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)

This paper discusses the relation between the natural deduction rules of deduction in sequent format and the provability valuation starting from Garson’s Local Expression Theorem, which is meant to establish that the natural deduction rules of inference enforce exactly the classical meanings of the propositional connectives if these rules are taken to be locally valid, i.e. if they are taken to preserve sequent satisfaction. I argue that the natural deduction rules for disjunction are in no better position than the (...) axiomatic calculi in uniquely determining the intended meaning of disjunction when the local models are used, if a satisfied sequent embeds, as a logical inferentialist should require, a formal derivability relation. This happens because, when governed by these rules and without additional semantic assumptions, the disjunction sign still expresses a non-extensional connective, i.e. a connective that, properly understood, has no unique logical characteristic. However, this is not a dead end for the logical inferentialists since both a multiple conclusions formalization of the disjunction operator and a bilateralist one do succeed in restoring the standard meaning of disjunction. (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.

Vann McGee has recently argued that Belnap’s criteria constrain the formal rules of classical natural deduction to uniquely determine the semantic values of the propositional logical connectives and quantifiers if the rules are taken to be open-ended, i.e., if they are truth-preserving within any mathematically possible extension of the original language. The main assumption of his argument is that for any class of models there is a mathematically possible language in which there is a sentence true in just those (...) models. I show that this assumption does not hold for the class of models of classical propositional logic. In particular, I show that the existence of non-normal models for negation undermines McGee’s argument. (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)

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)

The notion of an ℐ -matrix as a model of a given π -institution ℐ is introduced. The main difference from the approach followed so far in CategoricalAlgebraic Logic and the one adopted here is that an ℐ -matrix is considered modulo the entire class of morphisms from the underlying N -algebraic system of ℐ into its own underlying algebraic system, rather than modulo a single fixed -logical morphism. The motivation for introducing ℐ -matrices comes from a desire to formulate (...) a correspondence property for N -protoalgebraic π -institutions closer in spirit to the one for sentential logics than that considered in CAAL before. As a result, in the previously established hierarchy of syntactically protoalgebraic π -institutions, i. e., those with an implication system, and of protoalgebraic π -institutions, i. e., those with a monotone Leibniz operator, the present paper interjects the class of those π -institutions with the correspondence property, as applied to ℐ -matrices. Moreover, this work on ℐ -matrices enables us to prove many results pertaining to the local deduction-detachment theorems, paralleling classical results in Abstract Algebraic Logic formulated, first, by Czelakowski and Blok and Pigozzi. Those results will appear in a sequel to this paper. (shrink)

Second-order axiomatizations of certain important mathematical theories—such as arithmetic and real analysis—can be shown to be categorical. Categoricity implies semantic completeness, and semantic completeness in turn implies determinacy of truth-value. Second-order axiomatizations are thus appealing to realists as they sometimes seem to offer support for the realist thesis that mathematical statements have determinate truth-values. The status of second-order logic is a controversial issue, however. Worries about ontological commitment have been influential in the debate. Recently, Vann McGee has argued that one (...) can get some of the technical advantages of second-order axiomatizations—categoricity, in particular—while walking free of worries about ontological commitment. In so arguing he appeals to the notion of an open-ended schema—a schema that holds no matter how the language of the relevant theory is extended. Contra McGee, we argue that second-order quantification and open-ended schemas are on a par when it comes to ontological commitment. (shrink)

This book concerns the metasemantics of quantum mechanics (QM). Roughly, it pursues an investigation at an intersection of the philosophy of physics and the philosophy of language, and it offers a critical analysis of rival explanations of the semantic facts of standard QM. Two problems for such explanations are discussed: categoricity and permanence. New results include 1) a reconstruction of Einstein's incompleteness argument, which concludes that a local, separable, and categorical QM cannot exist, 2) a reinterpretation of Bohr's principle (...) of correspondence, grounded in the principle of permanence, 3) a meaning-variance argument for quantum logic, which follows a line of critical reflections initiated by Weyl, and 4) an argument for semantic indeterminacy leveled against inferentialism about QM, inspired by Carnap's work in the philosophy of classical logic. (shrink)

It is sometimes held that rules of inference determine the meaning of the logical constants: the meaning of, say, conjunction is fully determined by either its introduction or its elimination rules, or both; similarly for the other connectives. In a recent paper, Panu Raatikainen (2008) argues that this view - call it logical inferentialism - is undermined by some "very little known" considerations by Carnap (1943) to the effect that "in a definite sense, it is not true that the standard (...) rules of inference" themselves suffice to "determine the meanings of [the] logical constants" (p. 2). In a nutshell, Carnap showed that the rules allow for non-normal interpretations of negation and disjunction. Raatikainen concludes that "no ordinary formalization of logic ... is sufficient to `fully formalize' all the essential properties of the logical constants" (ibid.). We suggest that this is a mistake. Pace Raatikainen, intuitionists like Dummett and Prawitz need not worry about Carnap's problem. And although bilateral solutions for classical inferentialists - as proposed by Timothy Smiley and Ian Rumfitt - seem inadequate, it is not excluded that classical inferentialists may be in a position to address the problem too. (shrink)

It is sometimes held that rules of inference determine the meaning of the logical constants: the meaning of, say, conjunction is fully determined by either its introduction or its elimination rules, or both; similarly for the other connectives. In a recent paper, Panu Raatikainen argues that this view—call it logical inferentialism—is undermined by some “very little known” considerations by Carnap (1943) to the effect that “in a definite sense, it is not true that the standard rules of inference” themselves suffice (...) to “determine the meanings of [the] logical constants” (p. 2). In a nutshell, Carnap showed that the rules allow for non-normal interpretations of negation and disjunction. Raatikainen concludes that “no ordinary formalization of logic [... ] is sufficient to ‘fully formalize’ all the essential properties of the logical constants” (ibid.). We suggest that this is a mistake. Pace Raatikainen, intuitionists like Dummett and Prawitz need not worry about Carnap’s problem. And although bilateral solutions for classical inferentialists—as proposed by Timothy Smiley and Ian Rumfitt—seem inadequate, it is not excluded that classical inferentialists may be in a position to address the problem too. (shrink)

Carnap (1943) noted that categoricity fails even for propositional logic. He observed that some “non-normal” models exist for classical logical consequence. The immediate reaction to this phenomenon was to strengthen the formal machinery or impose semantical constraints to rule out nonnormal models, and as a result, their very structure was largely overlooked. In this paper, I examine these non-normal models and explore the possibility that they assign non-classical meanings to logical connectives. By relaxing some standards regarding semantics, we will (...) see that these models, in some sense, embed a non-deterministic matrix (Nmatrix) with interesting similarity to the strong Kleene’s 3-valued matrix. Then, I briefly explore its induced logics: two non-compositional counterparts of the well-known logics LP and K3. (shrink)

In this study two strands of inferentialism are brought together: the philosophical doctrine of Brandom, according to which meanings are generally inferential roles, and the logical doctrine prioritizing proof-theory over model theory and approaching meaning in logical, especially proof-theoretical terms.

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

Inferentialism is a philosophical approach premised on the claim that an item of language acquires meaning in virtue of being embedded in an intricate set of social practices normatively governed by inferential rules. Inferentialism found its paradigmatic formulation in Robert Brandom's landmark book Making it Explicit, and over the last two decades it has established itself as one of the leading research programs in the philosophy of language and the philosophy of logic. While Brandom's version of inferentialism has received wide (...) attention in the philosophical literature, thinkers friendly to inferentialism have proposed and developed new lines of inquiry that merit wider recognition and critical appraisal. From Rules to Meaning brings together new essays that systematically develop, compare, assess and critically react to some of the most pertinent recent trends in inferentialism. The book's four thematic sections seek to apply inferentialism to a number of core issues, including the nature of meaning and content, reconstructing semantics, rule-oriented models and explanations of social practices and inferentialism's historical influence and dialogue with other philosophical traditions. With contributions from a number of distinguished philosophers--including Robert Brandom and Jaroslav Peregrin--this volume is a major contribution to the philosophical literature on the foundations of logic and language. (shrink)

Studies of categorical induction typically examine how belief in a premise (e.g., Falcons have an ulnar artery) projects on to a conclusion (e.g., Robins have an ulnar artery). We study induction in cases in which the premise is uncertain (e.g., There is an 80% chance that falcons have an ulnar artery). Jeffrey's rule is a normative model for updating beliefs in the face of uncertain evidence. In three studies we tested the descriptive validity of Jeffrey's rule and a (...) related probability theorem, the rule of total probability. Although these rules provided good approximations to mean judgments in some cases, the results from regression and correlation analyses suggest that participants focus on the parts of these rules that are associated with the highest overall probability. We relate our findings to rational models of judgment. (shrink)

In bilateral systems for classical logic, assertion and denial occur as primitive signs on formulas. Such systems lend themselves to an inferentialist story about how truth-conditional content of connectives can be determined by inference rules. In particular, for classical logic there is a bilateral proof system which has a property that Carnap in 1943 called categoricity. We show that categorical systems can be given for any finite many-valued logic using $n$-sided sequent calculus. These systems are understood as a further development (...) of bilateralism—call it multilateralism. The overarching idea is that multilateral proof systems can incorporate the logic of a variety of denial speech acts. So against Frege we say that denial is not the negation of assertion and, with Mark Twain, that denial is more than a river in Egypt. (shrink)

Between the time of the Crusades and the end of the Latin political presence in the East, Cyprus, Crete, and Rhodes benefited from an important cultural cross-fertilization. As part of the maritime routes linking Eastern and Western ports, and home to cosmopolitan populations, those islands of the Byzantine Empire respectively came under the control of the Franks, Venice and the Hospitallers in 1191, 1204 and 1306. During the rule of the Franks in Cyprus, of the Venetians in Crete and (...) of the Hospitallers in Rhodes, the Greeks still made up the largest part of the local population. Preserving their identity, they never ceased to consider themselves members of the ecumene of the Eastern Roman Empire. Nevertheless, their attachment to Byzantine culture and their own local traditions did not prevent them from opening to outside models. (shrink)

Aim of the article is to present our research dealing with virtual reality modeling and education activities. It involves a chain of development steps: taking pictures of objects, collecting information, creation of 3D models and panoramas, setting up the interactive virtual reality environment along with educational support and testing the user experience with students. Methodology/methods of work: the creation of 3D models and panoramas is done by taking pictures of real objects and using them as textures. The education materials are (...) created as interactive presentations and the usability testing of these presentations should be planned. Scientific goal is to measure the quality of interaction by this education support in order to derive rules testing effectiveness of support, interactive applications and presentations. Findings: 3D models and other educational support materials may be created voluntarily by teachers. The open source library for many different models should be created by whole community provided suitable interfaces of modeling tools will be arranged. Conclusions: Complex educational and interactive materials including virtual reality and panoramas are satisfactory when combined with easy 3D models that are made from taken pictures, which could be made by teachers themselves and/or open communities by existing cameras. (shrink)

Inferentialism is a theory in the philosophy of language which claims that the meanings of expressions are constituted by inferential roles or relations. Instead of a traditional model-theoretic semantics, it naturally lends itself to a proof-theoretic semantics, where meaning is understood in terms of inference rules with a proof system. Most work in proof-theoretic semantics has focused on logical constants, with comparatively little work on the semantics of non-logical vocabulary. Drawing on Robert Brandom’s notion of material inference and Greg Restall’s (...) bilateralist interpretation of the multiple conclusion sequent calculus, I present a proof-theoretic semantics for atomic sentences and their constituent names and predicates. The resulting system has several interesting features: (1) the rules are harmonious and stable; (2) the rules create a structure analogous to familiar model-theoretic semantics; and (3) the semantics is compositional, in that the rules for atomic sentences are determined by those for their constituent names and predicates. (shrink)

A model of reality is called separable if the state of a composite system is equal to the union of the states of its parts, located in different regions of space. Spekkens has argued that it is trivial to reproduce the predictions of quantum mechanics using a separable ontological model, provided one allows for arbitrary violations of ‘dynamical locality’. However, since dynamical locality is strictly weaker than local causality, this leaves open the question of whether an ontological model (...) for quantum mechanics can be both separable and dynamically local. We answer this question in the affirmative, using an ontological model based on previous work by Deutsch and Hayden. Although the original formulation of the model avoids Bell’s theorem by denying that measurements result in single, definite outcomes, we show that the model can alternatively be cast in the framework of ontological models, where Bell’s theorem does apply. We find that the resulting model violates local causality, but satisfies both separability and dynamical locality, making it a candidate for the ‘most local’ ontological model of quantum mechanics. (shrink)

"Open Democracy envisions what true government by mass leadership could look like."—Nathan Heller, New Yorker How a new model of democracy that opens up power to ordinary citizens could strengthen inclusiveness, responsiveness, and accountability in modern societies To the ancient Greeks, democracy meant gathering in public and debating laws set by a randomly selected assembly of several hundred citizens. To the Icelandic Vikings, democracy meant meeting every summer in a field to discuss issues until consensus was reached. Our contemporary (...) representative democracies are very different. Modern parliaments are gated and guarded, and it seems as if only certain people—with the right suit, accent, wealth, and connections—are welcome. Diagnosing what is wrong with representative government and aiming to recover some of the lost openness of ancient democracies, Open Democracy presents a new paradigm of democracy in which power is genuinely accessible to ordinary citizens. Hélène Landemore favors the ideal of “representing and being represented in turn” over direct-democracy approaches. Supporting a fresh nonelectoral understanding of democratic representation, Landemore recommends centering political institutions around the “open mini-public”—a large, jury-like body of randomly selected citizens gathered to define laws and policies for the polity, in connection with the larger public. She also defends five institutional principles as the foundations of an open democracy: participatory rights, deliberation, the majoritarian principle, democratic representation, and transparency. Open Democracy demonstrates that placing ordinary citizens, rather than elites, at the heart of democratic power is not only the true meaning of a government of, by, and for the people, but also feasible and, today more than ever, urgently needed. (shrink)

Maximum likelihood (ML) estimation of categorical multitrait-multimethod (MTMM) data is challenging because the likelihood involves high-dimensional integrals over the crossed method and trait factors, with no known closed-form solution. The purpose of the study is to introduce three newly developed ML methods that are eligible for estimating MTMM models with categorical responses: Variational maximization-maximization (e.g., Rijmen and Jeon, 2013 ), alternating imputation posterior (e.g., Cho and Rabe-Hesketh, 2011 ), and Monte Carlo local likelihood (e.g., Jeon et al., under revision). (...) Each method is briefly described and its applicability for MTMM models with categorical data are discussed. (shrink)

A categorical ontology of space and time is presented for emergent biosystems, super-complex dynamics, evolution and human consciousness. Relational structures of organisms and the human mind are naturally represented in non-abelian categories and higher dimensional algebra. The ascent of man and other organisms through adaptation, evolution and social co-evolution is viewed in categorical terms as variable biogroupoid representations of evolving species. The unifying theme of local-to-global approaches to organismic development, evolution and human consciousness leads to novel patterns of relations (...) that emerge in super- and ultra- complex systems in terms of colimits of biogroupoids, and more generally, as compositions of local procedures to be defined in terms of locally Lie groupoids. Solutions to such local-to-global problems in highly complex systems with ‘broken symmetry’ may be found with the help of generalized van Kampen theorems in algebraic topology such as the Higher Homotopy van Kampen theorem (HHvKT). Primordial organism structures are predicted from the simplest metabolic-repair systems extended to self-replication through autocatalytic reactions. The intrinsic dynamic ‘asymmetry’ of genetic networks in organismic development and evolution is investigated in terms of categories of many-valued, Łukasiewicz–Moisil logic algebras and then compared with those obtained for (non-commutative) quantum logics. The claim is defended in this essay that human consciousness is unique and should be viewed as an ultra-complex, global process of processes. The emergence of consciousness and its existence seem dependent upon an extremely complex structural and functional unit with an asymmetric network topology and connectivities—the human brain—that developed through societal co-evolution, elaborate language/symbolic communication and ‘virtual’, higher dimensional, non-commutative processes involving separate space and time perceptions. Philosophical theories of the mind are approached from the theory of levels and ultra-complexity viewpoints which throw new light on previous representational hypotheses and proposed semantic models in cognitive science. Anticipatory systems and complex causality at the top levels of reality are also discussed in the context of the ontological theory of levels with its complex/entangled/intertwined ramifications in psychology, sociology and ecology. The presence of strange attractors in modern society dynamics gives rise to very serious concerns for the future of mankind and the continued persistence of a multi-stable biosphere. A paradigm shift towards non-commutative, or non-Abelian, theories of highly complex dynamics is suggested to unfold now in physics, mathematics, life and cognitive sciences, thus leading to the realizations of higher dimensional algebras in neurosciences and psychology, as well as in human genomics, bioinformatics and interactomics. (shrink)

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

There are many results in the literature where superstablity-like independence notions, without any categoricity assumptions, have been used to show the existence of larger models. In this paper we show that stability is enough to construct larger models for small cardinals assuming a mild locality condition for Galois types. Theorem 0.1. Suppose $\lambda <2^{\aleph _0}$. Let ${\mathbf {K}}$ be an abstract elementary class with $\lambda \geq {\operatorname {LS}}({\mathbf {K}})$. Assume ${\mathbf {K}}$ has amalgamation in $\lambda $, no maximal model in (...) $\lambda $, and is stable in $\lambda $. If ${\mathbf {K}}$ is $(<\lambda ^+, \lambda )$ -local, then ${\mathbf {K}}$ has a model of cardinality $\lambda ^{++}$. The set theoretic assumption that $\lambda <2^{\aleph _0}$ and model theoretic assumption of stability in $\lambda $ can be weakened to the model theoretic assumptions that $|{\mathbf {S}}^{na}(M)|< 2^{\aleph _0}$ for every $M \in {\mathbf {K}}_\lambda $ and stability for $\lambda $ -algebraic types in $\lambda $. This is a significant improvement of Theorem 0.1, as the result holds on some unstable abstract elementary classes. (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)

An open question regarding how people develop their models of the world is how new candidates are generated for consideration out of infinitely many possibilities. We discuss the role that evolutionary mechanisms play in this process. Specifically, we argue that when it comes to developing a global world model, innovation is necessarily incremental, involving the generation and selection among random local mutations and recombinations of (parts of) one's current model. We argue that, by narrowing and guiding exploration, this (...) feature of cognitive search is what allows human learners to discover better theories, without ever grappling directly with the problem of finding a “global optimum,” or best possible world model. We suggest this aspect of cognitive processing works analogously to how blind variation and selection mechanisms drive biological evolution. We propose algorithms developed for program synthesis provide candidate mechanisms for how human minds might achieve this. We discuss objections and implications of this perspective, finally suggesting that a better process-level understanding of how humans incrementally explore compositional theory spaces can shed light on how we think, and provide explanatory traction on fundamental cognitive biases, including anchoring, probability matching, and confirmation bias. (shrink)

Logical inferentialism maintains that the formal rules of inference fix the meanings of the logical terms. The categoricity problem points out to the fact that the standard formalizations of classical logic do not uniquely determine the intended meanings of its logical terms, i.e., these formalizations are not categorical. This means that there are different interpretations of the logical terms that are consistent with the relation of logical derivability in a logical calculus. In the case of the quantificational logic, the categoricity (...) problem is generated by the finite nature of the standard calculi and one direction in which it can be solved is to strengthen the deductive systems by adding infinitary rules (such as the ω-rule), i.e., to construct a full formalization. Another main direction is to provide a natural semantics for the standard rules of inference, i.e., a semantics for which these rules are categorical. My aim in this paper is to analyze some recent approaches for solving the categoricity problem and to argue that a logical inferentialist should accept the infinitary rules of inference for the first order quantifiers, since our use of the expressions “all” and “there is” leads us beyond the concrete and finite reasoning, and human beings do sometimes employ infinitary rules of inference in their reasoning. (shrink)

In this article, we expand the analytical and theoretical foundations of the study of knowledge commons in the context of more classical agrarian commons, such as seed commons. We show that it is possible to overcome a number of criticisms of earlier work by Ostrom (Governing the commons. The evolution of institutions for collective action, Cambridge University Press, Cambridge, 1990) on natural commons and its excludability/rivalry matrix in addressing the inclusive social practices of “commoning”, defined as a way of living (...) and acting for the preservation of the commons. Our empirical analysis emphasizes, using the most recent advances in the IAD/SES framework, the distributed and collaborative knowledge governance in a French peasant seed network as a key driver for reintroducing cultivated agrobiodiversity and on-farm seed conservation of ancient and landrace varieties. These inclusive peasant seed groups developed alternative peer-to-peer models of collaborative peasant-led community-based breeding and grassroots innovations in the search for more resilient population varieties. Our results highlight the various models of collective action within the network and discuss the organizational tradeoffs of opting out of peasant seed activities and recreating a shared collective knowledge base on the benefits of restoring cultivated agrobiodiversity. It helps us better understand how modern peasant seed groups function as epistemic communities which contributes to envisioning alternative agricultural systems. (shrink)

Logical inferentialism maintains that the formal rules of inference fix the meanings of the logical terms. The categoricity problem points out to the fact that the standard formalizations of classical logic do not uniquely determine the intended meanings of its logical terms, i.e., these formalizations are not categorical. This means that there are different interpretations of the logical terms that are consistent with the relation of logical derivability in a logical calculus. In the case of the quantificational logic, the categoricity (...) problem is generated by the finite nature of the standard calculi and one direction in which it can be solved is to strengthen the deductive systems by adding infinitary rules (such as the ω-rule), i.e., to construct a full formalization. Another main direction is to provide a natural semantics for the standard rules of inference, i.e., a semantics for which these rules are categorical. My aim in this paper is to analyze some recent approaches for solving the categoricity problem and to argue that a logical inferentialist should accept the infinitary rules of inference for the first order quantifiers, since our use of the expressions “all” and “there is” leads us beyond the concrete and finite reasoning, and human beings do sometimes employ infinitary rules of inference in their reasoning. (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)

There has been considerable discussion in the past about the assumptions and basis of different ethical rules. For instance, it is commonplace to say that ethical rules are defaults rules, which means that they tolerate exceptions. Some authors argue that morality can only be grounded in particular cases while others defend the existence of general principles related to ethical rules. Our purpose here is not to justify either position, but to try to model general ethical rules with artificial intelligence formalisms (...) and to compute logical consequences of different ethical theories. More precisely, this is an attempt to show that progress in non-monotonic logics, which simulates default reasoning, could provide a way to formalize different ethical conceptions. From a technical point of view, the model developed in this paper makes use of the Answer Set Programming (ASP) formalism. It is applied comparatively to different ethical systems with respect to their attitude towards lying. The advantages of such formalization are two-fold: firstly, to clarify ideas and assumptions, and, secondly, to use solvers to derive consequences of different ethical conceptions automatically, which can help in a rigorous comparison of ethical theories. (shrink)

Wójcicki has provided a characterization of selfextensional logics as those that can be endowed with a complete local referential semantics. His result was extended by Jansana and Palmigiano, who developed a duality between the category of reduced congruential atlases and that of reduced referential algebras over a fixed similarity type. This duality restricts to one between reduced atlas models and reduced referential algebra models of selfextensional logics. In this paper referential algebraic systems and congruential atlas systems are introduced, which (...) abstract referential algebras and congruential atlases, respectively. This enables the formulation of an analog of Wójcicki’s Theorem for logics formalized as π-institutions. Moreover, the results of Jansana and Palmigiano are generalized to obtain a duality between congruential atlas systems and referential algebraic systems over a fixed categorical algebraic signature. In future work, the duality obtained in this paper will be used to obtain one between atlas system models and referential algebraic system models of an arbitrary selfextensional π-institution. Using this latter duality, the characterization of fully selfextensional deductive systems among the selfextensional ones, that was obtained by Jansana and Palmigiano, can be extended to a similar characterization of fully selfextensional π-institutions among appropriately chosen classes of selfextensional ones. (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)

This article attempts to open the debate on a territorial development perspective that fixes the interest in organized territories that are characterized by the existence of a community with a local identity, politically and administratively regulated. We conceive these territories as subjects of development promotion interventions. For this we propose a model for the creation of Municipal Development Forums, which with the participation of local actors, can generate a tailor-made Local Development Program. The development program will (...) consider a diagnosis of the productive profile of the territory and its relationship with the characteristics and origin of goods or services consumed at the municipality. Based on the diagnosis, the causes of the decoupling between the local production system and consumption needs can be detected. And with this, find a way to readjust local production systems to meet the demand for goods and services of the inhabitants, generating new companies and encouraging local consumption. (shrink)

Recent work by Wan and McLean has shown that all quantum measurements may be reduced to local position measurements. Using an array of particle detectors as the measuring apparatus we show how a model employing superselection rules and unitary evolution leads to a single detector triggering in each act of measurement. We also present an explicit model of particle detection as a unitary ionization process producing a single ion in the detector, subsequent amplification of which to the visible can (...) be described adequately in classical terms. (shrink)

The article proceeds from the concept of understanding local government in Russia as a projection of the potestar (pre-state) organization of public life, which transforms under the dominance of methods of state organization of public life, but retains its influence as the essential core of political and social interaction between people. The existing complex «state-municipal» mechanism for exercising power at the local level largely determines the forms and nature of political actors’ interaction at the regional level. State authorities, (...) which are focused on the totality of their own administrative influences, have not yet succeeded in fully integrating local self-government into the general public space of responsibility for powers, since local power is unbuilt initially on the principles of autogenesis, autochthony and autonomy. The current situation contributes to the archaicism of a part of social relations. For some time now, to work in the multidimensional reality of society, the authorities have begun to need analytical models to understand what is happening. As an explanatory theoretical basis for the modes of existence of local authorities developed and described by the author in the current stage of development of Russian statehood, a modal methodology is used, supplemented by the results of constructing model-oriented political ontologies, which reflect the aggregates of the layers and actors of the municipal sphere of sociopolitical reality at the regional level that are actually and actively present in the Russian society. The author is interested in searching the patterns that operate in the environment created and cultivated by the organizational and activity efforts of state structures against the background of sociocultural and other characteristics inherent in the regions. Possible working models for the needs of public administration arising from the modes offered by the author are still under development, since the reform of constitutional rules proposed by the country’s leadership raises the question of new forms of organizing the continuum of coexistence and interaction of local and state authorities in Russia. The article outlines the main research lines that indicate a number of features and reference points of the author’s approach in understanding the relationship between local government and municipal government, which may serve as an occasion for a new circle of discussions of specialists interested in the prospects for the development of political potential in the Russian municipal environment. (shrink)

This paper describes a simple, causally deterministic model of quantum measurement based on an amplitude threshold detection scheme. Surprisingly, it is found to reproduce many phenomena normally thought to be uniquely quantum in nature. To model an $N$ -dimensional pure state, the model uses $N$ complex random variables given by a scaled version of the wave vector with additive complex noise. Measurements are defined by threshold crossings of the individual components, conditioned on single-component threshold crossings. The resulting detection probabilities match (...) or approximate those predicted by quantum mechanics according to the Born rule. Nevertheless, quantum phenomena such as entanglement, contextuality, and violations of Bell’s inequality under local measurements are all shown to be exhibited by the model, thereby demonstrating that such phenomena are not without classical analogs. (shrink)

Khoussainov and Nerode [14] posed various open questions on model-theoretic properties of automatic structures. In this work we answer some of these questions by showing the following results: There is an uncountably categorical but not countably categorical theory for which only the prime model is automatic; There are complete theories with exactly 3,4,5,…3,4,5,… countable models, respectively, and every countable model is automatic; There is a complete theory for which exactly 2 models have an automatic presentation; If LOGSPACE=PLOGSPACE=P then there (...) is an uncountably categorical but not countably categorical theory for which the prime model does not have an automatic presentation but all the other countable models are automatic; There is a complete theory with countably many countable models for which the saturated model has an automatic presentation but the prime model does not have one. (shrink)

Journal of Mathematical Logic, Ahead of Print. We initiate the study of a generalization of Kim-independence, Conant-independence, based on the notion of strong Kim-dividing of Kaplan, Ramsey and Shelah. A version of Conant-independence was originally introduced to prove that all [math] theories are [math]. We introduce an axiom on stationary independence relations, essentially generalizing the “freedom” axiom in some of the free amalgamation theories of Conant, and show that this axiom provides the correct setting for carrying out arguments of Chernikov, (...) Kaplan and Ramsey on [math] theories relative to a stationary independence relation. Generalizing Conant’s results on free amalgamation to the limits of our knowledge of the [math] hierarchy, we show using methods from Conant as well as our previous work that any theory where the equivalent conditions of this local variant of [math] holds is either [math] or [math] and is either simple or [math]. We observe that these theories give an interesting class of examples of theories where Conant-independence is symmetric. This includes all of Conant’s examples, the small cycle-free random graphs of Shelah, and the (finite-language) [math]-categorical Hrushovski constructions of Evans and Wong. We then answer Conant’s question on the existence of non-modular free amalgamation theories. We show that the generic functional structures of Kruckman and Ramsey are examples of non-modular free amalgamation theories. We also show that any free amalgamation theory is [math] or [math], while an [math] free amalgamation theory is simple if and only if it is modular. Finally, we show that every theory where Conant-independence is symmetric is [math]. Therefore, symmetry for Conant-independence gives the next known neostability-theoretic dividing line on the [math] hierarchy beyond [math]. We explain the connection to some established open questions. (shrink)

The standard inferentialist approaches to modal logic tend to suffer from not being able to uniquely characterize the modal operators, require that introduction and elimination rules be interdefined, or rely on the introduction of possible-world like indexes into the object language itself. In this paper I introduce a hypersequent calculus that is flexible enough to capture many of the standard modal logics and does not suffer from the above problems. It is therefore an ideal candidate to underwrite an inferentialist theory (...) of meaning for modal operators. Here I treat specifically the modal logics K, D, T, S4, B, and S5. I show that the calculi are adequate for each set of models, and show that they meet a large set of criteria that are generally thought necessary for a calculus to underwrite a theory of meaning. (shrink)

A new case of Shelah's eventual categoricity conjecture is established: Let be an abstract elementary class with amalgamation. Write and. Assume that is H2‐tame and has primes over sets of the form. If is categorical in some, then is categorical in all. The result had previously been established when the stronger locality assumptions of full tameness and shortness are also required. An application of the method of proof of the mentioned result is that Shelah's categoricity conjecture holds in the context (...) of homogeneous model theory (this was known, but our proof gives new cases): If D be a homogeneous diagram in a first‐order theory T and D is categorical in a, then D is categorical in all. (shrink)

The aim of this study was to examine the predictions of three theories of human logical reasoning, (a) mental model theory, (b) formal rules theory (e.g., PSYCOP), and (c) the probability heuristics model, regarding the inferences people make for extended categorical syllogisms. Most research with extended syllogisms has been restricted to the quantifier “All” and to an asymmetrical presentation. This study used three-premise syllogisms with the additional quantifiers that are used for traditional categorical syllogisms as well as additional syllogistic figures. (...) The predictions of the theories were examined using overall accuracy as well as a multinomial tree modelling technique. The results demonstrated that all three theories were able to predict response selections at high levels. However, the modelling analyses showed that the probability heuristics model did the best in both Experiments 1 and 2. (shrink)

Genericity is the idea that the same program can work at many different data types. Longo, Milstead and Soloviev proposed to capture the inability of generic programs to probe the structure of their instances by the following equational principle: if two generic programs, viewed as terms of type , are equal at any given instance A[T], then they are equal at all instances. They proved that this rule is admissible in a certain extension of System F, but finding a (...) semantically motivated model satisfying this principle remained an open problem.In the present paper, we construct a categorical model of polymorphism, based on game semantics, which contains a large collection of generic types. This model builds on two novel constructions:•A direct interpretation of variable types as games, with a natural notion of substitution of games. This allows moves in games A[T] to be decomposed into the generic part from A, and the part pertaining to the instance T. This leads to a simple and natural notion of generic strategy.•A “relative polymorphic product” which expresses quantification over the type variable Xi in the variable type A with respect to a “universe” which is explicitly given as an additional parameter B. We then solve a recursive equation involving this relative product to obtain a universe in a suitably “absolute” sense.Full Completeness for ML types is proved for this model. (shrink)

The essay first makes some observations on the general interrelationship between the logical writings of Albert and Buridan. Second, it gives an account of a ‘semantic logical model’ for analyzing complex subject terms in some basic categorical propositions which is defended by Albert of Saxony, and briefly recounts Buridan's criticisms of that model. Finally, the essay maintains that the Albertian model is typically compatible with, and a further development of, what is called by a late-fourteenth century anonymous scholar ‘the English- (...) class='Hi'>Rule’ but the ‘determinable/determinate’ grammatical model defended by Buridan is not. (shrink)

We initiate the study of a generalization of Kim-independence, Conant-independence, based on the notion of strong Kim-dividing of Kaplan, Ramsey and Shelah. A version of Conant-independence was originally introduced to prove that all [Formula: see text] theories are [Formula: see text]. We introduce an axiom on stationary independence relations, essentially generalizing the “freedom” axiom in some of the free amalgamation theories of Conant, and show that this axiom provides the correct setting for carrying out arguments of Chernikov, Kaplan and Ramsey (...) on [Formula: see text] theories relative to a stationary independence relation. Generalizing Conant’s results on free amalgamation to the limits of our knowledge of the [Formula: see text] hierarchy, we show using methods from Conant as well as our previous work that any theory where the equivalent conditions of this local variant of [Formula: see text] holds is either [Formula: see text] or [Formula: see text] and is either simple or [Formula: see text]. We observe that these theories give an interesting class of examples of theories where Conant-independence is symmetric. This includes all of Conant’s examples, the small cycle-free random graphs of Shelah, and the (finite-language) [Formula: see text]-categorical Hrushovski constructions of Evans and Wong. We then answer Conant’s question on the existence of non-modular free amalgamation theories. We show that the generic functional structures of Kruckman and Ramsey are examples of non-modular free amalgamation theories. We also show that any free amalgamation theory is [Formula: see text] or [Formula: see text], while an [Formula: see text] free amalgamation theory is simple if and only if it is modular. Finally, we show that every theory where Conant-independence is symmetric is [Formula: see text]. Therefore, symmetry for Conant-independence gives the next known neostability-theoretic dividing line on the [Formula: see text] hierarchy beyond [Formula: see text]. We explain the connection to some established open questions. (shrink)

Adopting a different method from the previous scholars, this article deduces the remaining 23 valid syllogisms just taking the syllogism AEE-4 as the basic axiom. The basic idea of this study is as follows: firstly, make full use of the trichotomy structure of categorical propositions to formalize categorical syllogisms. Then, taking advantage of the deductive rules in classical propositional logic and the basic facts in the generalized quantifier theory, we deduce the remaining 23 valid categorical syllogisms by taking just one (...) syllogism (that is, AEE-4) as the basic axiom. This article not only reveals the reducible relations between the syllogism AEE-4 and the other 23 valid syllogisms, but also establishes a concise formal axiomatic system for categorical syllogistic logic. We hope that the results and methods will provide a good mathematical paradigm for studying other kinds of syllogistic logics, and that the project will appeal to specialists in logic, linguistic semantics, computational semantics, cognitive science and artificial intelligence. (shrink)

Having been supressed and denied their rights by successive Syrian governments over the years, Syrian Kurds are now asserting a de facto autonomy. Since the withdrawal of the Syrian President's forces from the ethnically Kurdish areas in the early months of the current civil war, the inhabitants have declared a self-rule government along the lines of the Kurdistan regional government in northern Iraq. For Syrian Kurds, the creation of a small autonomous region is a dream fulfilled, albeit one unrecognized (...) by the international community. Some 15% to 17% of the Syrian population is Kurdish. Whether they can achieve statehood will depend on a reading of international law and on how the international community reacts. There are certain aspects which differentiate Kurdish self-rule in Syria from its counterpart in Iraq. View HTML Send article to KindleTo send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle. Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply. Find out more about the Kindle Personal Document Service.Kurdish Regional Self-rule Administration in Syria: A new Model of Statehood and its Status in International Law Compared to the Kurdistan Regional Government in IraqVolume 17, Issue 3LOQMAN RADPEY DOI: https://doi.org/10.1017/S1468109916000190Your Kindle email address Please provide your Kindle email.@free.kindle.com@kindle.com Available formats PDF Please select a format to send. By using this service, you agree that you will only keep articles for personal use, and will not openly distribute them via Dropbox, Google Drive or other file sharing services. Please confirm that you accept the terms of use. Cancel Send ×Send article to Dropbox To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about sending content to Dropbox. Kurdish Regional Self-rule Administration in Syria: A new Model of Statehood and its Status in International Law Compared to the Kurdistan Regional Government in IraqVolume 17, Issue 3LOQMAN RADPEY DOI: https://doi.org/10.1017/S1468109916000190Available formats PDF Please select a format to send. By using this service, you agree that you will only keep articles for personal use, and will not openly distribute them via Dropbox, Google Drive or other file sharing services. Please confirm that you accept the terms of use. Cancel Send ×Send article to Google Drive To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about sending content to Google Drive. Kurdish Regional Self-rule Administration in Syria: A new Model of Statehood and its Status in International Law Compared to the Kurdistan Regional Government in IraqVolume 17, Issue 3LOQMAN RADPEY DOI: https://doi.org/10.1017/S1468109916000190Available formats PDF Please select a format to send. By using this service, you agree that you will only keep articles for personal use, and will not openly distribute them via Dropbox, Google Drive or other file sharing services. Please confirm that you accept the terms of use. Cancel Send ×Export citation Request permission. (shrink)