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)

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

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

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)

his paper is an excerpt from a larger project that aims to open a new pathway into Spinoza's Ethics by formally reconstructing an initial fragment of this text. The semantic backbone of the project is a custom-made Spinozian model theory that lays out some of the formal prerequisites for more ne-grained investigations into Spinoza's fundamental ontology and modal metaphysics. We implement Spinoza's theory of attributes using many-sorted models with a rich system of identity that allows us to (...) clarify the puzzling status of such logical principles as the Substitution of Identicals and Transitivity of Identity in Spinoza's thought. The intensional structure of our Spinozian models also captures his proposal that states of a airs can be necessitated or excluded by the essences of particular things, an essence-relative modality that should be of interest to philosophers who have sought to rehabilitate the concept of essence in contemporary analytic metaphysics. (shrink)

Provability, Computability and Reflection.

Model theory is the branch of mathematical logic looking at the relationship between mathematical structures and logic languages. These formal languages are free from the ambiguities of natural languages, and are becoming increasingly important in areas such as computing, philosophy and linguistics. This book provides a clear introduction to the subject for both mathematicians and the non-specialists now needing to learn some model theory.

he model theory of groups of unitriangular matrices over rings is studied. An important tool in these studies is a new notion of a quasiunitriangular group. The models of the theory of all unitriangular groups are algebraically characterized; it turns out that all they are quasiunitriangular groups. It is proved that if R and S are domains or commutative associative rings then two quasiunitriangular groups over R and S are isomorphic only if R and S are isomorphic (...) or antiisomorphic. This algebraic result is new even for ordinary unitriangular groups. The groups elementarily equivalent to a single unitriangular group UTn are studied. If R is a skew field, they are of the form UTn, for some S ≡ R. In general, the situation is not so nice. Examples are constructed demonstrating that such a group need not be a unitriangular group over some ring; moreover, there are rings P and R such that UTn ≡ UTn, but UTn cannot be represented in the form UTn for S ≡ R. We also study the number of models in a power of the theory of a unitriangular group. In particular, we prove that, for any communicative associative ring R and any infinite power λ, I = I). We construct an associative ring such that I = 3 and I) = 2. We also study models of the theory of UTn in the case of categorical R. (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.

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)

The current paper studies the formal properties of the Global Reflection Principle, to wit the assertion “All theorems of$\mathrm {Th}$are true,” where$\mathrm {Th}$is a theory in the language of arithmetic and the truth predicate satisfies the usual Tarskian inductive conditions for formulae in the language of arithmetic. We fix the gap in Kotlarski’s proof from [15], showing that the Global Reflection Principle for Peano Arithmetic is provable in the theory of compositional truth with bounded induction only ($\mathrm {CT}_0$). (...) Furthermore, we extend the above result showing that$\Sigma _1$-uniform reflection over a theory of uniform Tarski biconditionals ($\mathrm {UTB}^-$) is provable in$\mathrm {CT}_0$, thus answering the question of Beklemishev and Pakhomov [2]. Finally, we introduce the notion of a prolongable satisfaction class and use it to study the structure of models of$\mathrm {CT}_0$. In particular, we provide a new model-theoretical characterization of theories of finite iterations of uniform reflection and present a new proof characterizing the arithmetical consequences of$\mathrm {CT}_0$. (shrink)

The purpose of this paper is to outline some recent progress in descriptive inner model theory, a branch of set theory which studies descriptive set theoretic and inner model theoretic objects using tools from both areas. There are several interlaced problems that lie on the border of these two areas of set theory, but one that has been rather central for almost two decades is the conjecture known as the Mouse Set Conjecture. One particular motivation (...) for resolving MSC is that it provides grounds for solving the inner model problem which dates back to 1960s. There have been some new partial results on MSC and the methods used to prove the new instances suggest a general program for solving the full conjecture. It is then our goal to communicate the ideas of this program to the community at large. (shrink)

In this paper we set out the basic model theory of differential fields of characteristic 0, which have finitely many commuting derivations. We give axioms for the theory of differentially closed differential fields with m derivations and show that this theory is ω-stable, model complete, and quantifier-eliminable, and that it admits elimination of imaginaries. We give a characterization of forking and compute the rank of this theory to be ω m + 1.

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)

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 study model theory of fields with actions of a fixed finite group scheme. We prove the existence and simplicity of a model companion of the theory of such actions, which generalizes our previous results about truncated iterative Hasse–Schmidt derivations [13] and about Galois actions [14]. As an application of our methods, we obtain a new model complete theory of actions of a finite group on fields of finite imperfection degree.

We study the model theory of fields k carrying a henselian valuation with real closed residue field. We give a criteria for elementary equivalence and elementary inclusion of such fields involving the value group of a not necessarily definable valuation. This allows us to translate theories of such fields to theories of ordered abelian groups, and we study the properties of this translation. We also characterize the first-order definable convex subgroups of a given ordered abelian group and prove (...) that the definable real valuation rings of k are in correspondence with the definable convex subgroups of the value group of a certain real valuation of k. (shrink)

George Lakoff (in his book Women, Fire, and Dangerous Things(1987) and the paper "Cognitive semantics" (1988)) champions some radical foundational views. Strikingly, Lakoff opposes realism as a metaphysical position, favoring instead some supposedly mild form of idealism such as that recently espoused by Hilary Putnam, going under the name "internal realism." For what he takes to be connected reasons, Lakoff also rejects truth conditional model-theoretic semantics for natural language. This paper examines an argument, given by Lakoff, against realism and (...) MTS. We claim that Lakoff's argument has very little, if any, impact for linguistic semantics. (shrink)

A Steiner triple system (STS) is a set S together with a collection B of subsets of S of size 3 such that any two elements of S belong to exactly one element of B. It is well known that the class of finite STS has a Fraïssé limit M_F. Here, we show that the theory T of M_F is the model completion of the theory of STSs. We also prove that T is not small and it (...) has quantifier elimination, TP2, NSOP1, elimination of hyperimaginaries and weak elimination of imaginaries. (shrink)

In this paper, I discuss the discovery of the DNA structure by Francis Crick and James Watson, which has provoked a large historical literature but has yet not found entry into philosophical debates. I want to redress this imbalance. In contrast to the available historical literature, a strong emphasis will be placed upon analysing the roles played by theory, model, and evidence and the relationship between them. In particular, I am going to discuss not only Crick and Watson's (...) well-known model and Franklin's x-ray diffraction pictures (the evidence) but also the less well known theory of helical diffraction, which was absolutely crucial to Crick and Watson's discovery. The insights into this groundbreaking historical episode will have consequences for the ‘new’ received view of scientific models and their function and relationship to theory and world. The received view, dominated by works by Cartwright and Morgan and Morrison ([1999]), rather than trying to put forth a ‘theory of models’, is interested in questions to do with (i) the function of models in scientific practice and (ii) the construction of models. In regard to (i), the received view locates the model (as an idealized, simplified version of the real system under investigation) between theory and the world and sees the model as allowing the application of the former to the latter. As to (ii) Cartwright has argued for a phenomenologically driven view and Morgan and Morrison ([1999]) for the ‘autonomy’ of models in the construction process: models are determined neither by theory nor by the world. The present case study of the discovery of the DNA structure strongly challenges both (i) and (ii). In contrast to claim (i) of the received view, it was not Crick and Watson's model but rather the helical diffraction theory which served a mediating purpose between the model and the x-ray diffraction pictures. In particular, Cartwright's take on (ii) is refuted by a comparison of Franklin's bottom-up approach with Crick and Watson's top-down approach in constructing the model. The former led to difficulties, which only a strong confidence in the structure incorporated in the model could circumvent. How to Get to the Structure 1.1 X-ray diffraction and its synthesis 1.2 Model building and Pauling's panache 1.3 The structure of proteins 1.3.1 A failed inference to the best explanation 1.3.2 The misleading 5.1 Å spot in proteins and how to get rid of it 1.3.3 Derived predictions from Pauling's alpha-helix of protein molecules The CCV Theory of Helical X-Ray Diffraction 2.1 The role of the CCV theory in the discovery of the DNA structure Killing the Helix 3.1 Appreciating all evidence—in vain Conclusion Epilogue: Chargaff's Ratios CiteULike Connotea Del.icio.us What's this? (shrink)

Thirty years after the conference that gave rise to The Structure of Scientific Theories, there is renewed interest in the nature of theories and models. However, certain crucial issues from thirty years ago are reprised in current discussions; specifically: whether the diversity of models in the science can be captured by some unitary account; and whether the temporal dimension of scientific practice can be represented by such an account. After reviewing recent developments we suggest that these issues can be accommodated (...) within the partial structures formulation of the semantic or model-theoretic approach. (shrink)

Specially selected articles from the Journal of Advanced Nursing have been updated where appropriate by the original author. Models, Theories and Concepts brings together international authorities in their specialist fields to consider the gaps occurring between theory and practice, as well as the evaluation of a selection of models and emerging theories.

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)

This book gives a comprehensive overview of central themes of finite model theory – expressive power, descriptive complexity, and zero-one laws – together with selected applications relating to database theory and artificial intelligence, especially constraint databases and constraint satisfaction problems. The final chapter provides a concise modern introduction to modal logic, emphasizing the continuity in spirit and technique with finite model theory. This underlying spirit involves the use of various fragments of and hierarchies within first-order, (...) second-order, fixed-point, and infinitary logics to gain insight into phenomena in complexity theory and combinatorics. The book emphasizes the use of combinatorial games, such as extensions and refinements of the Ehrenfeucht-Fraissé pebble game, as a powerful way to analyze the expressive power of such logics, and illustrates how deep notions from model theory and combinatorics, such as o-minimality and treewidth, arise naturally in the application of finite model theory to database theory and AI. Students of logic and computer science will find here the tools necessary to embark on research into finite model theory, and all readers will experience the excitement of a vibrant area of the application of logic to computer science. (shrink)

In his Ph.D. thesis [7], L. van den Dries studied the model theory of fields with finitely many orderings and valuations where all open sets according to the topology defined by an order or a valuation is globally dense according with all other orderings and valuations. Van den Dries proved that the theory of these fields is companionable and that the theory of the companion is decidable .In this paper we study the case where the fields (...) are expanded with finitely many orderings and an independent derivation. We show that the theory of these fields still admits a model companion in the language Lequation image = {+, –, ·, D, <1, …, model companion by CODFm and give a geometric axiomatization of this theory which uses basic notions of algebraic geometry and some generalized open subsets which appear naturally in this context. This axiomatization allows to recover the one given in [4] for the theory CODF of closed ordered differential fields. Most of the technics we use here are already present in [2] and [4].Finally, we prove that it is possible to describe the completions of CODFm and to obtain quantifier elimination in a slightly enriched language. This generalizes van den Dries' results in the “derivation free” case. (shrink)

This paper establishes model-theoretic properties of \, a variation of monadic first-order logic that features the generalised quantifier \. We will also prove analogous versions of these results in the simpler setting of monadic first-order logic with and without equality and \, respectively). For each logic \ we will show the following. We provide syntactically defined fragments of \ characterising four different semantic properties of \-sentences: being monotone and continuous in a given set of monadic predicates; having truth preserved (...) under taking submodels or being truth invariant under taking quotients. In each case, we produce an effectively defined map that translates an arbitrary sentence \ to a sentence \ belonging to the corresponding syntactic fragment, with the property that \ is equivalent to \ precisely when it has the associated semantic property. As a corollary of our developments, we obtain that the four semantic properties above are decidable for \-sentences. (shrink)

We explore some topics in the model theory of sheaves of modules. First we describe the formal language that we use. Then we present some examples of sheaves obtained from quivers. These, and other examples, will serve as illustrations and as counterexamples. Then we investigate the notion of strong minimality from model theory to see what it means in this context. We also look briefly at the relation between global, local and pointwise versions of properties related (...) to acyclicity. (shrink)

It is regrettably common for theorists to attempt to characterize the Humean dictum that one can’t get an ‘ought’ from an ‘is’ just in broadly logical terms. We here address an important new class of such approaches which appeal to model-theoretic machinery. Our complaint about these recent attempts is that they interfere with substantive debates about the nature of the ethical. This problem, developed in detail for Daniel Singer’s and Gillian Russell and Greg Restall’s accounts of Hume’s dictum, is (...) of a general type arising for the use of model-theoretic structures in cashing out substantive philosophical claims: the question of whether an abstract model-theoretic structure successfully interprets something often involves taking a stand on non-trivial issues surrounding the thing. In the particular case of Hume’s dictum, given reasonable conceptual or metaphysical claims about the ethical, Singer’s and Russell and Restall’s accounts treat obviously ethical claims as descriptive and vice versa. Consequently, their model-theoretic characterizations of Hume’s dictum are not metaethically neutral. This encourages skepticism about whether model-theoretic machinery suffices to provide an illuminating distinction between the ethical and the descriptive. (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.

We study the theory of a Hilbert space H as a module for a unital C*-algebra ${\mathcal{A}}$ from the point of view of continuous logic. We give an explicit axiomatization for this theory and describe the structure of all the representations which are elementary equivalent to it. Also, we show that this theory has quantifier elimination and we characterize the model companion of the incomplete theory of all non-degenerate representations of ${\mathcal{A}}$ . Finally, we show (...) that there is an homeomorphism between the space of types of norm less than 1 in this model companion, and the space of quasistates of the C*-algebra ${\mathcal{A}}$. (shrink)

About 25 years ago, it came to light that a single combinatorial property determines both an important dividing line in model theory and machine learning. The following years saw a fruitful exchange of ideas between PAC-learning and the model theory of NIP structures. In this article, we point out a new and similar connection between model theory and machine learning, this time developing a correspondence between stability and learnability in various settings of online learning. (...) In particular, this gives many new examples of mathematically interesting classes which are learnable in the online setting. (shrink)

We initiate the study of model theory in the absence of the Axiom of Choice, using the Axiom of Determinateness as a powerful substitute. We first show that, in this context, L ω 1 ω is no more powerful than first-order logic. The emphasis then turns to upward Lowenhein-Skolem theorems; ℵ 1 is the Hanf number of first-order logic, of L ω 1 ω , and of a strong fragment of L ω 1 ω . The main technical (...) innovation is the development of iterated ultrapowers using infinite supports; this requires an application of infinite-exponent partition relations. All our theorems can be proven from hypotheses weaker than AD. (shrink)

Vector spaces over unspecified fields can be axiomatized as one-sorted structures, namely, abelian groups with the relation of parallelism. Parallelism is binary linear dependence. When equipped with the n-ary relation of linear dependence for some positive integer n, a vector-space is existentially closed if and only if it is n-dimensional over an algebraically closed field. In the signature with an n-ary predicate for linear dependence for each positive integer n, the theory of infinite-dimensional vector spaces over algebraically closed fields (...) is the model-completion of the theory of vector spaces. (shrink)

I see model theory as becoming increasingly detached from set theory, and the Tarskian notion of set-theoretic model being no longer central to model theory. In much of modern mathematics, the set-theoretic component is of minor interest, and basic notions are geometric or category-theoretic. In algebraic geometry, schemes or algebraic spaces are the basic notions, with the older “sets of points in affine or projective space” no more than restrictive special cases. The basic notions (...) may be given sheaf-theoretically, or functorially. To understand in depth the historically important affine cases, one does best to work with more general schemes. The resulting relativization and “transfer of structure” is incomparably more flexible and powerful than anything yet known in “set-theoretic model theory”.It seems to me now uncontroversial to see the fine structure of definitions as becoming the central concern of model theory, to the extent that one can easily imagine the subject being called “Definability Theory” in the near future.Tarski's set-theoretic foundational formulations are still favoured by the majority of model-theorists, and evolution towards a more suggestive language has been perplexingly slow. None of the main texts uses in any nontrivial way the language of category theory, far less sheaf theory or topos theory. Given that the most notable interactions of model theory with geometry are in areas of geometry where the language of sheaves is almost indispensable, this is a curious situation, and I find it hard to imagine that it will not change soon, and rapidly. (shrink)

We prove some results about the model theory of fields with a derivation of the Frobenius map, especially that the model companion of this theory is axiomatizable by axioms used by Wood in the case of the theory $\operatorname {DCF}_p$ and that it eliminates quantifiers after adding the inverse of the Frobenius map to the language. This strengthens the results from [4]. As a by-product, we get a new geometric axiomatization of this model companion. (...) Along the way we also prove a quantifier elimination result, which holds in a much more general context and we suggest a way of giving “one-dimensional” axiomatizations for model companions of some theories of fields with operators. (shrink)

Take a formula of first-order logic which is a logical consequence of some other formulae according to model theory, and in all those formulae replace schematic letters with English expressions. Is the argument resulting from the replacement valid in the sense that the premisses could not have been true without the conclusion also being true? Can we reason from the model-theoretic concept of logical consequence to the modal concept of validity? Yes, if the model theory (...) is the standard one for sentential logic; no, if it is the standard one for the predicate calculus; and yes, if it is a certain model theory for free logic. These conclusions rely inter alia on some assumptions about possible worlds, which are mapped into the models of model theory. Plural quantification is used in the last section, while part of the reasoning is relegated to an appendix that includes a proof of completeness for a version of free logic. (shrink)