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)

The third author has shown that Shelah's eventual categoricity conjecture holds in universal classes: class of structures closed under isomorphisms, substructures, and unions of chains. We extend this result to the framework of multiuniversal classes. Roughly speaking, these are classes with a closure operator that is essentially algebraic closure (instead of, in the universal case, being essentially definable closure). Along the way, we prove in particular that Galois (orbital) types in multiuniversal classes are determined by their finite restrictions, generalizing a (...) result of the second author. (shrink)

Hellman [2003] raises interesting challenges to categorical structuralism. He starts citing Awodey [1996] which, as Hellman sees, is not intended as a foundation for mathematics. It offers a structuralist framework which could denned in any of many different foundations. But Hellman says Awodey's work is 'naturally viewed in the context of Mac Lane's repeated claim that category theory provides an autonomous foundation for mathematics as an alternative to set theory' (p. 129). Most of Hellman's paper 'scrutinizes the formulation of (...) category theory' specifically in 'its alleged role as providing a foundation' (p. 130). I will also focus on the foundational question. Page numbers in parentheses are page references to Hellman [2003]. (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)

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

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.

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)

Let be an abstract elementary class with amalgamation, and Lowenheim Skolem number LS. We prove that for a suitable Hanf number gc0 if χ0 < λ0 λ1, and is categorical inλ1+ then it is categorical in λ0.

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 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 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)

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

The 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)

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)

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)

We give an axiomatic framework for the non-modular simple 0-categorical structures constructed by Hrushovski. This allows us to verify some of their properties in a uniform way, and to show that these properties are preserved by iterations of the construction.

The hypothetical notion of consequence is normally understood as the transmission of a categorical notion from premisses to conclusion. In model-theoretic semantics this categorical notion is 'truth', in standard proof-theoretic semantics it is 'canonical provability'. Three underlying dogmas, (I) the priority of the categorical over the hypothetical, (II) the transmission view of consequence, and (III) the identification of consequence and correctness of inference are criticized from an alternative view of proof-theoretic semantics. It is argued that consequence is (...) a basic semantical concept which is directly governed by elementary reasoning principles such as definitional closure and definitional reflection, and not reduced to a categorical concept. This understanding of consequence allows in particular to deal with non-wellfounded phenomena as they arise from circular definitions. (shrink)

This paper largely engages with Brian Ellis’s description of categorical dimensions as put forward in his paper in this volume. The New Essentialism advocated by Ellis posits the ontologically-robust existence of both dispositional and categorical properties. I have argued that the distinction that Ellis draws between the two is unpersuasive, and that the causal role of categorical dimensions—what they do—is inseparable from what they are. This observation is reinforced by the fact that absolute physical quantities permit re-interpretations (...) of measurement that remove a clear differentiation between categoricity and dispositionality. Distinguishing between ‘pure power’ and ‘dispositionality’, I further argue that: i) there are no ontologically-robust categorical properties, although their apparent existence is explicable as higher-order and supervenient; ii) that the fundamental ingredients of the world may be accounted for in terms of pure-power that is neither categorical nor dispositional; and iii) that the categorical-dispositional distinction arises only at the higher-order level of objects, and does not in any case constitute ontologically-robust partitioning of reality. -/- . (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.

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)

This scarce antiquarian book is a facsimile reprint of the original. Due to its age, it may contain imperfections such as marks, notations, marginalia and flawed pages. Because we believe this work is culturally important, we have made it available as part of our commitment for protecting, preserving, and promoting the world's literature in affordable, high quality, modern editions that are true to the original work.

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

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.

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

In this paper, we give a classification of ℵ0-categorical coloured linear orders, generalizing Rosenstein's characterization of ℵ0-categorical linear orderings. We show that they can all be built from coloured singletons by concatenation and ℚn-combinations . We give a method using coding trees to describe all structures in our list.

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)

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

We characterize Δ20-categoricity in Boolean algebras and linear orderings under some extra effectiveness conditions. We begin with a study of the relativized notion in these structures.

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 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)

In his article 'Rejection' (1996), Timothy Smiley had shown how a logical system allowing rules of rejection could provide a categorical axiomatization of the classical propositional calculus. This paper shows how rules of rejection, when placed in a multiple conclusion setting, can also provide categorical axiomatizations of a range of non-classical calculi which permit truth-value gaps, among them the calculus in Smiley's own 'Sense without denotation' (1960).

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)

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

Category theory has become central to certain aspects of theoretical physics. Bain has recently argued that this has significance for ontic structural realism. We argue against this claim. In so doing, we uncover two pervasive forms of category-theoretic generalization. We call these ‘generalization by duality’ and ‘generalization by categorifying physical processes’. We describe in detail how these arise, and explain their significance using detailed examples. We show that their significance is two-fold: the articulation of high-level physical concepts, and the generation (...) of new models. 1 Introduction2 Categories and Structuralism 2.1 Categories: abstract and concrete 2.2 Structuralism: simple and ontic3 Bain’s Two Strategies 3.1 A first strategy for defending Objectless 3.2 A second strategy for defending Objectless4 Two Forms of Categorical Generalization 4.1 Generalization by duality 4.2 Generalization by categorification 4.3 The role of category theory in physics5 Conclusion. (shrink)

This article examines the traditional and modern doctrines of categorical propositions and argues that both doctrines have serious problems. While the doctrines disagree about existential imports...

We prove that every ω-categorical, generically stable group is nilpotent-by-finite and that every ω-categorical, generically stable ring is nilpotent-by-finite.

Protoalgebraic logics are characterized by the monotonicity of the Leibniz operator on their theory lattices and are at the lower end of the Leibniz hierarchy of abstract algebraic logic. They have been shown to be the most primitive among those logics with a strong enough algebraic character to be amenable to algebraic study techniques. Protoalgebraic π-institutions were introduced recently as an analog of protoalgebraic sentential logics with the goal of extending the Leibniz hierarchy from the sentential framework to the π-institution (...) framework. Many properties of protoalgebraic logics, studied in the sentential logic framework by Blok and Pigozzi, Czelakowski, and Font and Jansana, among others, have already been adapted in previous work by the author to the categorical level. This work aims at further advancing that study by exploring in this new level some more properties of protoalgebraic sentential logics. (shrink)

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

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

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.

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.