This paper is a critical response to Andreas Bartels’ sophisticated defense of a structural account of scientific representation. We show that, contrary to Bartels’ claim, homomorphism fails to account for the phenomenon of misrepresentation. Bartels claims that homomorphism is adequate in two respects. First, it is conceptually adequate, in the sense that it shows how representation differs from misrepresentation and non-representation. Second, if properly weakened, homomorphism is formally adequate to accommodate misrepresentation. We question both claims. First, we (...) show that homomorphism is not the right condition to distinguish representation from misrepresentation and non-representation: a “representational mechanism” actually does all the work, and it is independent of homomorphism – as of any structural condition. Second, we test the claim of formal adequacy against three typical kinds of inaccurate representation in science which, by reference to a discussion of the notorious billiard ball model, we define as abstraction, pretence, and simulation. We first point out that Bartels equivocates between homomorphism and the stronger condition of epimorphism, and that the weakened form of homomorphism that Bartels puts forward is not a morphism at all. After providing a formal setting for abstraction, pretence and simulation, we show that for each morphism there is at least one form of inaccurate representation which is not accommodated. We conclude that Bartels’ theory – while logically laying down the weakest structural requirements – is nonetheless formally inadequate in its own terms. This should shed serious doubts on the plausibility of any structural account of representation more generally. (shrink)

In this paper, we introduce the notion of intuitionistic fuzzy PMS-subalgebras under homomorphism and Cartesian product and investigate several properties. We study the homomorphic image and inverse image of the intuitionistic fuzzy PMS-subalgebras of a PMS-algebra, which are also intuitionistic fuzzy PMS-subalgebras of a PMS-algebra, and find some other interesting results. Furthermore, we also prove that the Cartesian product of intuitionistic fuzzy PMS-subalgebras is again an intuitionistic fuzzy PMS-subalgebra and characterize it in terms of its level sets. Finally, we (...) consider the strongest intuitionistic fuzzy PMS-relations on an intuitionistic fuzzy set in a PMS-algebra and demonstrate that an intuitionistic fuzzy PMS-relation on an intuitionistic fuzzy set in a PMS-algebra is an intuitionistic fuzzy PMS-subalgebra if and only if the corresponding intuitionistic fuzzy set in a PMS-algebra is an intuitionistic fuzzy PMS-subalgebra of a PMS-algebra. (shrink)

In an earlier paper, we introduced the following pre-order on the subgroups of a given Polish group: if G is a Polish group and \ are subgroups, we say H is homomorphism reducible to L iff there is a continuous group homomorphism \ such that \\). We previously showed that there is a \ subgroup L of the countable power of any locally compact Polish group G such that every \ subgroup of \ is homomorphism reducible to (...) L. In the present work, we show that this fails in the countable power of the group of increasing homeomorphisms of the unit interval. (shrink)

In this article, we investigate the status of the homomorphism preservation property amongst restricted classes of finite relational structures and algebraic structures. We show that there are many homomorphism-closed classes of finite lattices that are definable by a first-order sentence but not by existential positive sentences, demonstrating the failure of the homomorphism preservation property for lattices at the finite level. In contrast to the negative results for algebras, we establish a finite-level relativised homomorphism preservation theorem in (...) the relational case. More specifically, we give a complete finite-level characterisation of first-order definable finitely generated anti-varieties relative to classes of relational structures definable by sentences of some general forms. When relativisation is dropped, this gives a fresh proof of Atserias’s characterisation of first-order definable constraint satisfaction problems over a fixed template, a well known special case of Rossman’s Finite Homomorphism Preservation Theorem. (shrink)

Suppose G is a definably connected, definable group in an o-minimal expansion of an ordered group. We show that the o-minimal universal covering homomorphism equation image: equation image→ G is a locally definable covering homomorphism and π1 is isomorphic to the o-minimal fundamental group π of G defined using locally definable covering homomorphisms.

Man unterscheidet in der Organisation verschiedener Tiere folgende Kategorien von Parallelismen: 1) die Homologien, welche auf einen gemeinsamen Ursprung der betreffenden Organe hinweisen, 2) die Analogien, die als eine Folge von ähnlichen Funktionen oder der äusseren Wirkungen sekundär enstehen, 3) die Homomorphien, wodurch ich diejenigen Übereinstimmungen im Körperbau von verschiedenen Tieren bezeichne, welche auf Grund der allgemeinen Gesetze der Morphogenese zustande kommen. Das Studium der Homologien ist historischer Art, dasjenige der Homomorphien und Analogien typologischer Art.Man kann weiter unterscheiden: eine allgemeine (...) Homomorphie, welche sich im Tierreiche überall, wo dazu günstige Bedingungen vorhanden sind, offenbart , und eine spezielle Homomorphie, welche als eine der Möglichkeiten bei einer begrenzten Anzahl von Tierformen hervortritt . Die Grenze zwischen diesen beiden Kategorien ist natürlich nicht sehr deutlich, ebenso wie manchmal die Grenze zwischen Homologien, Homomorphien und Analogien. Alle diese Erscheinungen beruhen letzten Endes auf morphogenetischen Gesetzen, können also als Homomorphien im weitesten Sinne des Wortes bezeichnet werden.Dans l'organisation des divers animaux on peut distinguer les parallélismes suivants: 1) les homologies qui dénotent la descendance des porteurs des organes similaires d'un seul et même ancêtre, 2) les analogies qui se forment sous l'influence d'une fonction analogue ou des mêmes conditions extérieures, et 3) les homomorphies qui représentent la manifestation des propriétés fondamentales de la substance animée. Les homologies sont les phénomènes de l'ordre historique, tandis que les homomorphies et les analogies doivent être étudiées d'après une méthode typologique.Si une similitude homomorphique des formes se manifeste partout dans le règne animal ou au moins dans tous les cas où se trouvent des conditions favorables, il faut parler d'une homomorphie générale . Si au contraire on observe le parallélisme seulement dans les cas isolés, comme p. ex. les yeux composés chez les Arthropodes, les Vers et les Mollusques, il faut parler de l'homomorphie spéciale. La limite entre ces deux catégories des phénomènes n'est pas très précise. De même il est difficile parfois de définir strictement si nous avons à faire à une homologie, à une analogie ou à une homomorphie. Tous ces phénomènes-là dépendent finalement des lois morphogénètiques. C'est pourquoi qu'on peut les indiquer comme des homomorphies dans le sens le plus étendu de ce mot. (shrink)

The case against analytical isomorphism is made within an evolutionary framework. The relevance to neural filling-in is discussed. Homomorphism is argued for as a conceptually superior substitute for isomorphism, and the implications for the personal/subpersonal distinction are explored.

Consciousness and Mind presents David Rosenthal's influential work on the nature of consciousness. Central to that work is Rosenthal's higher-order-thought theory of consciousness, according to which a sensation, thought, or other mental state is conscious if one has a higher-order thought that one is in that state. The first four essays develop various aspects of that theory. The next three essays present Rosenthal's homomorphism theory of mental qualities and qualitative consciousness, and show how that theory fits with and helps (...) sustain the HOT theory. A crucial feature of homomorphism theory is that it individuates and taxonomizes mental qualities independently of the way we're conscious of them, and indeed independently of our being conscious of them at all. So the theory accommodates the qualitative character not only of conscious sensations and perceptions, but also of those which fall outside our stream of consciousness. Rosenthal argues that, because this account of mental qualities makes no appeal to consciousness, it enables us to dispel such traditional quandaries as the alleged conceivability of undetectable quality inversion, and to disarm various apparent obstacles to explaining qualitative consciousness and understanding its nature. Six further essays build on the HOT theory to explain various important features of consciousness, among them the complex connections that hold in humans between consciousness and speech, the self-interpretative aspect of consciousness, and the compelling sense we have that consciousness is unified. Two of the essays, one an extended treatment of homomorphism theory, appear here for the first time. There is also a substantive introduction, which draws out the connections between the essays and highlights their implications. (shrink)

Angell's logic of analytic containment AC has been shown to be characterized by a 9-valued matrix NC by Ferguson, and by a 16-valued matrix by Fine. We show that the former is the image of a surjective homomorphism from the latter, i.e., an epimorphic image. The epimorphism was found with the help of MUltlog, which also provides a tableau calculus for NC extended by quantifiers that generalize conjunction and disjunction.

I show that the partial truth of a sentence in a partial structure is equivalent to the truth of that sentence in an expansion of a structure that corresponds naturally to the partial structure. Further, a mapping is a partial homomorphism/partial isomorphism between two partial structures if and only if it is a homomorphism/isomorphism between their corresponding structures. It is a corollary that the partial truth of a sentence in a partial structure is equivalent to the truth of (...) a specific Ramsey sentence in a corresponding structure. Hence the partial structures approach can be expressed in standard first or second-order model theory, and it can be captured in the received view on scientific theories as developed by Carnap and Hempel. (shrink)

This is a survey of work on set-theoretical invariance criteria for logicality. It begins with a review of the Tarski-Sher thesis in terms, first, of permutation invariance over a given domain and then of isomorphism invariance across domains, both characterized by McGee in terms of definability in the language L∞,∞. It continues with a review of critiques of the Tarski-Sher thesis, and a proposal in response to one of those critiques via homomorphism invariance. That has quite divergent characterization results (...) depending on its formulation, one in terms of FOL, the other by Bonnay in terms of L∞,∞, both without equality. From that we move on to a survey of Bonnay’s work on similarity relations between structures and his results that single out invariance with respect to potential isomorphism among all such. Turning to the critique that calls for sameness of meaning of a logical operation across domains, the paper continues with a result showing that the isomorphism invariant operations that are absolutely definable with respect to KPU−Inf are exactly those definable in full FOL; this makes use of an old theorem of Manders. The concluding section is devoted to a critical discussion of the arguments for set-theoretical criteria for logicality. (shrink)

In this paper we extend the theory of neutrosophic rings and neutrosophic fields to soft sets and construct soft neutrosophic rings and soft neutrosophic fields. We also extend neutrosophic ideal theory to form soft neutrosophic ideal over a neutrosophic ring and soft neutrosophic ideal of a soft neutrosophic ring . We have given many examples to illustrate the theory of soft neutrosophic rings and soft neutrosophic fields and display many properties of of these. At the end of this paper we (...) gave soft neutrosophic ring homomorphism. (shrink)

We define notions of homomorphism, submodel, and sandwich of Kripke models, and we define two syntactic operators analogous to universal and existential closure. Then we prove an intuitionistic analogue of the generalized (dual of the) Lyndon-Łoś-Tarski Theorem, which characterizes the sentences preserved under inverse images of homomorphisms of Kripke models, an intuitionistic analogue of the generalized Łoś-Tarski Theorem, which characterizes the sentences preserved under submodels of Kripke models, and an intuitionistic analogue of the generalized Keisler Sandwich Theorem, which characterizes (...) the sentences preserved under sandwiches of Kripke models. We also define several intuitionistic formula hierarchies analogous to the classical formula hierarchies $\forall_n (= \Pi^0_n)$ and $\exists_n (=\Sigma^0_n)$ , and we show how our generalized syntactic preservation theorems specialize to these hierarchies. Each of these theorems implies the corresponding classical theorem in the case where the Kripke models force classical logic. (shrink)

The paper starts with an examination and critique of Tarski’s wellknown proposed explication of the notion of logical operation in the type structure over a given domain of individuals as one which is invariant with respect to arbitrary permutations of the domain. The class of such operations has been characterized by McGee as exactly those definable in the language L∞,∞. Also characterized similarly is a natural generalization of Tarski’s thesis, due to Sher, in terms of bijections between domains. My main (...) objections are that on the one hand, the Tarski-Sher thesis thus assimilates logic to mathematics, and on the other hand fails to explain the notion of same logical operation across domains of different sizes. A new notion of homomorphism invariant operation over functional type structures (with domains M0 of individuals and {T, F} at their base) is introduced to accomplish the latter. The main result is that an operation is definable from.. (shrink)

We study the lattice of extensions of four-valued Belnap–Dunn logic, called super-Belnap logics by analogy with superintuitionistic logics. We describe the global structure of this lattice by splitting it into several subintervals, and prove some new completeness theorems for super-Belnap logics. The crucial technical tool for this purpose will be the so-called antiaxiomatic (or explosive) part operator. The antiaxiomatic (or explosive) extensions of Belnap–Dunn logic turn out to be of particular interest owing to their connection to graph theory: the lattice (...) of finitary antiaxiomatic extensions of Belnap–Dunn logic is isomorphic to the lattice of upsets in the homomorphism order on finite graphs (with loops allowed). In particular, there is a continuum of finitary super-Belnap logics. Moreover, a non-finitary super-Belnap logic can be constructed with the help of this isomorphism. As algebraic corollaries we obtain the existence of a continuum of antivarieties of De Morgan algebras and the existence of a prevariety of De Morgan algebras which is not a quasivariety. (shrink)

In their constructive reviews, Frances Egan, Randy Gallistel and Steven Gross have raised some important problems for the account of content advanced by Nicholas Shea in Representation in Cognitive Science. Here the author addresses their main challenges. Egan argues that the account includes an unrecognised pragmatic element; and that it makes contents explanatorily otiose. Gallistel raises questions about homomorphism and correlational information. Gross puts the account to work to resolve a dispute about probabilistic contents in perception, but argues that (...) a question remains about whether probabilities are found in the content or instead in the manner of representation. (shrink)

Given classical (2 valued) structures and and a homomorphism h of onto , it is shown how to construct a (non-degenerate) 3-valued counterpart of . Classical sentences that are true in are non-false in . Applications to number theory and type theory (with axiom of infinity) produce finite 3-valued models in which all classically true sentences of these theories are non-false. Connections to relevant logic give absolute consistency proofs for versions of these theories formulated in relevant logic (the proof (...) for number theory was obtained earlier by R. K. Meyer and suggested the present abstract development). (shrink)

An interesting feature of some sets of representations is that their structure mirrors the structure of the items they represent. Founding an account of representational content on isomorphism, homomorphism or structural resemblance has proven elusive, however, largely because these relations are too liberal when the candidate structure over representational vehicles is unconstrained. Furthermore, in many cases where there is a clear isomorphism, it is not relied on in the way the representations are used. That points to a potential resolution: (...) that an isomorphism must be used, hence usable, if it is to be an ingredient in a theory of content. This paper argues that the class of exploitable isomorphisms can indeed play a content-constituting role. (shrink)

Recent work in the philosophy of science has generated an apparent conflict between theories attempting to explicate the nature of scientific representation. On one side, there are what one might call 'informational' views, which emphasize objective relations (such as similarity, isomorphism, and homomorphism) between representations (theories, models, simulations, diagrams, etc.) and their target systems. On the other side, there are what one might call 'functional' views, which emphasize cognitive activities performed in connection with these targets, such as interpretation and (...) inference. The main sources of the impression of conflict here are arguments by some functionalists to the effect that informational theories are flawed: it is suggested that relations typically championed by informational theories are neither necessary nor sufficient for scientific representation, and that any theory excluding functions is inadequate. In this paper I critically examine these arguments, and contend that, as it turns out, informational and functional theories are importantly complementary. (shrink)

In this paper we defend structural representations, more specifically neural structural representation. We are not alone in this, many are currently engaged in this endeavor. The direction we take, however, diverges from the main road, a road paved by the mathematical theory of measure that, in the 1970s, established homomorphism as the way to map empirical domains of things in the world to the codomain of numbers. By adopting the mind as codomain, this mapping became a boon for all (...) those convinced that a representation system should bear similarities with what was being represented, but struggled to find a precise account of what such similarities mean. The euforia was brief, however, and soon homomorphism revealed itself to be affected by serious weaknesses, the primary one being that it included systems embarrassingly alien to representations. We find that the defense attempts that have followed, adopt strategies that share a common format: valid structural representations come as “homomorphism plus X”, with various “X”, provided in descriptive format only. Our alternative direction stems from the observation of the overlooked departure from homomorphism as used in the theory of measure and its later use in mental representations. In the former case, the codomain or the realm of numbers, is the most suited for developing theorems detailing the existence and uniqueness of homomorphism for a wide range of empirical domains. In the latter case, the codomain is the realm of the mind, possibly more vague and more ill-defined than the empirical domain itself. The time is ripe for articulating the mapping between represented domains and the mind in formal terms, by exploiting what is currently known about coding mechanisms in the brain. We provide a sketch of a possible development in this direction, one that adopts the theory of neural population coding as codomain. We will show that our framework is not only not in disagreement with the “plus X” proposals, but can lead to natural derivation of several of the “X”. (shrink)

In this paper, we study natural language constructions which were first examined by Barwise: The richer the country, the more powerful some of its officials. Guided by Barwise’s observations, we suggest that conceivable interpretations of such constructions express the existence of various similarities between partial orders such as homomorphism or embedding. Semantically, we interpret the constructions as polyadic generalized quantifiers restricted to finite models. We extend the results obtained by Barwise by showing that similarity quantifiers are not expressible in (...) elementary logic over finite models. We also investigate whether the proposed readings are sound from the cognitive perspective. We prove that almost all similarity quantifiers are intractable. This leads us to first-order variants, which only approximate the strong readings, but are cognitively more plausible. Driven by the question of ambiguity, we recall Barwise’s argumentation in favour of strong readings, enriching it with some arguments of our own. Given that Barwise-like sentences are indeed ambiguous, we use a generalized Strong Meaning Hypothesis to derive predictions for their verification. Finally, we propose a hypothesis according to which conflicting pressures of communication and cognition might give rise to an ambiguous construction, provided that different semantic variants of the construction withstand different pressures involved in its usage. (shrink)

It is generally accepted that, in the cognitive and neural sciences, there are both computational and mechanistic explanations. We ask how computational explanations can integrate into the mechanistic hierarchy. The problem stems from the fact that implementation and mechanistic relations have different forms. The implementation relation, from the states of an abstract computational system to the physical, implementing states is a homomorphism mapping relation. The mechanistic relation, however, is that of part/whole; the explaining features in a mechanistic explanation are (...) the components of the explanandum phenomenon and their causal organization. Moreover, each component in one level of mechanism is constituted and explained by components of an underlying level of mechanism. Hence, it seems, computational variables and functions cannot be mechanistically explained by the medium-dependent states and properties that implement them. How then, do the computational and the implementational integrate to create the mechanistic hierarchy? After explicating the general problem, we further demonstrate it through a concrete example, of reinforcement learning, in the cognitive and neural sciences. We then examine two possible solutions. On one solution, the mechanistic hierarchy embeds at the same levels computational and implementational properties. This picture fits with the view that computational explanations are mechanistic sketches. On the other solution, there are two separate hierarchies, one computational and another implementational, which are related by the implementation relation. This picture fits with the view that computational explanations are functional and autonomous explanations. It is less clear how these solutions fit with the view that computational explanations are full-fledged mechanistic explanations. Finally, we argue that both pictures are consistent with the reinforcement learning example, but that scientific practice does not align with the view that computational models are merely mechanistic sketches. (shrink)

Given a Polish groupG, let$E(G)$be the right coset equivalence relation$G^{\omega }/c(G)$, where$c(G)$is the group of all convergent sequences inG. The connected component of the identity of a Polish groupGis denoted by$G_0$.Let$G,H$be locally compact abelian Polish groups. If$E(G)\leq _B E(H)$, then there is a continuous homomorphism$S:G_0\rightarrow H_0$such that$\ker (S)$is non-archimedean. The converse is also true whenGis connected and compact.For$n\in {\mathbb {N}}^+$, the partially ordered set$P(\omega )/\mbox {Fin}$can be embedded into Borel equivalence relations between$E({\mathbb {R}}^n)$and$E({\mathbb {T}}^n)$.

In this paper we will study the properties of the least extension n(Λ) of a given intermediate logic Λ by a strong negation. It is shown that the mapping from Λ to n(Λ) is a homomorphism of complete lattices, preserving and reflecting finite model property, frame-completeness, interpolation and decidability. A general characterization of those constructive logics is given which are of the form n(Λ). This summarizes results that can be found already in [13, 14] and [4]. Furthermore, we determine (...) the structure of the lattice of extensions of n(LC). (shrink)

I define sublaltices of quantum propositions that can be taken as having determinate (but perhaps unknown) truth values for a given quantum state, in the sense that sufficiently many two-valued maps satisfying a Boolean homomorphism condition exist on each determinate sublattice to generate a Kolmogorov probability space for the probabilities defined by the slate. I show that these sublattices are maximal, subject to certain constraints, from which it follows easily that they are unique. I discuss the relevance of this (...) result for the measurement problem, relating it to an early proposal by Jauch and Piron for defining a new notion of state for quantum systems, to a recent uniqueness proof by Clifton for the sublattice of propositions specified as determinate by modal interpretations of quantum mechanics that exploit the polar decompostion theorem, and to my own previous suggestions for interpreting quantum mechanics without the projection postulate. (shrink)

Perceptual experience seems to involve distinct intentional and qualitative features. Inasmuch as one can visually perceive that there is a Coke can in front of one, perceptual experience must be intentional. But such experiences seem to differ from paradigmatic intentional states in having introspectible qualitative character. Peacocke argues that a perceptual experience’s qualitative character is determined by intrinsic, nonrepresentational properties. But and also argues that perceptual experiences have nonconceptual representational content in addition to conceptual content and nonrepresentational sensational properties. He (...) thus distinguishes between conceptual, nonrepresentational, and nonconceptual but representational aspects of perceptual experience. I will argue that Peacocke posits too much. Contrary to his arguments, the sensational properties Peacocke claims are nonrepresentational are best construed as representational; they are best explained in terms of their relation to the perceptible properties they enable us to perceive. Since sensational properties are arguably nonconceptual, they are best construed as nonconceptual representational properties. I offer the Homomorphism View of sensory qualities, pioneered by Sellars , as a unified account of qualitative character and nonconceptual sensory representation. According to this view, a sensory quality represents a perceptible stimulus property in virtue of resembling and differing from other sensory qualities in ways parallel to the ways the stimulus property resembles and differs from other perceptible properties. (shrink)

It is known that a countable $\omega $ -categorical structure interprets all finite structures primitively positively if and only if its polymorphism clone maps to the clone of projections on a two-element set via a continuous clone homomorphism. We investigate the relationship between the existence of a clone homomorphism to the projection clone, and the existence of such a homomorphism which is continuous and thus meets the above criterion.

In this paper we defend structural representations, more specifically neural structural representation. We are not alone in this, many are currently engaged in this endeavor. The direction we take, however, diverges from the main road, a road paved by the mathematical theory of measure that, in the 1970s, established homomorphism as the way to map empirical domains of things in the world to the codomain of numbers. By adopting the mind as codomain, this mapping became a boon for all (...) those convinced that a representation system should bear similarities with what was being represented, but struggled to find a precise account of what such similarities mean. The euforia was brief, however, and soon homomorphism revealed itself to be affected by serious weaknesses, the primary one being that it included systems embarrassingly alien to representations. We find that the defense attempts that have followed, adopt strategies that share a common format: valid structural representations come as “homomorphism plus X”, with various “X”, provided in descriptive format only. Our alternative direction stems from the observation of the overlooked departure from homomorphism as used in the theory of measure and its later use in mental representations. In the former case, the codomain or the realm of numbers, is the most suited for developing theorems detailing the existence and uniqueness of homomorphism for a wide range of empirical domains. In the latter case, the codomain is the realm of the mind, possibly more vague and more ill-defined than the empirical domain itself. The time is ripe for articulating the mapping between represented domains and the mind in formal terms, by exploiting what is currently known about coding mechanisms in the brain. We provide a sketch of a possible development in this direction, one that adopts the theory of neural population coding as codomain. We will show that our framework is not only not in disagreement with the “plus X” proposals, but can lead to natural derivation of several of the “X”. (shrink)

We show that the existential preservation theorem fails for two-variable first-order logic FO2. It is known that for all k ≥ 3, FOk does not have an existential preservation theorem, so this settles the last open case, answering a question of Andreka, van Benthem, and Németi. In contrast, we prove that the homomorphism preservation theorem holds for FO2.

We prove the Compact Domination Conjecture for groups definable in linear o-minimal structures. Namely, we show that every definably compact group G definable in a saturated linear o-minimal expansion of an ordered group is compactly dominated by (G/G 00, m, π), where m is the Haar measure on G/G 00 and π : G → G/G 00 is the canonical group homomorphism.

By elaborating on the results presented in Lógica cuántica, Nmatrices y adecuación I, here we discuss the notions of adequacy and truth functionality in quantum logic from the point of view of a non-deterministic semantics based on Nmatrices. We present a proof of the impossibility of providing a functional semantics for the quantum lattice. An advantage of our proof is that it is independent of the number of truth values involved, generalizing previous works. Due to the impossibility of defining adequate (...) interpreting sets for the conjunction and disjunction (as was shown in our previous work), it follows that a homomorphism between the lattice of quantum projections and a Boolean algebra of n-elements cannot exist. (shrink)

In this paper, dynamic behavior analysis has been discussed for a class of switched complex-valued neural networks with interval parameter uncertainties and impulse disturbance. Sufficient conditions for guaranteeing the existence, uniqueness, and global robust exponential stability of the equilibrium point have been obtained by using the homomorphism mapping theorem, the scalar Lyapunov function method, the average dwell time method, and M-matrix theory. Since there is no result concerning the stability problem of switched neural networks defined in complex number domain, (...) the stability results we describe in this paper generalize the existing ones. The effectiveness of the proposed results is illustrated by a numerical example. (shrink)

SummaryAfter giving examples for different relations of analogy among different objects the gist of analogy relations are interpreted as homomorphism and isomorphism . The main purpose of the paper is to give a number of precise definitions for different kinds of analogy . In chapter 7 definitions are proposed for analogy relations between theories and common features of the relations T1 is anologous to T2 and T1 is interpretable in T2 are discussed. Chapter 8 compares analogy and transformation and (...) chapter 9 proves some theorems on analogy. (shrink)

Perceptual-role theories of mental qualities hold that we can discover the nature of a being’s mental qualities by investigating that being’s capacity to make perceptual discriminations. Many advocates of perceptual-role theories hold that the best explanation of these capacities is that mental quality spaces are homomorphic to the spaces of the physical properties that they help to discriminate. This paper disputes this thesis on largely empirical grounds, and offers an alternative. The alternative explains interesting patterns in our perception of color (...) differences and similarities across edges, emphasizing the importance of visual discontinuities. Further, it provides a simple explanation of the intransitivity of indiscriminability. The homomorphism thesis can explain this intransitivity only with the addition of an independently unsupported empirical hypothesis. Also, in contrast to the homomorphism thesis, the alternative view is perfectly consistent with empirical data that suggest that the visual system uses the dynamic allocation of a limited range of responses in order to discriminate a wider range of physical stimuli and that we are much better at detecting changes in relative luminance than changes in absolute luminance. Moreover, it does all this while using a more efficient strategy for encoding sensory information. (shrink)

Though deceptively simple and plausible on the face of it, Craig's interpolation theorem has proved to be a central logical property that has been used to reveal a deep harmony between the syntax and semantics of first order logic. Craig's theorem was generalized soon after by Lyndon, with application to the characterization of first order properties preserved under homomorphism. After retracing the early history, this article is mainly devoted to a survey of subsequent generalizations and applications, especially of many-sorted (...) interpolation theorems. Attention is also paid to methodological considerations, since the Craig theorem and its generalizations were initially obtained by proof-theoretic arguments while most of the applications are model-theoretic in nature. The article concludes with the role of the interpolation property in the quest for "reasonable" logics extending first-order logic within the framework of abstract model theory. (shrink)

These two volumes contain all of my articles published between 1956 and 1975 which might be of interest to readers in the English-speaking world. The first three essays in Vol. 1 deal with historical themes. In each case I as far as possible, meets con have attempted a rational reconstruction which, temporary standards of exactness. In The Problem of Universals Then and Now some ideas of W.V. Quine and N. Goodman are used to create a modern sketch of the history (...) of the debate on universals beginning with Plato and ending with Hao Wang's System L. The second article concerns Kant's Philosophy of Science. By analyzing his position vis-a-vis I. Newton, Christian Wolff, and D. Hume, it is shown that for Kant the very notion of empirical knowledge was beset with a funda mental logical difficulty. In his metaphysics of experience Kant offered a solution differing from all prior as well as subsequent attempts aimed at the problem of establishing a scientific theory. The last of the three historical papers utilizes some concepts of modern logic to give a precise account of Wittgenstein's so-called Picture Theory of Meaning. E. Stenius' interpretation of this theory is taken as an intuitive starting point while an intensional variant of Tarski's concept of a relational system furnishes a technical instrument. The concepts of inodel world and of logical space, together with those of homomorphism and isomorphism be tween model worlds and between logical spaces, form the conceptual basis of the reconstruction. (shrink)

The author's purpose is to read the main work of Euclid “with modern eyes” and to find out what knowledge a mathematician of today, familiar with the works of V. D. Waerden and Bourbaki, can gain by studying Euclid's “theory of magnitudes”, and what new insight into Greek mathematics occupation with this subject can provide. The task is to analyse and to axiomatize by modern means (i) in a narrower sense Book V. of the Elements, i.e. the theory of proportion (...) of Eudoxus, (ii) in a wider sense the whole sphere of magnitudes which Euclid applies in his Elements. This procedure furnishes a clear picture of the inherent structure of his work, thereby making visible specific characteristics of Greek mathematics. After a clarification of the preconditions and a short survey of the historical development of the theory of proportions (Part I of this work), an exact analysis of the definitions and propositions of Book V. of the Elements is carried out in Part II. This is done “word by word”. The author applies his own system of axioms, set up in close accordance with Euclid, which permits one to deduce all definitions and propositions of Euclid's theory of magnitudes (especially those of Books V. and VI.). In this way gaps and “tacit assumptions” in the work become clearly visible; above all, the logical structure of the system of magnitudes given by Euclid becomes evident: not “ratio” — like something sui generis — is the governing concept of Book V., but magnitudes and their relation “of having a ratio” form the base of the theory of proportions. These magnitudes represent a well defined structure, a so-called “Eudoxic Semigroup” with the numbers as operators; it can easily be imbedded in a general theory of magnitudes equally applicable to geometry and physics. The transition to ratios — a step not executed by Euclid — is examined in Part III; it turns out to be particularly unwieldy. An elegant way opens up by interpreting proportion as a mapping of totally ordered semigroups. When closely examined, this mapping proves to be an isomorphism, thus suggesting the application of the modern theory of homomorphism. This theory permits a treatment of the theory of proportions as developed by Eudoxus and Euclid which is hardly surpassable in brevity and elegance in spite of its close affinity to Euclid. The generalization to a “classically” founded theory of magnitudes is now self-evident. (shrink)

Starting from the familiar observation that no straightforward treatment of pure quotation can be compositional in the standard (homomorphism) sense, we introduce general compositionality, which can be described as compositionality that takes linguistic context into account. A formal notion of linguistic context type is developed, allowing the context type of a complex expression to be distinct from those of its constituents. We formulate natural conditions under which an ordinary meaning assignment can be non-trivially extended to one that is sensitive (...) to context types and satisfies general compositionality. As our main example we work out a Fregean treatment of pure quotation, but we also indicate that the method applies to other kinds of context, e.g. intensional contexts. (shrink)

The aim of this paper is to defend the structural concept of representation, as defined by homomorphisms, against its main objections, namely: logical objections, the objection from misrepresentation, theobjection from failing necessity, and the copy theory objection. The logical objections can be met by reserving the relation.

Cooper storage is a widespread technique for associating sentences with their meanings, used in diverse linguistic and computational linguistic traditions. This paper encodes the data structures and operations of cooper storage in the simply typed linear \-calculus, revealing the rich categorical structure of a graded applicative functor. In the case of finite cooper storage, which corresponds to ideas in current transformational approaches to syntax, the semantic interpretation function can be given as a linear homomorphism acting on a regular set (...) of trees, and thus generation can be done in polynomial time. (shrink)

It is shown within Bishop's constructive mathematics that, under one extra, classically automatic, hypothesis, a continuous homomorphism from R onto a compact metric abelian group is periodic, but that the existence of the minimum value of the period is not derivable.

The definition of “model of a system” in terms of a homomorphism of the states of the system is evaluated and an alternative definition in terms of sequence generators is proposed. Sequence generators are finite graphs whose points represent complete states of a system. Sequence generators include finite automata and other information processing systems as special cases. It is shown how to define models in terms of a projection operator which applies to any sequence generator which has an output (...) projection and yields a new sequence generator. A model produced by the projection operator is embedded in the system it models. The notion of embedding is discussed informally and some questions raised about the relations of deterministic, indeterministic, and probabilistic models and systems. (shrink)

The logical system Hyperproof and the computer implementation of it--both created by Jon Barwise and John Etchemendy--present a radical new approach to modeling and teaching about reasoning. Hyperproof is a heterogeneous proof system that uses both sentences and diagrams as steps in proofs. This dissertation addresses important logical, philosophical, and pedagogical issues that Hyperproof raises. We formalize the syntax and semantics of Hyperproof, show that the major inference rules are valid, and give completeness results for four subsystems of Hyperproof. We (...) address philosophical issues raised by the logical analysis, including the importance of the presence of homomorphism between diagrams and the worlds they represent. We also apply the constructivist educational philosophy to create a detailed explanation of how to teach analytic reasoning with Hyperproof, and we discuss how Hyperproof can be augmented to teach about reasoning with different types of domains. (shrink)

Distributive bounded lattices with a dual homomorphism as unary operation, called Ockham algebras, were firstly studied by Berman (1977). The varieties of Boolean algebras, De Morgan algebras, Kleene algebras and Stone algebras are some of the well known subvarieties of Ockham algebra. In this paper, new results about the congruence lattice of Ockham algebras are given. From these results and Urquhart's representation theorem for Ockham algebras a complete characterization of the subdirectly irreducible Ockham algebras is obtained. These results are (...) particularized for a large number of subvarieties of Ockham algebras. For these subvarieties a full description of their subdirectly irreducible algebras is given as well. (shrink)

In his Scientific Representation. Paradoxes of Perspective, Bas van Fraassen offers a pragmatic account of scientific representation and representation tout court. In this paper I examine the three conditions for a user to succeed in representing a target in some context: identification of the target of the representational action, representing the target as such and correctly representing it in some respects. I argue that success on these three counts relies on the supposed truth of some predicative assertions, and thus that (...) truth is more fundamental than representation. I do this in the framework of a version of the so-called “structural” account of representation according to which the establishment of a homomorphism by the user between a structure abstracted from the intended target and some relevant structure of the representing artefact is a necessary condition of success for representing the target in some respects. Finally, on the basis of a correspondence view of truth, I show that it is possible to address what van Fraassen calls “the loss of reality objection”. (shrink)

A relational structure \ is called reversible iff each bijective homomorphism from \ onto \ is an isomorphism, and linear orders are prototypical examples of such structures. One way to detect new reversible structures of a given relational language L is to notice that the maximal or minimal elements of isomorphism-invariant sets of interpretations of the language L on a fixed domain X determine reversible structures. We isolate certain syntactical conditions providing that a satisfiable \-theory defines a class of (...) interpretations having extreme elements on a fixed domain and detect several classes of reversible structures. For some of these classes, we characterize the corresponding reversible extreme interpretations. In particular, we characterize the reversible countable ultrahomogeneous graphs. (shrink)