters of properties were identified together with their relative relevance for each target entity (two that relate to metabolism and catalysis, one to genetics, one to evolvability, one to structure, and—rather unexpectedly—three that concern interactions with the environment broadly construed). While aligning with intuitions—for instance about viruses being less alive than bacteria—these quantitative results also reveal differential degrees of performance that have so far remained elusive or overlooked. Taken together, these analyses provide a conceptual “lifeness space” that makes it possible to move away from a categorical construal of life by empirically assessing the relative lifeness of more-or-less alive entities. (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)

Drawing from a Sellarsian realist-naturalist epistemology, we trace different levels of cognitive hierarchy procedures through which a representational system learns to update its own states and improve its ‘map-making’ capabilities from pre-conscious operations which modulate base-localization functions, to patterns of epistemic revision and integration at the conceptual and theoretical levels, producing a nomological double of its world. We show how ontological theorization becomes diachronically coordinated with and constrained by empirical science, and how the formal-quantitative kernel of scientific theories corresponds to (...) qualitative-conceptual determinations at the structural level. Following Johanna Seibt’s characterization of ontology as a theory of categorial inference, we trace the preservation of inferential semantic structure across ontological theories in relation to model languages and provide provisional indications to coordinate Seibt’s account with a convergent realist assessment of systematic modeling, defining the epistemological conditions for articulating the preservation of formal structure in theories toward a limit-point of enquiry. (shrink)

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)

Kant has argued that moral requirements are categorical. Kant's claim has been challenged by some contemporary philosophers; this article defends Kant's doctrine. I argue that Kant's claim captures the unique feature of moral requirements. The main arguments against Kant's claim focus on one condition that a categorical imperative must meet: to be independent of desires. I argue that there is another important, but often ignored, condition that a categorical imperative must meet, and this second condition is crucial (...) to understanding why moral requirements are not hypothetical. I also argue that the claim that moral requirements are not categorical because they depend on desires for motivation is beside the point. The issue of whether moral requirements are categorical is not an issue about whether moral desires or feelings are necessary for moral motivation but are rather an issue about the ground of moral desires or moral feelings. Moral requirements are categorical because they are requirements of reason, and reason makes moral desires or feelings possible. (shrink)

We develop a general covariant categorical modeling theory of natural systems’ behavior based on the fundamental functorial processes of representation and localization-globalization. In the first part of this study we analyze the process of representation. Representation constitutes a categorical modeling relation that signifies the semantic bidirectional process of correspondence between natural systems and formal symbolic systems. The notion of formal systems is substantiated by algebraic rings of observable attributes of natural systems. In this perspective, the distinction between simple (...) and complex systems is reflected in the appropriate qualification of their corresponding rings of observables. The crucial distinguishing requirement with respect to the coordinatizing rings of observables has to do with the property of global commutativity. The global information structure representing the behavior of a complex system is modeled functorially in terms of its Spectrum functor. Due to the fact that, the fundamental process of representation is bidirectional by construction, it admits a precise categorical formulation in terms of the syntactic language of adjoint functors, constituting thus, a categorical adjunction. The left adjoint functor of this adjunction signifies the process of encoding the information related with phenomena of natural systems in terms of coordinatizing rings of observables, whereas, the right adjoint functor signifies the inverse process of information decoding, which, can be used for making predictions about the behavior of natural systems. (shrink)

We develop a general covariant categorical modeling theory of natural systems' behavior based on the fundamental functorial processes of representation and localization-globalization. In the second part of this study we analyze the semantic bidirectional process of localization-globalization. The notion of a localization system of a complex information structure bears a dual role: Firstly, it determines the appropriate categorical environment of base reference contexts for considering the operational modeling of a complex system's behavior, and secondly, it specifies the global (...) compatibility conditions of local contextual information. A localization system acts on the global information structure of a complex system, partitions it into sorts, and eventually, forces the consistent sheaf-theoretic fibering of the latter over the base category of commutative reference contexts. In this manner, the sheafification of the Spectrum functor of a complex information structure takes place by imposing on the uniform and homologous fibered structure of elements of the Spectrum presheaf the following two requirements of coherence in relation to the localization-globalization process: [i]. Compatibility of information under restriction from the global to the local level, and [ii]. Compatibility of information under extension from the local to the global level. Correspondingly, the options of local and global receive a concrete mathematical meaning with respect to a suitable notion of topology (categorical Grothendieck topology) defined on the base category of commutative reference contexts. Finally, the accurate functorial process modeling of a complex system's behavior, respecting the processes of representation and localization-globalization, is being effectuated by means of establishing a categorical dual equivalence between the category of complex information structures, and the topes of sheaves over the base category of partial or local information carriers, equipped with the categorical topology of epimorphosis families. (shrink)

We investigate effective categoricity of computable equivalence structures . We show that is computably categorical if and only if has only finitely many finite equivalence classes, or has only finitely many infinite classes, bounded character, and at most one finite k such that there are infinitely many classes of size k. We also prove that all computably categorical structures are relatively computably categorical, that is, have computably enumerable Scott families of existential formulas. Since all computable equivalence structures (...) are relatively categorical, we further investigate when they are categorical. We also obtain results on the index sets of computable equivalence structures. (shrink)