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A classic exposition of Kant's ethical thought.

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)

A classic result in the foundations of Yang-Mills theory, due to J. W. Barrett ["Holonomy and Path Structures in General Relativity and Yang-Mills Theory." Int. J. Th. Phys. 30, ], establishes that given a "generalized" holonomy map from the space of piece-wise smooth, closed curves based at some point of a manifold to a Lie group, there exists a principal bundle with that group as structure group and a principal connection on that bundle such that the holonomy map corresponds to (...) the holonomies of that connection. Barrett also provided one sense in which this "recovery theorem" yields a unique bundle, up to isomorphism. Here we show that something stronger is true: with an appropriate definition of isomorphism between generalized holonomy maps, there is an equivalence of categories between the category whose objects are generalized holonomy maps on a smooth, connected manifold and whose arrows are holonomy isomorphisms, and the category whose objects are principal connections on principal bundles over a smooth, connected manifold. This result clarifies, and somewhat improves upon, the sense of "unique recovery" in Barrett's theorems; it also makes precise a sense in which there is no loss of structure involved in moving from a principal bundle formulation of Yang-Mills theory to a holonomy, or "loop", formulation. (shrink)

A theory of categorization is presented in which knowledge of causal relationships between category features is represented in terms of asymmetric and probabilistic causal mechanisms. According to causal‐model theory, objects are classified as category members to the extent they are likely to have been generated or produced by those mechanisms. The empirical results confirmed that participants rated exemplars good category members to the extent their features manifested the expectations that causal knowledge induces, such as correlations between feature pairs that are (...) directly connected by causal relationships. These expectations also included sensitivity to higher‐order feature interactions that emerge from the asymmetries inherent in causal relationships. Quantitative fits of causal‐model theory were superior to those obtained with extensions to traditional similarity‐based models that represent causal knowledge either as higher‐order relational features or “prior exemplars” stored in memory. (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 and Zermelo’s quasi-categoricity (...) theorem for second-order —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)

Journal of Social Philosophy, EarlyView.

If categorical equivalence is a good criterion of theoretical equivalence, then it would seem that if some class of mathematical structures is represented as a category, then any other class of structures categorically equivalent to it will have the same representational capacities. Hudetz (2019a) has presented an apparent counterexample to this claim; in this note, I argue that the counterexample fails.

Fefermans argument is indeed convincing in a certain context, it can be dissolved entirely by modifying the context appropriately.

Humans are poorer at identifying smells and communicating about them, compared to other sensory domains. They also cannot easily organize odor sensations in a general conceptual space, where geometric distance could represent how similar or different all odors are. These two generalities are more or less accepted by psychologists, and they are often seen as connected: If there is no conceptual space for odors, then olfactory identification should indeed be poor. We propose here an important revision to this conclusion: We (...) believe that the claim that there is no odor space is true only if by odor space, one means a conceptual space representing all possible odor sensations, in the paradigmatic sense used for instance for color. However, in a less paradigmatic sense, local conceptual spaces representing a given subset of odors do exist. Thus the absence of a global odor space does not warrant the conclusion that there is no olfactory conceptual map at all. Here we show how a localist account provides a new interpretation of experts and cross‐cultural categorization studies: Rather than being exceptions to the poor olfactory identification and communication usually seen elsewhere, experts and cross‐cultural categorization are here taken to corroborate the existence of local conceptual spaces. (shrink)

A common view in the philosophy of mind and philosophy of psychology is that there is an ideally correct way of categorizing the structures and operations of the mind, and that the goal of neuroscience and psychology is to find this correct categorizational scheme. Categories which cannot find a place within this correct framework ought to be eliminated from scientific practice. In this paper, I argue that this general idea runs counter to productive scientific practices. Such a view ignores the (...) plurality of aims and goals that neuroscientists and psychologists have in studying mental phenomena, and the necessity of employing distinct classificatory frameworks to achieve them. (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)

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)

First published in 1984, this is a study of categorization practices: how people categorize each other and their actions; how they describe, infer, and judge. The book presents a sociological analysis and description of practical activities and makes a cogent contribution to the study of how the moral order actually works in practical communicative contexts. Among the issues dealt with are: collectivity categorizations, the organization of lists and descriptions, moral attribution and inferences, and the relationship between standards of morality and (...) standards of rationality. (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)

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)

In proof-theoretic semantics, model-theoretic validity is replaced by proof-theoretic validity. Validity of formulae is defined inductively from a base giving the validity of atoms using inductive clauses derived from proof-theoretic rules. A key aim is to show completeness of the proof rules without any requirement for formal models. Establishing this for propositional intuitionistic logic raises some technical and conceptual issues. We relate Sandqvist’s (complete) base-extension semantics of intuitionistic propositional logic to categorical proof theory in presheaves, reconstructing categorically the soundness (...) and completeness arguments, thereby demonstrating the naturality of Sandqvist’s constructions. This naturality includes Sandqvist’s treatment of disjunction that is based on its second-order or elimination-rule presentation. These constructions embody not just validity, but certain forms of objects of justifications. This analysis is taken a step further by showing that from the perspective of validity, Sandqvist’s semantics can also be viewed as the natural disjunction in a category of sheaves. (shrink)

The categorical imperative can be construed as a universalization test for moral permissibility. False negatives of the categorical imperative would be maxims failing this test, despite the permissibility of their actions; maxims like: ‘I’ll withdraw all my savings on April 15th’. Examples of purported false negatives familiar from the literature can be grouped into three general categories, and dispatched by applying category-specific methods for proper formulation of their maxims, or for proper testing. Methods for reformulating failing maxims, such (...) as the addition of appropriate conditional clauses, do not generate false-positive counterexamples in other instances. (shrink)

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)

This essay starts from a consideration of Deleuze's theory of time. It begins with the empty form of time. But the essay's aim is to understand Deleuze's reversal of Platonism in his 1968 Difference and Repetition. There is no question that the stakes of the reversal of Platonism are ontological. But I argue that what is really at stake is a movement of demoralisation. The essay proceeds in three steps. First, we determine what sufficient reason or grounding is, for Deleuze. (...) Sufficient reason is struck with an irreducible ambiguity. It is this ambiguity in sufficient reason that allows it to be taken advantage of, to be used by representation and good sense for a moral purpose. The second part of the essay will therefore concern ‘the moralisation of sufficient reason’. Its focus will be good sense. But then, third, we must understand Deleuze's ‘demoralisation of sufficient reason’, which necessarily passes through others. Like sufficient reason, others are ambiguous, at once lending themselves to what cancels differences, and opening the way towards difference and intensity. The third step focuses on what Deleuze calls ‘the ethics of intensive quantities’. In the Conclusion, I examine Deleuze's famous, almost cliché, definition of ethics as not being unworthy of the event and, through the empty form of time, I connect it to Kant's formalistic ethics. (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)

The indexing systems used to systematise our knowledge about a domain tend to have an evaluative character: they represent some things as more important, general, complex, or central than others. They are also imperfect and can misrepresent something as more or less important, etc., than it really is. Such distortions mostly result from mistakes made due to lack of time or resources. In some cases they follow systematic patterns which can reveal the implicit judgements and values shared within a community (...) who maintains and uses an index. I focus on the example of PhilPapers, the largest database of philosophy texts available, to show how the arrangement of categories and the way items are assigned to them, can effectively marginalise certain topics, authors, and entire traditions. I draw attention to such issues as: ordering, size and depth of categories, the use of miscellaneous categories, localising indexes in category names, and assigning items to some but not other categories. I suggest that such structuring of the index can have an impact on users, normalising marginalisation and contributing to the perpetuation of inequalities. I conclude by offering some suggestions for improvement which might help our databases flourish and become even more useful. (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)