λ-calculi are of interest to logicians and computer scientists but have largely escaped philosophical commentary, perhaps because they appear narrowly technical or uncontroversial or both. I argue that even within logic λ-expressions need to be understood correctly, as functors signifying functions in intension within a categorical or typed language. λ-expressions are not names but pure viable binders generating functors, and as such they are of use in giving explicit definitions. But λ is applicable outside logic and computer science, anywhere where (...) the notions of complex whole, substitution, abstraction and structure make sense. To illustrate this, two domains are considered. One is somewhat frivolous: the study of flags; the other is very serious: manufacturing engineering. In each case we can employ λ-abstraction to describe substitutions within a structure, and in the latter case there is even a practical need for such a notation. (shrink)

Hailed by the Bulletin of the American Mathematical Society as "easy to use and a pleasure to read," this research monograph is recommended for students and professionals interested in model theory and definability theory. The sole prerequisite is a familiarity with the basics of logic, model theory, and set theory. 1974 edition.

We need not propose, as Carey does, a radical discontinuity between core cognition, which is responsible for abstract structure, and language and which are responsible for learning and conceptual change. From a probabilistic models view, conceptual structure and learning reflect the same principles, and they are both in place from the beginning.

Walter Benjamin foreshadowed many of the aesthetic theories, currently playing a fundamental role in the production and interpretation of art. By emphasising the role of the expressive character of art, or rather the category of expressivity itself, Benjamin defined art as a language. His aesthetics was characterised by the continuous interaction of two almost reciprocal projects: the theoretical critique of art which is based on an understanding of historical processes, and the understanding of historical processes which is formed by the (...) critical experience of art. We find a fundamental similarity between Benjamin’s dialectical character of the aesthetic sign and Lotman’s double-sidedness of the artwork. In classifying the system of art as a language, both theoreticians space out the structure ofart and determine it as the intersection of the synchronic and the diachronic aesthetic discourse. The paper follows the traces of the transition of modern painting from its representational status to an autonomous signification, that is, from being a symbolic expression to a discourse in the grammatological meaning of écriture. Parallel to this transition which resulted into the process of abstraction in painting, there can be observed a shift in the cultural values of art which had its critical bearing upon the world secured not by connections of likeness, but by virtue of the very independence of its values. The abstract form of the modern painting has been the declaration of the language of art as an exemplary realm. What must be expressed and experienced within this realm was (1) the critical reflection on the human condition, and (2) representing the society in so far as art maintained a moral independence from those conditions. This dialectic between the autonomous and social character of art has left deep impacts on the language of painting, a complexity, which has been made transparent through the various semiotic analytic approaches of the aesthetic sign. The paper discusses the processual character of the modern painting and demonstrates briefly the deficiency in the structural analysis of the painting language, encouraging its synthesis with the dynamical character of cultural products as we find it in the Lotmanian culture theory. (shrink)

My aim in this chapter is to extend the Realist account of the foundations of linguistics offered by Postal, Katz and others. I first argue against the idea that naive Platonism can capture the necessary requirements on what I call a ‘mixed realist’ view of linguistics, which takes aspects of Platonism, Nominalism and Mentalism into consideration. I then advocate three desiderata for an appropriate ‘mixed realist’ account of linguistic ontology and foundations, namely (1) linguistic creativity and infinity, (2) linguistics as (...) a theory of types (and not tokens) and (3) independence but structural respect between language and the linguistic competence thereof. My own brand of mixed realism, what I call ante rem realism, is defended along the lines of an ante rem or non-eliminative structuralism, the likes of which has been offered for mathematics by Resnik (1997) and Shapiro (1997). In other words, grammars describe a mind-independent (but not necessarily unconnected) linguistic reality in terms of linguistic patterns or structures also known as natural languages. I further amend this picture to allow for the possibility of a naturalistic account of language acquisition and evolution by arguing against a particular view of the type-token distinction. (shrink)

An abstract framework for structured arguments is presented, which instantiates Dung's ('On the Acceptability of Arguments and its Fundamental Role in Nonmonotonic Reasoning, Logic Programming, and n- Person Games', Artificial Intelligence , 77, 321-357) abstract argumentation frameworks. Arguments are defined as inference trees formed by applying two kinds of inference rules: strict and defeasible rules. This naturally leads to three ways of attacking an argument: attacking a premise, attacking a conclusion and attacking an inference. To resolve such attacks, (...) preferences may be used, which leads to three corresponding kinds of defeat: undermining, rebutting and undercutting defeats. The nature of the inference rules, the structure of the logical language on which they operate and the origin of the preferences are, apart from some basic assumptions, left unspecified. The resulting framework integrates work of Pollock, Vreeswijk and others on the structure of arguments and the nature of defeat and extends it in several respects. Various rationality postulates are proved to be satisfied by the framework, and several existing approaches are proved to be a special case of the framework, including assumption-based argumentation and DefLog. (shrink)

Beginning at the end of the nineteenth century, systematic scientific abstracting played a crucial role in reconfiguring the sciences on an international scale. For mathematicians, the 1931 launch of the Zentralblatt für Mathematik and 1940 launch of Mathematical Reviews marked and intensified a fundamental transformation, not just to the geographic scale of professional mathematics but to the very nature of mathematicians’ research and theories. It was not an accident that mathematical abstracting in this period coincided with an embrace across mathematical (...) research fields of a distinctive form of symbolic and conceptual abstraction. This essay examines the historical, institutional, embodied, and conceptual bases of mathematical abstracting and abstraction in the mid-twentieth century, placing them in historical context within the first half of the twentieth century and then examining their consequences and legacies for the second half of the twentieth century and beyond. Focused on scale, media, and the relationship between mathematical knowledge and its forms of articulation, my analysis connects the changing social structure of modern mathematical research communities to their changing domains of investigation and resources for representation and collective understanding. (shrink)

In this paper, I present a class of ‘structural’ abstraction principles, and describe how they are suggested by some features of Cantor's and Dedekind's approach to abstraction. Structural abstraction is a promising source of mathematically tractable new axioms for the neo-logicist. I illustrate this by showing, first, how a theorem of Shelah gives a sufficient condition for consistency in the structural setting, solving what neo-logicists call the ‘bad company’ problem for structural abstraction. Second, I show how, in the structural setting, (...) we can measure the logical strength of abstraction principles using categories of interpretations between theories. (shrink)

In this article I argue that structure in chemistry is a creature of abstraction: attending selectively to structural similarities, we neglect differences. There are different ways to abstract, so abstraction is interest dependent. So is structure. First, there are two different and mutually irreducible notions of structure in chemistry: bond structure and geometrical structure. Second, structure is relative to scale : the same substance has different structures at different scales, and relationships of structural sameness and difference vary across (...) the scales. However, these facts have no tendency to undermine structure’s claim to reality, or its metaphysical seriousness. (shrink)

We present an abstract model of rationality that focuses on structural properties of attitudes. Rationality requires coherence between your attitudes, such as your beliefs, values, and intentions. We define three 'logical' conditions on attitudes: consistency, completeness, and closedness. They parallel the familiar logical conditions on beliefs, but contrast with standard rationality conditions like preference transitivity. We establish a formal correspondence between our logical conditions and standard rationality conditions. Addressing John Broome's programme 'rationality through reasoning', we formally characterize how you (...) can (not) become more logical by reasoning. Our analysis connects rationality with logic, and enables logical talk about multi-attitude psychology. (shrink)

While we are commonly told that the distinctive method of mathematics is rigorous proof, and that the special topic of mathematics is abstract structure, there has been no agreement among mathematicians, logicians, or philosophers as to just what either of these assertions means. John P. Burgess clarifies the nature of mathematical rigor and of mathematical structure, and above all of the relation between the two, taking into account some of the latest developments in mathematics, including the rise of experimental (...) mathematics on the one hand and computerized formal proofs on the other hand. Along the way, a great many historical developments in mathematics, philosophy, and logic are surveyed. Yet very little in the way of background knowledge on the part of the reader is presupposed. (shrink)

Rhythmic structure in speech is characterized by sequences of stressed and unstressed syllables. A large body of literature suggests that speakers of English attempt to achieve rhythmic harmony by evenly distributing stressed syllables throughout prosodic phrases. The question remains as to how speakers plan metrical structure during speech production and whether it is planned independently of phonemes. To examine this, we designed a tongue twister task consisting of disyllabic word pairs with overlapping phonological segments and either matching or non‐matching metrical (...) structure. Results showed that speakers had more difficulty producing metrically regular word pairs, compared to irregular pairs; that is, word pairs with irregular meter had faster productions and fewer speech errors in this production task. This finding of metrical regularity inhibiting production is inconsistent with an abstract metrical structure that is planned independently of phonemes at the point of phonological encoding. (shrink)

L’objectif de cet article est de montrer que la réflexion engagée par Jules Vuillemin sur les structures algébriques dans les années 1960 lui permettra de dégager une conception structurale de l’abstraction, dont il tirera une compréhension nomologique de l’objectivité. Ces deux concepts seront alors convoqués bien des années plus tard pour proposer une construction de l’espace représentatif, que Vuillemin conçoit comme une réfutation de l’idéalisme matériel de Berkeley. La possibilité de retracer la genèse de nos notions spatiales et de (...) leur garantir une certaine objectivité serait ainsi rendue possible par un emprunt aux méthodes de l’algèbre moderne, conformément à l’esprit de La Philosophie de l’algèbre. (shrink)

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

The partial structures program of da Costa, French and others offers a unified framework within which to handle a wide range of issues central to contemporary philosophy of science. I argue that the program is inadequately equipped to account for simple cases where idealizations are used to construct abstract, mathematical models of physical systems. These problems show that da Costa and French have not overcome the objections raised by Cartwright and Suárez to using model-theoretic techniques in the philosophy (...) of science. However, my concerns arise independently of the more controversial assumptions that Cartwright and Suárez have employed. (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)

The partial structures program of da Costa, French and others offers a unified framework within which to handle a wide range of issues central to contemporary philosophy of science. I argue that the program is inadequately equipped to account for simple cases where idealizations are used to construct abstract, mathematical models of physical systems. These problems show that da Costa and French have not overcome the objections raised by Cartwright and Suárez to using model‐theoretic techniques in the philosophy (...) of science. However, my concerns arise independently of the more controversial assumptions that Cartwright and Suárez have employed. (shrink)

Linnebo and Pettigrew have recently developed a version of non-eliminative mathematical structuralism based on Fregean abstraction principles. They recognize that this version of structuralism is vulnerable to the well-known problem of non-rigid structures. This paper offers a solution to the problem for this version of structuralism. The solution involves expanding the languages used to describe mathematical structures. We then argue that this solution is philosophically acceptable to those who endorse mathematical structuralism based on Fregean abstraction principles.

I this paper, I draw on recent research on the radically embodied and perceptual bases of conceptualization in linguistics and cognitive science to develop a new way of reading and evaluating abstract concepts in social theory. I call this approach Sociological Idea Analysis. I argue that, in contrast to the traditional view of abstract concepts, which conceives them as amodal “presuppositions” removed from experience, abstract concepts are irreducibly grounded in experience and partake of non-negotiable perceptual-symbolic features from (...) which a non-propositional “logic” naturally follows. This implies that uncovering the imagistic bases of allegedly abstract notions should be a key part of theoretical evaluation of concepts in social theory. I provide a case study of the general category of “structure” in the social and human sciences to demonstrate the analytic utility of the approach. (shrink)

Our ability to deal with abstract concepts is one of the most intriguing faculties of human cognition. Still, we know little about how such concepts are formed, processed, and represented in mind. Current views are presented in their most recent and advanced form in this special issue, and directly compared and discussed in a lively debate, reported at the end of each chapter. The main results are reported in the editors’ introduction.

Carneades is a recently proposed formalism for structured argumentation with varying proof standards, inspired by legal reasoning, but more generally applicable. Its distinctive feature is that each statement can be given its own proof standard, which is claimed to allow a more natural account of reasoning under burden of proof than existing formalisms for structured argumentation, in which proof standards are defined globally. In this article, the two formalisms are formally related by translating Carneades into the ASPIC+ framework for structured (...) argumentation. Since ASPIC+ is defined to generate Dung-style abstract argumentation frameworks, this in effect translates Carneades graphs into abstract argumentation frameworks. For this translation, we prove a formal correspondence and show that certain rationality postulates hold. It is furthermore proved that Carneades always induces a unique Dung extension, which is the same in all of Dung's semantics, allowing us to generalise Carneades to cycle-containing structures. (shrink)

DP hypothesis of Abney (1987), the syntactic unit that had formerly been known as noun phrase should in fact be analyzed as a phrase headed by a determiner, hence the label DP. Quite independently of this syntactic development, Partee (1987) proposed a type shifting paradigm for the semantic analysis of nominals (now called DPs). In Partee's proposal DPs are ambiguous between a referential reading of type e, a predicative reading of type et and a quantificational reading of type (et)t. DP (...) meanings can flexibly move between their different readings due to covert application of semantic operators. The present paper proposes some strong relationships between these syntactic and semantic paradigms. It is argued that the structure of the DP affects its semantics in that the NP level within the DP is purely predicative and the DP level itself is purely quantificational. However, the intermediate D' level is flexible between the predicate/quantifier semantic categories, due to the covert application of semantic operators at this level. Partee's assumption, adopted from Discourse Representation Theory and more traditional approaches in philosophical logic, that some DPs need to have a (discourse) referential reading, is withdrawn. Instead of Partee's type shifting operators between the three semantic categories she assumes, two operators are used between predicates and quantifiers. The choice function operation of Reinhart (1997) and Winter (1997) is used as a general operator from predicates to quantifiers. The minimum operator of Winter (1996) is used as a general operator from quantifiers to predicates. These two operations, referred to as category shifting operators , account for most of the Partee data and substantially extend the theory of flexibility to treat some intricate phenomena in the domains of coordination, plurality and scope. (shrink)

A major objective of cognitive science is to understand the nature of the abstract representations and computational processes responsible for our ability to reason, speak, perceive, and interact with the world. In addition, a commitment to a materialist resolution of the mind–body problem requires that we search for the manner in which these representations and processes are neurally instantiated in the brain. Given this dual aim, one might proceed in one of two ways: (1) from the bottom up, commencing (...) with the study of how low‐level information and computations are encoded in the neuroanatomy of the brain, in the hope of working upwards toward an understanding of the properties of higher‐level cognitive processes; (2) from the top down, using behavioral data from a variety of sources to provide an abstract characterization of cognitive processes and then utilizing these results to guide our search for the neural mechanisms of cognition. At present, our understanding of the neural basis of higher‐level cognition is virtually negligible. Thus, it is probably fair to say that a contemporary study of any of the domains discussed in part II of this volume may be carried out only in a top‐down fashion. (shrink)

The abstract found at the beginning of most journal articles has increasingly become an essential part of the article. It tends to be the first part of the article to be read and, to some extent, it `sells' the article. Acquiring the skills of writing an abstract is therefore important to novice writers to enter the discourse community of their discipline. Based on 30 abstracts from three journals, the present study aims at exploring not only the rhetorical moves (...) of abstracts in the fields of applied linguistics and educational technology, but also the linguistic realizations of moves and authorial stance in different abstract moves. The results show that there are three obligatory moves in abstracts in these two disciplines — Presenting the research, Describing the methodology, and Summarizing the results. The results also indicate that a combination of certain linguistic features such as grammatical subjects, verb tense and voice can help distinguish moves in the abstract. The findings of the study have some pedagogical implications for academic writing courses for graduate students, especially students from non-English backgrounds. (shrink)

We generalise the Blok–Jónsson account of structural consequence relations, later developed by Galatos, Tsinakis and other authors, in such a way as to naturally accommodate multiset consequence. While Blok and Jónsson admit, in place of sheer formulas, a wider range of syntactic units to be manipulated in deductions (including sequents or equations), these objects are invariablyaggregatedvia set-theoretical union. Our approach is more general in that nonidempotent forms of premiss and conclusion aggregation, including multiset sum and fuzzy set union, are considered. (...) In their abstract form, thus,deductive relationsare defined as additional compatible preorderings over certain partially ordered monoids. We investigate these relations using categorical methods and provide analogues of the main results obtained in the general theory of consequence relations. Then we focus on the driving example of multiset deductive relations, providing variations of the methods of matrix semantics and Hilbert systems in Abstract Algebraic Logic. (shrink)

TWO OPPOSING VIEWS ABOUT THE RELATIONSHIP between concepts and the world can be found in the history of philosophy. One view—deriving from Immanuel Kant and endorsed by Karl Popper, among many others—claims that in forming and using concepts we structure the world. Concepts produce or increase order. Hence, the world, insofar as it is knowable by human beings, is necessarily a conceptually structured world. The second, still older view—represented by the Aristotelian tradition and by John Locke, for example—holds that concepts (...) are formed by abstracting from the particularities of the world. By leaving out the accidental or irrelevant features of particular entities, we abstract a concept as a general representative of a kind. (shrink)

The study of structure systems, an abstraction of the concept of first-order structures, is continued. Structure systems have algebraic systems as their algebraic reducts and their relational component consists of a collection of relation systems on the underlying functors. An analog of the expansion of a first-order structure by constants is presented. Furthermore, analogs of the Diagram Lemma and the Reduction Operator Lemma from the theory of equality-free first-order structures are provided in the framework of structure systems. (© (...) 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim). (shrink)

Abstract: Laws of computer science are prescriptive in nature but can have descriptive analogs in the physical sciences. Here, we describe a law of conservation of information in network programming, and various laws of computational motion (invariants) for programming in general, along with their pedagogical utility. Invariants specify constraints on objects in abstract computational worlds, so we describe language and data abstraction employed by software developers and compare them to Floridi's concept of levels of abstraction. We also consider (...) Floridi's structural account of reality and its fit for describing abstract computational worlds. Being abstract, such worlds are products of programmers' creative imaginations, so any "laws" in these worlds are easily broken. The worlds of computational objects need laws in the form of self-prescribed invariants, but the suspension of these laws might be creative acts. Bending the rules of abstract reality facilitates algorithm design, as we demonstrate through the example of search trees. (shrink)