Stemming from the works of Petr Hájek on mathematical fuzzy logic, graded model theory has been developed by several authors in the last two decades as an extension of classical model theory that studies the semantics of many-valued predicate logics. In this paper we take the first steps towards an abstract formulation of this model theory. We give a general notion of abstract logic based on many-valued models and prove six Lindström-style characterizations of (...) maximality of first-order logics in terms of metalogical properties such as compactness, abstract completeness, the Löwenheim–Skolem property, the Tarski union property, and the Robinson property, among others. As necessary technical restrictions, we assume that the models are valued on finite MTL-chains and the language has a constant for each truth-value. (shrink)

The ability to reason and think in a logical manner forms the basis of learning for most mathematics, computer science, philosophy and logic students. Based on the author's teaching notes at the University of Maryland and aimed at a broad audience, this text covers the fundamental topics in classical logic in an extremely clear, thorough and accurate style that is accessible to all the above. Covering propositional logic, first-order logic, and second-order logic, as well as proof theory, computability (...) class='Hi'>theory, and model theory, the text also contains numerous carefully graded exercises and is ideal for a first or refresher course. (shrink)

In this paper we consider a claim that in the natural world there is no fact of the matter about the spatio-temporal separation of events. In order to make sense of such a notion and construct useful models of the world, it is proposed to use elements of a non-classical logic. Specifically, we focus here on causality, as a concept tightly related with the assumption of there being distinct, separate events, proposing a model according to which it can be (...) considered to be spatio-temporally graded. It is outlined how this can be described using the formalism of fuzzy sets theory, with the degree of causality varying between 1, that is no separation between causes and effects, and 0, that is perfect separation between causes and their effects as in classical ’billiard balls’ models of physical systems, namely such based on the notion of ideal mathematical point. Our model posits that subjective moments of time are like fuzzy sets, with their extension determined by local degrees of causality, resulting from information integration processes extended gradually in space and time. This, we argue, is how a notion of causality could be, to a certain degree, spared and reconciled with a variant of Bergsonian duration theory as formulated in the theory of continuous change. Relation of the proposed viewpoint to other theories, as well as possible solutions it suggests to various problems, in particular the measurement problem, are also discussed. (shrink)

Admiration is often described as having a singular motivational profile – the disposition to imitate. This article provides a developmental assessment of admiration’s action-potential, proposing a series of stages between (1) naïve imitation, a basic mimetic impulse, and (2) non-imitative virtuous actions. The process is marked by an increasing ability to represent the actions and desires of another, becoming the middle term between the learner and the exemplar. This developmental assessment is necessary because the leading accounts of moral development today (...) lean on the idea of imitation as essential to the process of virtue acquisition without providing an explanation of how imitation works, psychologically speaking. Moreover, these accounts treat imitation as a static disposition, rather than one that matures over time. Insight regarding this developmental progression can provide us with a better sense of how to educate using exemplars in order to advance a learner from admiration to moral virtue. This article also fills in gaps in the admiration literature concerning how we regard inimitable excellences and contends that it may not be beneficial to emulate an exemplar’s motivations, in addition to her actions. (shrink)

There are several relations which may fall short of genuine identity, but which behave like identity in important respects. Such grades of discrimination have recently been the subject of much philosophical and technical discussion. This paper aims to complete their technical investigation. Grades of indiscernibility are defined in terms of satisfaction of certain first-order formulas. Grades of symmetry are defined in terms of symmetries on a structure. Both of these families of grades of discrimination have been studied in some detail. (...) However, this paper also introduces grades of relativity, defined in terms of relativeness correspondences. This paper explores the relationships between all the grades of discrimination, exhaustively answering several natural questions that have so far received only partial answers. It also establishes which grades can be captured in terms of satisfaction of object-language formulas and draws connections with definability theory. (shrink)

Aristotle is oftentimes viewed through a strictly philosophical lens as heir to Plato and has having introduced logical rigor where an emphasis on the theory of Forms formerly prevailed. It must be appreciated that Aristotle was the son of a physician, and that his inculcation of the thought of other Greek philosophers addressing health and the natural elements led to an extremely broad set of biologically- and medically-related writings. As this article proposes, Aristotle deepened the fourfold theory of (...) the elements with anatomic and physiologic observations. In books like History of Animals, Parts of Animals, and the lost Anatomai, he actively dissected organisms and recorded his findings. The corpus of medically-related literature Aristotle developed had a direct influence on subsequent Greek thinkers, including Galen, and on Medieval Islamic and modern Western practitioners such as William Harvey. Aristotle's ideas continue to influence modern medical thinkers in Europe and America through both his interpretation by European philosophers gaining the attention of medical humanists, and his writings' enduring impact on medical researchers balancing scientific with more personalistic approaches to medicine. (shrink)

This paper addresses theoretical problems arising from the vagueness of language terms, and intuitions of the vagueness of the concepts to which they refer. It is argued that the central intuitions of prototype theory are sufficient to account for both typicality phenomena and psychological intuitions about degrees of membership in vaguely defined classes. The first section explains the importance of the relation between degrees of membership and typicality (or goodness of example) in conceptual categorization. The second and third section (...) address arguments advanced by Osherson and Smith (1997), and Kamp and Partee (1995), that the two notions of degree of membership and typicality must relate to fundamentally different aspects of conceptual representations. A version of prototype theory—the Threshold Model—is proposed to counter these arguments and three possible solutions to the problems of logical selfcontradiction and tautology for vague categorizations are outlined. In the final section graded membership is related to the social construction of conceptual boundaries maintained through language use. (shrink)

We study classes of graded structures satisfying the properties of amalgamation, joint embedding and hereditariness. Given appropriate conditions, we can build a graded analogue of the Fraïssé limit. Some examples such as the class of all finite weighted graphs or the class of all finite fuzzy orders (evaluated on a particular countable algebra) will be examined.

The estrangement between genetic scientists and theologians originating in the 1960s is reflected in novel combinations of human thought (subject) and genes (investigational object), paralleling each other through the universal process known in chaos theory as self-similarity. The clash and recombination of genes and knowledge captures what Philip Hefner refers to as irony, one of four voices he suggests transmit the knowledge and arguments of the religion-and-science debate. When viewed along a tangent connecting irony to leadership, journal dissemination, and (...) the activities of the "public intellectual" and the public at large, the sequence of voices is shown to resemble the passage of genetic information from DNA to mRNA, tRNA, and protein, and from cell nucleus to surrounding environment. In this light, Hefner's inquiry into the voices of Zygon is bound up with the very subject matter Zygon covers. (shrink)

It has become increasingly common to endorse and implement anonymous grading as a way of promoting fairness or equality of opportunity in the classroom. The American Philosophical Association currently recommends anonymous grading, as do the Canadian Philosophical Association, the British Philosophical Association, the Society for Women in Philosophy, and Minorities and Philosophy. Despite its increasing prevalence, the practice has received surprisingly little attention in applied ethics. This paper begins filling this gap. I start by clarifying the ‘Standard Argument’ from fairness (...) or equality of opportunity for anonymous grading. I then argue that the Standard Argument fails to support current models for anonymous grading. The issue: current models for anonymous grading are practically self-undermining if the aim is to reduce violations of equality of opportunity or fairness. So, if we want to address bias in grading because we care about equality of opportunity or fairness, we need to change how we do it. (shrink)

We introduce a notion of bisimulation for graded modal logic. Using this notion, the model theory of graded modal logic can be developed in a uniform manner. We illustrate this by establishing the finite model property and proving invariance and definability results.

Coherence and correspondence are classical contenders as theories of truth. In this paper we examine them instead as interacting factors in the dynamics of belief across epistemic networks. We construct an agent-based model of network contact in which agents are characterized not in terms of single beliefs but in terms of internal belief suites. Individuals update elements of their belief suites on input from other agents in order both to maximize internal belief coherence and to incorporate ‘trickled in’ elements (...) of truth as correspondence. Results, though often intuitive, prove more complex than in simpler models (Hegselmann & Krause 2002, 2006; Grim, Singer, Reade & Fisher 2015). The optimistic finding is that pressures toward internal coherence can exploit and expand on small elements of truth as correspondence is introduced into epistemic networks. Less optimistic results show that pressures for coherence can also work directly against the incorporation of truth, particularly when coherence is established first and new data is introduced later. (shrink)

Bringing Human Rights Education to US Classrooms presents ten research-based human rights projects powerfully implemented in a range of U.S. classrooms, from elementary school through community college and university. In these classrooms, the students--primarily young people of color who have experienced or witnessed human rights abuses such as discrimination and poverty--are exposed for the first time to thinking about their own lives and the world through an empowering human rights lens. Unique in integrating theory and classroom practice, and in (...) addressing human rights issues with special relevance for communities of color in the US, Katz and Spero provide indispensable guidance for those studying and teaching human rights. (shrink)

Adding certain cardinality quantifiers into first-order language will give substantially more expressive languages. Thus, many mathematical concepts beyond first-order logic can be handled. Since basic modal logic can be seen as the bisimular invariant fragment of first-order logic on the level of models, it has no ability to handle modally these mathematical concepts beyond first-order logic. By adding modalities regarding the cardinalities of successor states, we can, in principle, investigate modal logics of all cardinalities. Thus ways of exploring model-theoretic (...) logics can be transferred to modal logics. (shrink)

Adding certain cardinality quantifiers into first-order language will give substantially more expressive languages. Thus, many mathematical concepts beyond first-order logic can be handled. Since basic modal logic can be seen as the bisimular invariant fragment of first-order logic on the level of models, it has no ability to handle modally these mathematical concepts beyond first-order logic. By adding modalities regarding the cardinalities of successor states, we can, in principle, investigate modal logics of all cardinalities. Thus ways of exploring model-theoretic (...) logics can be transferred to modal logics. (shrink)

This book contains the material for a first course in pure model theory with applications to differentially closed fields.

Provability, Computability and Reflection.

We introduce the subject of modal model theory, where one studies a mathematical structure within a class of similar structures under an extension concept that gives rise to mathematically natural notions of possibility and necessity. A statement φ is possible in a structure (written φ) if φ is true in some extension of that structure, and φ is necessary (written φ) if it is true in all extensions of the structure. A principal case for us will be the (...) class Mod(T) of all models of a given theory T—all graphs, all groups, all fields, or what have you—considered under the substructure relation. In this article, we aim to develop the resulting modal model theory. The class of all graphs is a particularly insightful case illustrating the remarkable power of the modal vocabulary, for the modal language of graph theory can express connectedness, k-colorability, finiteness, countability, size continuum, size ℵ1, ℵ2, ℵω, ℶω, first ℶ-fixed point, first ℶ-hyper-fixed point, and much more. A graph obeys the maximality principle φ(a)→φ(a) with parameters if and only if it satisfies the theory of the countable random graph, and it satisfies the maximality principle for sentences if and only if it is universal for finite graphs. (shrink)

As a branch of statistical latent variable modeling, multidimensional item response theory plays an important role in psychometrics. Multidimensional graded response model is a key model for the development of multidimensional computerized adaptive testing with graded-response data and multiple traits. This paper explores how to automatically identify the item-trait patterns of replenished items based on the MGRM in MCAT. The problem is solved by developing an exploratory pattern recognition method for graded-response items based on (...) the least absolute shrinkage and selection operator, which is named LPRM-GR and facilitates the subsequent parameter estimation of replenished items and helps maintaining the effectiveness of item replenishment in MCAT. In conjunction with the proposed approach, the regular BIC and weighted BIC are applied, respectively, to select the optimal item-trait patterns. Simulation for evaluating the LPRM-GR in pattern recognition accuracy of replenished items and the corresponding item estimation accuracy is conducted under multiple conditions across different numbers with respect to dimensionality, response-category numbers, latent trait correlation, stopping rules, and item selection criteria. Results show that the proposed method with the two types of BIC both have good performance in pattern recognition for item replenishment in the two- to four-dimensional MCAT with the MGRM, for which the weighted BIC is generally superior to the regular BIC. The proposed method has relatively high accuracy and efficiency in identifying the patterns of graded-response items, and has the advantages of easy implementation and practical feasibility. (shrink)

What structure do paradigm cases of oppression and domination, like racism and capitalism, have? Most theories of oppression and domination take them to have a binary structure. There are the oppressors and the oppressed, the dominators and the dominated. I argue that a better model for many paradigm cases of oppression and domination is a structure of graded inequality. Such a structure comprises multiple groups arranged in hierarchically ascending and descending order. A model of graded inequality (...) has both descriptive and normative advantages over a binary model. It better illuminates some mechanisms by which oppression and domination stabilize themselves. Understanding those mechanisms allows us better to destabilize them. Yet there still may be, I suggest, some good reasons for adopting the binary model in certain situations for certain purposes. The larger methodological upshot of the paper is to connect more closely social philosophy to moral psychology. (shrink)

One of the most difficult challenges of mature legal systems is the need to balance the conflicting demands of stability and flexibility. The demand for flexibility is at odds with the principle of impartiality, which is considered a cornerstone of the rule of law. In the present article, I explore the way in which the law copes with this dilemma by developing the idea of tolerance of incoherence. I argue that tolerance of incoherence emerges from the interplay between the inferential (...) and lexical-semantic rules that determine the meaning of legal speech acts. I base this argument on an inferential model of speech acts, which I develop through a discussion of graded speech acts, and on the idea that the use of speech acts is governed by multiple and potentially conflicting conventions. I show how this tolerance allows the law to resolve the tension between dynamism and traditionality, and discuss its sociological and moral implications. (shrink)

An algebraization of the notion of topology has been proposed more than 70 years ago in a classical paper by McKinsey and Tarski, leading to an area of research still active today, with connections to algebra, geometry, logic and many applications, in particular, to modal logics. In McKinsey and Tarski’s setting the model theoretical notion of homomorphism does not correspond to the notion of continuity. We notice that the two notions correspond if instead we consider a preorder relation \( (...) \sqsubseteq \) defined by \(a \sqsubseteq b\) if _a_ is contained in the topological closure of _b_, for _a_, _b_ subsets of some topological space. A _specialization poset_ is a partially ordered set endowed with a further coarser preorder relation \( \sqsubseteq \). We show that every specialization poset can be embedded in the specialization poset naturally associated to some topological space, where the order relation corresponds to set-theoretical inclusion. Specialization semilattices are defined in an analogous way and the corresponding embedding theorem is proved. Specialization semilattices have the amalgamation property. Some basic topological facts and notions are recovered in this apparently very weak setting. The interest of these structures arises from the fact that they also occur in many rather disparate contexts, even far removed from topology. (shrink)

It is widely thought that causal cognition underpins technical reasoning. Here we suggest that understanding causal cognition as a thinking system that includes theory of mind (i.e., social cognition) can be a productive theoretical tool for the field of evolutionary cognitive archaeology. With this contribution, we expand on an earlier model that distinguishes seven grades of causal cognition, explicitly presenting it together with a new analysis of the theory of mind involved in the different grades. We then (...) suggest how such thinking may manifest in the archaeological or stone tool record and techno-behaviors of the last three million years or so. Our thesis is threefold: (a) theory of mind is an integral element of causal cognition; (b) generally speaking, the more advanced causal cognition is, the more it is dependent on theory of mind; and (c) the evolution of causal cognition depends more and more on mental representations of hidden variables. Ultimately, the final or seventh grade of causal cognition allows us to reason from a network of hidden variables that, amongst other things, enables the learning, manufacture, and use of complex technological systems. It also facilitates the seamless mapping of knowledge between personal (egocentric), physical, and social networks that allows for newly devised and innovative technical and social outcomes. (shrink)

The semantic view on theories has been much in vogue over four decades as the successor of the syntactic view. In the present paper, I take issue with this approach by arguing that theories and models must be separated and that a theory should be considered to be a linguistic systems consisting of a vocabulary and a set of rules for the use of that vocabulary.

Most words in natural languages are polysemous; that is, they have related but different meanings in different contexts. This one‐to‐many mapping of form to meaning presents a challenge to understanding how word meanings are learned, represented, and processed. Previous work has focused on solutions in which multiple static semantic representations are linked to a single word form, which fails to capture important generalizations about how polysemous words are used; in particular, the graded nature of polysemous senses, and the flexibility (...) and regularity of polysemy use. We provide a novel view of how polysemous words are represented and processed, focusing on how meaning is modulated by context. Our theory is implemented within a recurrent neural network that learns distributional information through exposure to a large and representative corpus of English. Clusters of meaning emerge from how the model processes individual word forms. In keeping with distributional theories of semantics, we suggest word meanings are generalized from contexts of different word tokens, with polysemy emerging as multiple clusters of contextually modulated meanings. We validate our results against a human‐annotated corpus of polysemy focusing on the gradedness, flexibility, and regularity of polysemous sense individuation, as well as behavioral findings of offline sense relatedness ratings and online sentence processing. The results provide novel insights into how polysemy emerges from contextual processing of word meaning from both a theoretical and computational point of view. (shrink)

It is widely thought that causal cognition underpins technical reasoning. Here we suggest that understanding causal cognition as a thinking system that includes theory of mind (i.e., social cognition) can be a productive theoretical tool for the field of evolutionary cognitive archaeology. With this contribution, we expand on an earlier model that distinguishes seven grades of causal cognition, explicitly presenting it together with a new analysis of the theory of mind involved in the different grades. We then (...) suggest how such thinking may manifest in the archaeological or stone tool record and techno-behaviors of the last three million years or so. Our thesis is threefold: (a) theory of mind is an integral element of causal cognition; (b) generally speaking, the more advanced causal cognition is, the more it is dependent on theory of mind; and (c) the evolution of causal cognition depends more and more on mental representations of hidden variables. Ultimately, the final or seventh grade of causal cognition allows us to reason from a network of hidden variables that, amongst other things, enables the learning, manufacture, and use of complex technological systems. It also facilitates the seamless mapping of knowledge between personal (egocentric), physical, and social networks that allows for newly devised and innovative technical and social outcomes. (shrink)

Therefore, the text is divided into three parts: an introduction into mathematical logic (Chapter 1), model theory (Chapters 2 and 3), and the model theoretic ...

The philosophy of linguistics is a rich philosophical domain which encompasses various disciplines. One of the aims of this thesis is to unite theoretical linguistics, the philosophy of language, the philosophy of science and the ontology of language. Each part of the research presented here targets separate but related goals with the unified aim of bringing greater clarity to the foundations of linguistics from a philosophical perspective. Part I is devoted to the methodology of linguistics in terms of scientific modelling. (...) I argue against both the Conceptualist and Platonist interpretations of linguistic theory by means of three grades of mathematical involvement for linguistic grammars. Part II explores the specific models of syntactic and semantics by an analogy with the harder sciences. In Part III, I develop a novel account of linguistic ontology and in the process comment on the type-token distinction, the role and connection with mathematics and the nature of linguistic objects. In this research, I offer a structural realist interpretation of linguistic methodology with a nuanced structuralist picture for its ontology. This proposal is informed by historical and current work in theoretical linguistics as well as philosophical views on ontology, scientific modelling and mathematics. (shrink)

CONTINUOUS MODEL THEORY CHAPTER I TOPOLOGICAL PRELIMINARIES. Notation Throughout the monograph our mathematical notation does not differ drastically from ...

Philosophy and model theory frequently meet one another. Philosophy and Model Theory aims to understand their interactions -/- Model theory is used in every ‘theoretical’ branch of analytic philosophy: in philosophy of mathematics, in philosophy of science, in philosophy of language, in philosophical logic, and in metaphysics. But these wide-ranging appeals to model theory have created a highly fragmented literature. On the one hand, many philosophically significant mathematical results are found only in (...) mathematics textbooks: these are aimed squarely at mathematicians; they typically presuppose that the reader has a significant background in mathematics; and little clue is given as to their philosophical significance. On the other hand, the philosophical applications of these results are then scattered across disconnected pockets of papers, separated by decades or more. -/- The first aim of Philosophy and Model Theory, then, is to consider the philosophical uses of model theory. On a technical level, we try to show how philosophically significant results connect to one another, and also to state the best version of a result for philosophical purposes. On a philosophical level, we show how similar dialectical situations arise repeatedly, across fragmented debates in varied philosophical areas. -/- The second aim of Philosophy and Model Theory, though, is to consider the philosophy of model theory. Model theory itself is rarely taken as the subject matter of philosophising (contrast this, say, with the philosophy of biology, or the philosophy of set theory). But model theory is a beautiful part of pure mathematics, and worthy of philosophical study in its own right. (shrink)

Applying the graded response model within the item response theory framework, the present study analyzes the psychometric properties of Karwowski’s creative self-efficacy scale. With an ethnically diverse sample of US college students, the results suggested that the six items of the CSE scale were well fitted to a latent unidimensional structure. The scale also had adequate measurement precision or reliability, high levels of item discrimination, and an appropriate range of item difficulty. Gender-based differential item functioning analyses confirmed (...) that there were no differences in the measurement results of the scale concerning gender. Additionally, openness to experience was found to be positively related to the CSE scale scores, providing some support for the scale’s convergent validity. Collectively, these results confirmed the psychometric soundness of the CSE scale for measuring CSE and also identified avenues for future research. (shrink)

Finite model theory is currently not one of the hot topics in the philosophy and history of mathematics, not even in the philosophy and history of mathematical logic. The philosophy of mathematics and mathematical logic has concentrated on infinite structures, closely related to foundational issues. In that context, finite models deserved only marginal attention because it was taken for granted that the study of finite structures is trivial compared to the study of infinite structures. In retrospect, research on (...) finite structures turned out to be neither trivial nor irrelevant but had an important impact on theoretical computer science. A closer look at the early history of the spectrum problem shows that the birth of finite model theory has roots in formal logic, at a time when computer science did not yet exist. (shrink)

We continue the analysis of foundations of positive model theory as introduced by Ben Yaacov and Poizat. The objects of this analysis are $h$-inductive theories and their models, especially the “positively” existentially closed ones. We analyze topological properties of spaces of types, introduce forms of quantifier elimination, and characterize minimal completions of arbitrary $h$-inductive theories. The main technical tools consist of various forms of amalgamations in special classes of structures.

Major shifts in the field of model theory in the twentieth century have seen the development of new tools, methods, and motivations for mathematicians and philosophers. In this book, John T. Baldwin places the revolution in its historical context from the ancient Greeks to the last century, argues for local rather than global foundations for mathematics, and provides philosophical viewpoints on the importance of modern model theory for both understanding and undertaking mathematical practice. The volume also (...) addresses the impact of model theory on contemporary algebraic geometry, number theory, combinatorics, and differential equations. This comprehensive and detailed book will interest logicians and mathematicians as well as those working on the history and philosophy of mathematics. (shrink)

We develop positive model theory, which is a non first order analogue of classical model theory where compactness is kept at the expense of negation. The analogue of a first order theory in this framework is a compact abstract theory: several equivalent yet conceptually different presentations of this notion are given. We prove in particular that Banach and Hilbert spaces are compact abstract theories, and in fact very well-behaved as such.

We shift attention from the development of model theory for demarcated languages to the development of this theory for fragments of a language. Although it is often assumed that model theory for demarcated languages is not compatible with a universalist conception of logic, no one has denied that model theory for fragments of a language can be compatible with that conception. It thus seems unwarranted to ignore the universalist tradition in the search for (...) the origins and development of model theory. This point is illustrated by Carnap’s early semantics and model theory, which he developed within a type theoretical framework and which stand out both for their universalistic treatment and for certain idiosyncratic technicalities by which the construction is supported. One special property is that individuals are context relative in Carnap’s system. This leads to a model theory in which the model domains are more flexible than has been suggested in the literature. (shrink)