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 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)

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 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.

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)

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)

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)

One of the philosophical uses of Dedekind’s categoricity theorem for Peano Arithmetic is to provide support for semantic realism. To this end, the logical framework in which the proof of the theorem is conducted becomes highly significant. I examine different proposals regarding these logical frameworks and focus on the philosophical benefits of adopting open-ended schemas in contrast to second order logic as the logical medium of the proof. I investigate Pederson and Rossberg’s critique of the ontological advantages of open-ended (...) arithmetic when it comes to establishing the categoricity of Peano Arithmetic and show that the critique is highly problematic. I argue that Pederson and Rossberg’s ontological criterion deliver the bizarre result that certain first order subsystems of Peano Arithmetic have a second order ontology. As a consequence, the application of the ontological criterion proposed by Pederson and Rossberg assigns a certain type of ontology to a theory, and a different, richer, ontology to one of its subtheories. (shrink)

The categoricity problem for a system of logic reveals an asymmetry between the model-theoretic and the proof-theoretic resources of that logic. In particular, it reveals prima facie that the proof-theoretic instruments are insufficient for matching the envisaged model-theory, when the latter is already available. Among the proposed solutions for solving this problem, some make use of new proof-theoretic instruments, some others introduce new model-theoretic constrains on the proof-systems, while others try to use instruments from both sides. On the proof-theoretical (...) side, two main approaches for solving the categoricity problem for propositional classical logic consist in the enforcement of the formal systems of this logic by introducing rules of inference of a new kind. A multiple-conclusions logic allows rules of inference with more than one conclusion, while a bilateralist system contains rules of inference formulated in terms of assertion and rejection. Both these approaches reveal interesting formal features of logical reasoning. This paper analyses some of the advantages and limitations of these two approaches as solutions for a full formalization of logic, i.e., for a categorical formalization. (shrink)

This article surveys recent literature by Parsons, McGee, Shapiro and others on the significance of categoricity arguments in the philosophy of mathematics. After discussing whether categoricity arguments are sufficient to secure reference to mathematical structures up to isomorphism, we assess what exactly is achieved by recent ‘internal’ renditions of the famous categoricity arguments for arithmetic and set theory.

Consequence rleations over sets of "judgments" are defined by using "overdetermined" as well as "underdetermined" valuations. Some of these relations are shown to be categorical. And generalized soundness and completeness results are given for both multiple and single conclusion consequence relations.

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)

Suppose that X and Y can’t possibly exist apart in reality; then—by definition—there’s no real distinction between them, only a conceptual distinction. There’s a conceptual distinction between a rectilinear figure’s triangularity and its trilaterality, for example, but no real distinction. In fundamental metaphysics there is no real distinction between an object’s categorical properties and its dispositional properties. So too there is no real distinction between an object and its properties. And in fundamental metaphysics, for X and Y to be such (...) that there is no real distinction between them is for them to be identical. (shrink)

What is sexual orientation? The contemporary consensus among philosophers is that it is a disposition. Unsurprisingly, recent debates about the metaphysics of sexual orientation are almost entirely intramural. Behavioral dispositionalists argue that sexual orientation is a disposition to behave sexually. Desire dispositionalists argue that it is a disposition to desire sexually. We argue that sexual orientation is not best understood in terms of dispositions to behave or dispositions to desire before arguing that dispositions tout court fail to illuminate sexual orientation. (...) We then introduce and defend the idea that sexual orientation is best understood in terms of categorical phenomenal properties of sexual arousal. (shrink)

David Gauthier suggested that all genuine moral problems are Prisoners Dilemmas (PDs), and that the morally and rationally required solution to a PD is to co-operate. I say there are four other forms of moral problem, each a different way of agents failing to be in PDs because of the agents’ preferences. This occurs when agents have preferences that are malevolent, self-enslaving, stingy, or bullying. I then analyze preferences as reasons for action, claiming that this means they must not target (...) the impossible, they must be able to be acted on in the circumstances, their targets must be attainable, and having the preferences must make their targets more likely. For groups of agents to have a distribution of preferences, their preferences must jointly have those four features, this imposing a kind of universalizability requirement on possible preferences. I then claim that, if all agents began with preferences satisfying these requirements, their preferences would not be of the morally problematic sort (on pain, variously, of circularity or contradiction in the specification of their targets). Instead, they would be either morally innocent preferences, or ones which put the agents in PDs. And it would then be instrumentally rational for the agents to prefer mutual co-operation. Thus if all agents initially had rationally permissible preferences and made rational choices of actions and preferences thereafter, they would never acquire immoral preferences, and so never be rationally moved to immoral actions. Further, the states of affairs such agents would be moved to bring about would be compatible with what Rawls’ agents would chose behind a veil of ignorance. Morality therefore reduces to rationality; necessarily, the actions categorically required by morality are also categorically required by rationality. (shrink)

This essay discusses the difficulty to reconcile two paradigms about beliefs: the binary or categorical paradigm of yes/no beliefs and the probabilistic paradigm of degrees of belief. The possibility for someone to hold both types of belief simultaneously is challenged by the lottery paradox, and more recently by a general impossibility theorem by Dietrich and List (2018, 2021). The nature, relevance, and implications of the tension are explained and assessed.

The paper sketches an argument against modal monism, more specifically against the reduction of physical possibility to metaphysical possibility. The argument is based on the non-categoricity of quantum logic.

The primary concern of this paper is to outline an explanation of how Kant derives morality from reason. We all know that Kant thought that morality comprises a set of demands that are unconditionally and universally valid. In addition, he thought that to support this understanding of moral principles, one must show that they originate in reason a priori, rather than in contingent facts about human psychology, or the circumstances of human life. But do we really understand how he tries (...) to establish that moral principles originate in reason? In at least two passages in the second section of the Groundwork, Kant insists upon the importance of grounding the moral law in practical reason a priori, and subsequently states a conception of practical reason from which he appears to extract a formulation of the Categorical Imperative. The reasoning employed in these passages would appear to be of central importance to the overall argument of the Groundwork, but in each case the route travelled from the definition of practical reason to the ensuing formulation of the moral law is obscure. My goal is to work out a plausible reconstruction of this portion of Kant’s argument. At the very least, I hope that my interpretation will illuminate the distinctive structure of the Kantian approach to questions of justification in ethics. What I understand of Kant’s view leads me to believe that its aims and overall shape are different in important respects from what is often assumed. It also represents an approach to foundational issues in ethics, which provides an alternative to many contemporary attempts to ground morality in reason. (shrink)

This paper responds to recent work in the philosophy of Homotopy Type Theory by James Ladyman and Stuart Presnell. They consider one of the rules for identity, path induction, and justify it along ‘pre-mathematical’ lines. I give an alternate justification based on the philosophical framework of inferentialism. Accordingly, I construct a notion of harmony that allows the inferentialist to say when a connective or concept is meaning-bearing and this conception unifies most of the prominent conceptions of harmony through category theory. (...) This categorical harmony is stated in terms of adjoints and says that any concept definable by iterated adjoints from general categorical operations is harmonious. Moreover, it has been shown that identity in a categorical setting is determined by an adjoint in the relevant way. Furthermore, path induction as a rule comes from this definition. Thus we arrive at an account of how path induction, as a rule of inference governing identity, can be justified on mathematically motivated grounds. (shrink)

Allan Hazlett has recently developed an alternative to the most popular form of anti-realism about epistemic normativity, epistemic expressivism. He calls it “convention-relativism about epistemic discourse”. The view deserves more attention. In this paper, I give it attention in the form of an objection. Specifically, my objection turns on a distinction between inescapable and categorical norms. While I agree with Hazlett that convention-relativism is consistent with inescapable epistemic norms, I argue that it is not consistent with categorical epistemic norms. I (...) then argue that Hazlett's account of a controversial upshot of convention-relativism – namely, that epistemic discourse is not “normative” – should, but does not, adequately address the question of whether epistemic norms are categorical. This leads to a more general discussion of anti-realism in epistemology. (shrink)

Gottfried Leibniz embarked on a research program to prove all the Aristotelic categorical syllogisms by diagrammatic and algebraic methods. He succeeded in proving them by means of Euler diagrams, but didn’t produce a manuscript with their algebraic proofs. We demonstrate how key excerpts scattered across various Leibniz’s drafts on logic contained sufficient ingredients to prove them by an algebraic method –which we call the Leibniz-Cayley (LC) system– without having to make use of the more expressive and complex machinery of first-order (...) quantificational logic. In addition, we prove the classic categorical syllogisms again by a relational method –which we call the McColl-Ladd (ML) system– employing categorical relations studied by Hugh McColl and Christine Ladd. Finally, we show the connection of ML and LC with Boolean algebra, proving that ML is a consequence of LC, and that LC is a consequence of the Boolean lattice axioms, thus establishing Leibniz’s historical priority over George Boole in characterizing and applying (a sufficient fragment of) Boolean algebra to effectively tackle categorical syllogistic. (shrink)

In this paper I examine the relationship between the principle of right and the principle of morals [Sitten] in Kant’s Metaphysics of Morals. My interpretation denies that the principle of right is derived from the categorical imperative, but neither does it adhere to the independence thesis. I present a third way of understanding the relationship between the law of right and the universal law of morals: the latter is needed in order to formulate the former, but it is not sufficient. (...) The principle of right is obtained by applying the categorical imperative to the concept of right. I highlight the discussion on the subject within the late eighteenth-century German natural law tradition and similar arguments found in Achenwall. (shrink)

We present a colorful and novel discussion of mathematical techniques of visualizing a fourth spatial dimension. We first discuss notions of dimensionality including the homotopy dimension for objects in an (infinity,1)-topos. We try to visualize the fourth spatial dimension using color, and illustrate this with four-dimensional ice-cream. We apply categorical negative thinking to what we have called (infinity,1)-visual epistemology. The aim is that visualizations of higher spatial dimensions can occur functorially. We illustrate with five images five conjectural methods for how (...) to visualize the fourth spatial dimension. (shrink)

Both realist and anti-realist accounts of natural kinds possess prima facie virtues: realists can straightforwardly make sense of the apparent objectivity of the natural kinds, and anti-realists, their knowability. This paper formulates a properly anti-realist account designed to capture both merits. In particular, it recommends understanding natural kinds as ‘categorical bottlenecks,’ those categories that not only best serve us, with our idiosyncratic aims and cognitive capacities, but also those of a wide range of alternative agents. By endorsing an ultimately subjective (...) categorical principle, this view sidesteps epistemological difficulties facing realist views. Yet, it nevertheless identifies natural kinds that are fairly, though not completely, stance-independent or objective. (shrink)

In this paper, I consider an argument of Harvey Siegel's according to which there can be no hypothetical normativity anywhere unless there is categorical normativity in epistemology. The argument fails because it falsely assumes people must be bound by epistemic norms in order to have justified beliefs.

What are physical objects like when they are considered independently of their causal interactions? Many think that the answer to this question involves categorical properties– properties that make contributions to their bearers that are independent of any causal interactions those objects may enter into. In this paper, I examine two challenges that this solution poses to Physicalism. The first challenge is that, given that they are distinct from any of the scientifically described causal powers that they happen to convey, categorical (...) properties will not qualify as being ‘physical’ properties. Given the right definition of ‘physical’, this challenge can be overcome. I argue, however, that the only way we can have a positive grasp of the nature of categorical properties is via ‘acquaintance’– a non-physical relation. This second challenge to Physicalism cannot be overcome.1. (shrink)

We present colorful illustrations of particular properties of functorial diamonds, in the sense of Scholze; namely profinite reflections as categorical colors.We discuss sight as site using representable functors in the condensed formalism. We illuminate diamonds using our novel constructions of categorical Ozma and Cinderella, the site of Oz, and condensed Through the Looking-Glass.

In this chapter, I will present an empirical conjecture to the effect that some bodily actions are categorically perceived. These are bodily actions such as grasping or reaching for something, which I am going to call motor actions. My conjecture builds on one recently put forward about how the categorical perception of facial expressions of some emotions works. I shall motivate my own conjecture on the basis of both theoretical and empirical considerations, describe how it could be operationalised and what (...) explanatory gain could be obtained from it. (shrink)

The aim of this paper is to make sense of the Keynes–Johnson octagon of oppositions. We will discuss Keynes' logical theory, and examine how his view is reflected on this octagon. Then we will show how this structure is to be handled by means of a semantics of partition, thus computing logical relations between matching formulas with a semantic method that combines model theory and Boolean algebra.

Naturalism, as Binmore understands the term, is characterized by a scientific stance on moral behavior. Binmore claims that a naturalistic account of morality necessarily goes with the conviction “that only hypothetical imperatives make any sense”. In this paper it is argued that this claim is mistaken. First, as Hume’s theory of promising shows, naturalism in the sense of Binmore is very well compatible with acknowledging the importance of categorical imperatives in moral practice. Moreover, second, if Binmore’s own theory of moral (...) practice and its evolution is correct, then the actual moral practice does-and in fact must-incorporate norms, which have the form of a categorical imperative. Categorical imperatives are part of social reality and, therefore, any (normative) moral theory that adequately reflects moral practice must also include categorical imperatives. (shrink)

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

In this chapter, I critically examine two of the main approaches to colour categorization in cognitive science: the perceptual salience theory and linguistic relativism. I then turn to reviewing several decades of psychological research on colour categorical perception (CP). A careful assessment of relevant findings suggests that most of the experimental effects that have been understood in terms of CP actually fall on the cognition side of the perception-cognition divide: they are effects of colour language, for example, on memory or (...) decision-making. (shrink)

The type-level abstraction is a formal way to represent molecular structures in biological practice. Graphical representations of molecular structures of biological objects are also used to identify functional processes of things. This paper will reveal that category theory is a formal mathematical language not only to visualize molecular structures of biological objects as type-level abstraction formally but also to understand how to infer biological functions from the molecular structures of biological objects. Category theory is a toolkit to understand biological knowledge (...) at the type-level formally, not individual token-level, as well as typical heuristic strategies in molecular biology. (shrink)

This paper has two main aims. The first is to present a general approach for understanding “dispositional” and “categorical” properties; the second aim is to use this approach to criticize Russellian Monism. On the approach I suggest, what are usually thought of as “dispositional” and “categorical” properties are really just the extreme ends of a spectrum of options. The approach allows for a number of options between these extremes, and it is plausible, I suggest, that just about everything of scientific (...) interest falls in this middle ground. I argue that Russellian Monism depends for its plausibility on the unarticulated assumption that there are no properties in the middle ground. (shrink)

Proponents of cognitive penetration often argue for the thesis on the basis of combined intuitions about categorical perception and perceptual learning. The claim is that beliefs penetrate perceptions in the course of learning to perceive categories. I argue that this "diachronic" penetration thesis is false. In order to substantiate a robust notion of penetration, the beliefs that enable learning must describe the particular ability that subjects learn. However, they cannot do so, since in order to help with learning they must (...) instruct learners to employ previously existing abilities. I argue that a better approach recognizes that we can have sophisticated causal precursors to perceptual learning, but that the learning process itself must operate outside of cognitive influence. (shrink)

In this paper, I argue that current attempts at classifying life–mind continuity (LMC) feature several important ambiguities. We can resolve these ambiguities by distinguishing between the extensional, categorical, and systematic relationships that LMC might encompass. In Sect. 1, I begin by introducing the notion of LMC and the theory behind it. In Sect. 2, I show how different ideas of mind shape different approaches to continuity and how to achieve its aim. In Sect. 3, I canvas various canonical formulations and (...) classifications of LMC; I then demonstrate that they retain important ambiguities. Section 4 builds on this by arguing that we must conceive of the extensional and categorical aspects of continuity independently. In Sect. 5, I show further that current literature has underexplored multiple systematic aspects of continuity. I then take a constructive approach in Sect. 6 by providing a classification model for LMC based on extensional and categorical commitments. Here, I comment on aspects of the thesis omitted from the model but essential for a full classification and thorough comparison between various approaches to LMC. All of these arguments lay the foundation for more exhaustively classifying accounts of LMC. (shrink)

One of the traditional desiderata for a metaphysical theory of laws of nature is that it be able to explain natural regularities. Some philosophers have postulated governing laws to fill this explanatory role. Recently, however, many have attempted to explain natural regularities without appealing to governing laws. Suppose that some fundamental properties are bare dispositions. In virtue of their dispositional nature, these properties must be (or are likely to be) distributed in regular patterns. Thus it would appear that an ontology (...) including bare dispositions can dispense with governing laws of nature. I believe that there is a problem with this line of reasoning. In this essay, I’ll argue that governing laws are indispensable for the explanation of a special sort of natural regularity: those holding among categorical properties (or, as I’ll call them, categorical regularities). This has the potential to be a serious objection to the denial of governing laws, since there may be good reasons to believe that observed regularities are categorical regularities. (shrink)

Dispostions, such as solubility, cannot be reduced to categorical properties, such as molecular structure, without some element of dipositionaity remaining. Democritus did not reduce all properties to the geometry of atoms - he had to retain the rigidity of the atoms, that is, their disposition not to change shape when a force is applied. So dispositions-not-to, like rigidity, cannot be eliminated. Neither can dispositions-to, like solubility.

This paper demonstrates something that Kant notoriously claimed to be possible, but which Kant scholars today widely believe to be impossible: unification of all three formulations of the Categorical Imperative. Part 1 of this paper tells a broad-brush story of how I understand Kant’s theory of practical reason and morality, showing how the three formulations of the Categorical Imperative appear to be unified. Part 2 then provides clear textual support for each premise in the argument for my interpretation.

Tae Wan Kim, John Hooker, and Thomas Donaldson make an attempt, in recent articles, to solve the alignment problem. As they define the alignment problem, it is the issue of how to give AI systems moral intelligence. They contend that one might program machines with a version of Kantian ethics cast in deontic modal logic. On their view, machines can be aligned with human values if such machines obey principles of universalization and autonomy, as well as a deontic utilitarian principle. (...) Programming machines to do so might be useful, in their view, for applications such as future autonomous vehicles. Their proposal draws both on traditional logic-based and contemporary connectionist approaches, to fuse factual information with normative principles. I will argue that this approach makes demands of machines that go beyond what is currently feasible, and may extend past the limits of the possible for AI. I also argue that a deontological ethics for machines should place greater stress on the formula of humanity of the Kantian categorical imperative. On this principle, one ought never treat a person as a mere means. Recognition of what makes a person a person requires ethical insight. Similar insight is needed to tell treatment as a means from treatment as a mere means. The resources in Kim, Hooker, and Donaldson’s approach is insufficient for this reason. Hesitation regarding deployment of autonomous machines is warranted in light of these alignment concerns. (shrink)

Inductively defined structures are ubiquitous in mathematics; their specification is unambiguous and their properties are powerful. All fields of mathematical logic feature these structures prominently: the formula of a language, the set of theorems, the natural numbers, the primitive recursive functions, the constructive number classes and segments of the cumulative hierarchy of sets. -/- This dissertation gives a mathematical characterization of a species of inductively defined structures, called accessible domains, which include all of the above examples except the set of (...) theorems. The concept of an accessible domain comes from Wilfried Sieg's analysis of proof-theoretic practices, starting with his dissertation. In particular, he noticed the special epistemological character of elements of an accessible domain: they can always be uniquely identified with their build-up. Generally, the unique build-up of elements justifies the principles of induction and recursion. -/- I use category theory to give an abstract characterization of accessible domains. I claim that accessible domains are all instances of initial algebras for endofunctors. Grounded in the historical roots of Sieg's discussions, this dissertation shows how the properties of initial algebras for endofunctors and accessible domains coincide in a satisfying and natural way. -/- Filling out this characterization, I show how important examples of accessible domains fit into this broad characterization. I first characterize some accessible domains by relatively simple functors (e.g. finite, polynomial, those that preserve certain colimits) where we can see how iterating the functor can produce the accessible domain. Then I describe accessible domains that result from more involved specifications (e.g. ordinals and segments of the cumulative hierarchies associated with CZF, IZF, and ZF) by relying heavily on algebraic set theory. I end with a discussion of some of the methodological features of category theory in particular that helped characterize accessible domains. (shrink)

I show that the model-theoretic meaning that can be read off the natural deduction rules for disjunction fails to have certain desirable properties. I use this result to argue against a modest form of inferentialism which uses natural deduction rules to fix model-theoretic truth-conditions for logical connectives.

Kant famously claims that there is only a single supreme principle of morality: the Categorical Imperative. This claim is often treated with skepticism. After all, Kant proceeds to provide no fewer than six formulations of this purportedly single supreme principle—formulations which appear to differ significantly. But appearances can be deceptive. In this paper, I argue that Kant was right. There is only a single Categorical Imperative, and each of its formulations expresses the very same moral principle.

This essay explores the notion of moral exemplarity, positing that our morality is underpinned by moral exemplars – paradigmatic examples of virtuous individuals or actions. Theoretical precepts of moral exemplarity are explored across historical and contemporary contexts, including the philosophies of Plato, Aristotle, Stoic and Christian ethics, and recent works of Alexandro Ferrara and Linda Zagzebski. This essay debates the necessity of moral exemplars, the intrinsic moral and epistemic exemplarity, and the distinction between categorical and hypothetical exemplarity, as well as (...) referencing the paradoxical Kantian dismissals of moral exemplarity. It critiques current accounts of moral exemplarity and proposes a transcendental explanation, culminating in an examination of the “exemplarist categorical imperative.”. (shrink)

I develop a new theory of properties by considering two central arguments in the debate whether properties are dispositional or categorical. The first claims that objects must possess categorical properties in order to be distinct from empty space. The second argument, however, points out several untoward consequences of positing categorical properties. I explore these arguments and argue that despite appearances, their conclusions need not be in conflict with one another. In particular, we can view the second argument as supporting only (...) the claim that there is not a plurality of categorical properties, and not the stronger claim that there are no categorical properties whatsoever. I then develop a new account of properties which capitalizes on this insight. (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)

This paper explores the charge by Bruce Aune and Allen Wood that a gap exists in Kant's derivation of the Categorical Imperative. I show that properly understood, no such gap exists, and that the deduction of the Categorical Imperative is successful as it stands.

The conditional interpretation of general categorical statements like ‘All men are animals’ as universally quantified material conditionals ‘For all x, if x is F, then x is G’ suggests that the logical structure of law statements is conditional rather than categorical. Disregarding the problem that the universally quantified material conditional is trivially true whenever there are no xs that are F, there are some reasons to be sceptical of Frege’s equivalence between categorical and conditional expressions. -/- Now many philosophers will (...) claim that the material conditional interpretation of laws statements, dispositions ascriptions, or any causal claim is generally accepted as wrong and outdated. Still, there seem to be some basic logical assumptions that are shared by most of the participants in the debate on causal matters which at least stems from the traditional truth functional interpretation of conditionals. This is indicated by the vocabulary in the philosophical debate on causation, where one often speaks of ‘counterfactuals’, ‘possible worlds’ and ‘necessity’ without being explicit on whether or to what extent one accepts the logical-technical definition of these notions. To guarantee a non-Humean and non-extensional approach to causal relations, it is therefore important to be aware of the logical and metaphysical implications of the technical vocabulary. -/- In this paper we want to show why extensional logic cannot deal with causal relations. Via a logical analysis of law-like statements ‘All Fs are Gs’ we hope to throw some new light on interrelated notions like causation, laws, induction, hypotheticality and modality. If successful, our analysis should be of relevance for a deeper understanding of any type of causal relations, whether we understand them to be laws, dispositions, singulars or categoricals. (shrink)

In Anarchy, State, and Utopia, Robert Nozick made the argument that income redistribution violates the Kantian categorical imperative. Nozick’s retrospective enslavement argument is still used today in discussions about the moral justification of taxation. This article explicates four implicit premises of Nozick’s argument: the self-ownership principle, its fullness, the absence of restrictions on the appropriation of natural resources, and the absence of restrictions on the distribution of the fruits of cooperation. Without additional justification for each of these premises, Nozick’s argument (...) cannot show that income redistribution violates the categorical imperative. (shrink)