We introduce and study the model-theoretic notions of absolute connectedness and type-absolute connectedness for groups. We prove that groups of rational points of split semisimple linear groups (that is, Chevalley groups) over arbitrary infinite fields are absolutely connected and characterize connected Lie groups which are type-absolutely connected. We prove that the class of type-absolutely connected group is exactly the class of discretely topologized groups with the trivial Bohr compactification, that is, the class of minimally almost periodic groups.

This book brings together several directions of work in model theory between the late 1950s and early 1980s.

An introduction to the model-theoretic approach in the philosophy of science is given and it is argued that this program is further enhanced by the introduction of partial structures. It is then shown that this leads to a natural and intuitive account of both "iconic" and mathematical models and of the role of the former in science itself.

An introduction to the model-theoretic approach in the philosophy of science is given and it is argued that this program is further enhanced by the introduction of partial structures. It is then shown that this leads to a natural and intuitive account of both "iconic" and mathematical models and of the role of the former in science itself.

We study model theoretic tree properties and their associated cardinal invariants. In particular, we obtain a quantitative refinement of Shelah’s theorem for countable theories, show that [Formula: see text] is always witnessed by a formula in a single variable and that weak [Formula: see text] is equivalent to [Formula: see text]. Besides, we give a characterization of [Formula: see text] via a version of independent amalgamation of types and apply this criterion to verify that some examples in the literature are (...) indeed [Formula: see text]. (shrink)

Recent discussions of the nature of representation in science have tended to import pre-established decompositions from analyses of representation in the arts, language, cognition and so forth. Which of these analyses one favours will depend on how one conceives of theories in the first place. If one thinks of them in terms of an axiomatised set of logico-linguistic statements, then one might be naturally drawn to accounts of linguistic representation in which notions of denotation, for example, feature prominently. If, on (...) the other hand, one conceives of theories in non-linguistic terms, as in the model-theoretic approach, then one might look to analyses of representation in the arts where notions of resemblance tend to be brought to the fore. Thus van Fraassen, for example, has imported such an analysis into his discussion of representation in science and argued that an appropriate account of resemblance can be given in terms of the set-theoretic relation of isomorphism. This has been strongly criticised by Suarez, who argues that just as isomorphism cannot capture representation in art, so it is inappropriate in the scientific context as well. Similarly Hughes draws on Goodman`s rejection of resemblance in art in favour of denotation and, rather confusingly perhaps, favours the latter whilst also maintaining the model-theoretic view of theories. In this paper, I shall examine the debate in terms of four claims: 1. Isomorphism is not sufficient for representation; 2. Isomorphism is not necessary for representation; 3. Models represent but theories do not; 4. Models denote and do not resemble. Each of these claims will be questioned and I will conclude by suggesting that, through appropriate modifications, a form of isomorphism can serve to underpin representation in both the arts and science. (shrink)

In this paper I investigate Putnam’s model-theoretic argument from a transcendent standpoint, in spite of Putnam’s well-known objections to such a standpoint. This transcendence, however, requires ascent to something more like a Tarskian meta-level than what Putnam regards as a “God’s eye view”. Still, it is methodologically quite powerful, leading to a significant increase in our investigative tools. The result is a shift from Putnam’s skeptical conclusion to a new understanding of realism, truth, correspondence, knowledge, and theories, or certain aspects (...) thereof, based on, among other things, a better understanding of what models are designed (and not designed) to do. (shrink)

My model-theoretic realist account of science places linguistic systems and the corresponding non-linguistic structures at different stages of the scientific process. It is shown that science and its progress cannot be analysed in terms of only one of these strata. Philosophy of science literature offers mainly two approaches; to the structure of scientific knowledge analysed in terms of theories and their models, the "statement" and the "non-statement" approaches. In opposition to the statement approach's belief that scientific knowledge is embodied in (...) theories symbolic language) with direct interpretative links---via so-called "bridge principles"---to reality, the defenders of the non-statement approach believe in an analysis where the language in which the theory is formulated plays a much smaller role than the structures which satisfy that theory. ;The model-theoretic realism expounded here retains the notion of a scientific theory as a set of sentences, while simultaneously emphasising the interpretative role of the conceptual models of these theories. My criticism against the non-statement approach is based on the fact that merely "giving" the theory "in terms of" its mathematical structures leaves out any real interpretation of the nature and role of general terms in science. Against the statement approach's "direct" linking of general theoretical terms to reality, my approach interpolates models between theories and reality in the interpretative chain. ;The links between the general terms of scientific theories and their interpretations in the various models of the theory regulate the whole referential process. The terms of a theory are "general" in the sense that they are the result of certain abstractive conceptualisations of the object of scientific investigation and subsequent linguistic formulations of these conceptualisations. Their meanings can be "given back" only by interpreting them in the limited context, of the various conceptual models of their theory and, finally, by finding an isomorphic relation between some substructure of the conceptual model in question and some empirical conceptualisation of relevant experimental data. In this sense the notion of scientific "truth" becomes inextricably linked with that of articulated reference, as it---given its model-dependent nature---should be. (shrink)

In "Realism and Reason" Hilary Putnam has offered an apparently strong argument that the position of metaphysical realism provides an incoherent model of the relation of a correct scientific theory to the world. However, although Putnam's attack upon the notion of the "intended" interpretation of a scientific theory is sound, it is shown here that realism may be formulated in such a way that the realist need make no appeal to any "intended" interpretation of such a theory. Consequently, it can (...) be shown that realism is immune to Putnam's criticism and that attempts at reformulating this criticism are not likely to meet with success. (shrink)

Revenge arguments purport to show that any proposed solution to the semantic paradoxes generates new paradoxes that prove that solution to be inadequate. In this paper, I focus on revenge arguments that employ the model-theoretic semantics of a target theory and I argue, contra the current revenge-theoretic wisdom, that they can constitute genuine expressive limitations. I consider the anti-revenge strategy elaborated by Field and argue that it does not offer a way out of the revenge problem. More generally, I argue (...) that the difference between ‘standard’ and ‘revenge’ paradoxes is ill-conceived and should be abandoned. This will contribute to show that the theories that provide a uniform account of truth and other semantic notions are the ones best equipped to avoid the paradoxes altogether—‘standard’ and ‘revenge’ alike. (shrink)

In this book Emma Ruttkamp demonstrates the power of the full-blown employment of the model-theoretic paradigm in the philosophy of science. Within this paradigm she gives an account of sciences as process and product. She expounds the "received statement" and the "non-statement" views of science, and shows how the model-theoretic approach resolves the spurious tension between these views. In this endeavour she also engages the views of a number of contemporary philosophers of science with affinity to model theory. This text (...) can be read by specialists working in philosophy of science or formal semantics, by logicians working on the structure of theories, and by students in philosophy of science - this text offers a thorough introduction to non-statement accounts of sciences as well as a discussion of the traditional statement account of science. (shrink)

We present a framework for model theoretic forcing in a non first order context, and present some applications of this framework to Banach space theory.

We describe a model-theoretic approach to ordinal analysis via the finite combinatorial notion of an α-large set of natural numbers. In contrast to syntactic approaches that use cut elimination, this approach involves constructing finite sets of numbers with combinatorial properties that, in nonstandard instances, give rise to models of the theory being analyzed. This method is applied to obtain ordinal analyses of a number of interesting subsystems of first- and second-order arithmetic.

Discussions of representation in science tend to draw on examples from art. However, such examples need to be handled with care given a) the differences between works of art and scientific theories and b) the accommodation of these examples within certain philosophies of art. I shall examine the claim that isomorphism is neither necessary nor sufficient for representation and I shall argue that there exist accounts of representation in both art and science involving isomorphism which accommodate the apparent counterexamples and, (...) moreover, allow us to understand how “impossible” artistic objects and inconsistent scientific theories can be said to represent. (shrink)

We give a model-theoretic proof of the fact that for all infinite Abelian groups P ≠ NP in the sense of binary nondeterminism. This result has been announced 1994 by Christine Gabner.

We prove that for a finitely generated infinite nilpotent group $G$ with structure $(G,\cdot,\dots)$, the connected component ${G^*}^0$ of a sufficiently saturated extension $G^*$ of $G$ exists and equals \[ \bigcap_{n\in\N} \{g^n\colon g\in G^*\}. \] We construct an expansion of ${\mathbb Z}$ by a predicate $({\mathbb Z},+,P)$ such that the type-connected component ${{\mathbb Z}^*}^{00}_{\emptyset}$ is strictly smaller than ${{\mathbb Z}^*}^0$. We generalize this to finitely generated virtually solvable groups. As a corollary of our construction we obtain an optimality result for (...) the van der Waerden theorem for finite partitions of groups. (shrink)

This paper studies some model-theoretic properties of special groups of finite type. Special groups are a first-order axiomatization of the algebraic theory of quadratic forms, introduced by Dickmann and Miraglia, which is essentially equivalent to abstract Witt rings. More precisely, we consider elementary equivalence, saturation, elementary embeddings, quantifier elimination, stability and Morley rank.

This paper introduces three model-theoretic constructions for generalized Epstein semantics: reducts, ultramodels and $\textsf {S}$ -sets. We apply these notions to obtain metatheoretical results. We prove connective inexpressibility by means of a reduct, compactness by an ultramodel and definability theorem which states that a set of generalized Epstein models is definable iff it is closed under ultramodels and $\textsf {S}$ -sets. Furthermore, a corollary concerning definability of a set of models by a single formula is given on the basis of (...) the main theorem and the compactness theorem. We also provide an example of a natural set of generalized Epstein models which is undefinable. Its undefinability is proven by means of an $\textsf {S}$ -set. (shrink)

We analyze model-theoretic connected components in extensions of a given group by abelian groups which are defined by means of 2-cocycles with finite image. We characterize, in terms of these 2-cocycles, when the smallest type-definable subgroup of the corresponding extension differs from the smallest invariant subgroup. In some situations, we also describe the quotient of these two connected components. Using our general results about extensions of groups together with Matsumoto–Moore theory or various quasi-characters considered in bounded cohomology, we obtain new (...) classes of examples of groups whose smallest type-definable subgroup of bounded index differs from the smallest invariant subgroup of bounded index. This includes the first known example of a group with this property found by Conversano and Pillay, namely the universal cover of [Formula: see text], as well as various examples of different nature, e.g. some central extensions of free groups or of fundamental groups of closed orientable surfaces. As a corollary, we get that both non-abelian free groups and fundamental groups of closed orientable surfaces of genus [Formula: see text], expanded by predicates for all subsets, have this property, too. We also obtain a variant of the example of Conversano and Pillay for [Formula: see text] instead of [Formula: see text], which was not accessible by the previously known methods. (shrink)

Notions of k-asimulation and asimulation are introduced as asymmetric counterparts to k-bisimulation and bisimulation, respectively. It is proved that a first-order formula is equivalent to a standard translation of an intuitionistic propositional formula iff it is invariant with respect to k-asimulations for some k, and then that a first-order formula is equivalent to a standard translation of an intuitionistic propositional formula iff it is invariant with respect to asimulations. Finally, it is proved that a first-order formula is equivalent to a (...) standard translation of an intuitionistic propositional formula over the class of intuitionistic Kripke models iff it is invariant with respect to asimulations between intuitionistic models. (shrink)

We use model-theoretic methods described in [3] to obtain ordinal analyses of a number of theories of first- and second-order arithmetic, whose proof-theoretic ordinals are less than or equal to Γ0.

In the early days of natural language semantics, Donald Davidson issued a challenge to those, like Richard Montague, who would do semantics in a model-theoretic framework that gives a central role to a model-relative notion of truth. Davidson argued that no theory of this kind can claim to be an account of real truth conditions unless it first makes clear how the relativized notion relates to our ordinary non-relativized notion of truth. In the 1990s, Davidson’s challenge was developed by Etchemendy (...) into an argument against the model-theoretic account of logical consequence, one that also threatens the attempt to capture natural language entailment relations in modeltheoretic terms—one of the central desiderata of semantics. Nevertheless, the modeltheoretic framework has flourished within natural language semantics. But it has flourished without any consensus among semanticists as to how to answer Davidson’s challenge. The aim of this essay is to develop an answer. I argue that model-theoretic semantics is best understood as model-based science: a semantics for a natural language is a scientific model of truth conditions. This makes good sense of the way model-theoretic tools are used in natural language semantics. And it allows us to answer Davidson’s challenge by showing how a theory that employs a relativized notion of truth manages to tell us about ordinary truth conditions. Moreover, I argue that it helps us see how semantics can provide genuine explanations for natural language entailment and other truth-conditional phenomena. (shrink)

Abstract A general framework is proposed for accommodating the recent results of studies into ?natural? decision making. A crucial element of this framework is the notion of a ?partial structure?, recently introduced into the semantic approach to scientific theories. It is through the introduction of this element that connections can be made with certain problems regarding inconsistency and rationality in general.

We generalise the concept of clique width to structures of arbitrary signature and cardinality. We present characterisations of clique width in terms of decompositions of a structure and via interpretations in trees. Several model-theoretic properties of clique width are investigated including VC-dimension and preservation of finite clique width under elementary extensions and compactness.

[...] I will only investigate [Austin's] claims as challenges to present-day model theoretic semantics. My main point will be to draw a sharp line between the semantic and pragmatic aspects of performatives and thereby discover a gap in Austin’s treatment. This will in my view naturally lead to the proposal in Section 2, that is, to treating performatives as denoting changes in intensional models. The rest of Section 2 will be concerned with the status of felicity conditions and a tentative (...) extension of Montague’s PTQ. (shrink)

Abstract: Hilary Putnam has reformulated his model-theoretic argument as an argument against indirect realism in the philosophy of perception. This new argument is reviewed and defended. Putnam’s new focus on philosophical theories of perception (instead of metaphysical realism) makes better sense of his previous responses to the objection from the causal theory of reference. It is argued that the model-theoretic argument can also be construed as an argument that holders of a causal theory of reference should adopt direct realism in (...) the philosophy of perception. (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)

Let $\mathscr{L} = \{0, 1, +, \cdot, <\}$ be the usual first-order language of arithmetic. We show that Peano arithmetic is the least first-order L-theory containing IΔ0 + exp such that every complete extension T of it has a countable model K satisfying. (i) K has no proper elementary substructures, and (ii) whenever $L \prec K$ is a countable elementary extension there is $\bar{L} \prec L$ and $\bar{K} \subseteq_\mathrm{e} \bar{L}$ such that $K \prec_{\mathrm{cf}}\bar{K}$ . Other model-theoretic conditions similar to (i) (...) and (ii) are also discussed and shown to characterize Peano arithmetic. (shrink)

We use model-theoretic methods described in [3] to obtain ordinal analyses of a number of theories of first-and second-order arithmetic, whose proof-theoretic ordinals are less than or equal to $\Gamma_0$.

We point out how the "Fundamental Theorem of Stability Theory", namely the equivalence between the "non order property" and definability of types, proved by Shelah in the 1970s, is in fact an immediate consequence of Grothendieck's "Criteres de compacite" from 1952. The familiar forms for the defining formulae then follow using Mazur's Lemma regarding weak convergence in Banach spaces.

The major argument against Ramsey-style epistemic structural realism is the model-theoretic refinement of Newman’s objection against Russell, presented in Ketland : 409–424, 2004), where a technical result is interpreted as showing that the Ramsey-sentence approach collapses into instrumentalism. This paper addresses some questions raised by the application of model theory to the scientific realism debate. Firstly, I will suggest three different formal semantics for the positions in the debate. Then, some technicalities of Ketland’s result will be scrutinized in light of (...) comments by Zahar and Demopoulos. Finally, I will formalize Ketland’s argument by means of an intensional operator and focus on one problematic premise. The conclusion is that, with some adjustments, the Ramsey-sentence approach can represent an intermediate position between realism and instrumentalism, but the term “structuralism” does not suit it well. (shrink)

This contribution is concerned with a particular model theoretic semantics of the object language of quantum physics. The object language considered here comprises logically connected propositions, sequentially connected propositions and modal propositions. The model theoretic semantics arises from the already established dialogic semantics, if the pragmatic concept of the dialog-game is replaced by a "metaphysical" concept of the game. The game is determined by a game tree, the branches of which constitute a set, the set of "possible worlds" of an (...) individual quantum physical system. The semantic concepts like truth, falsity and valuation are defined with respect to this set. (shrink)

We bring together some facts about the weak pigeonhole principle from bounded arithmetic, complexity theory, cryptography and abstract model theory. We characterize the models of arithmetic in which WPHP fails as those which are determined by an initial segment and prove a conditional separation result in bounded arithmetic, that PV + lies strictly between PV and S21 in strength, assuming that the cryptosystem RSA is secure.

We define a subhierarchy of the infinitely deep languagesN described by Jaakko Hintikka and Veikko Rantala. We shall show that some model theoretic results well-known in the model theory of the ordinary infinitary languages can be generalized for these new languages. Among these are the downward Löwenheim-Skolem and o's theorems as well as some compactness properties.

Let= {0,1, +,·,<} be the usual first-order language of arithmetic. We show that Peano arithmetic is the least first-order-theory containingIΔ0+ exp such that every complete extensionTof it has a countable modelKsatisfying(i)Khas no proper elementary substructures, and(ii) wheneverL≻Kis a countable elementary extension there isandsuch that.Other model-theoretic conditions similar to (i) and (ii) are also discussed and shown to characterize Peano arithmetic.

The 3-valued paraconsistent logic Ciore was developed by Carnielli, Marcos and de Amo under the name LFI2, in the study of inconsistent databases from the point of view of logics of formal inconsistency (LFIs). They also considered a first-order version of Ciore called LFI2*. The logic Ciore enjoys extreme features concerning propagation and retropropagation of the consistency operator: a formula is consistent if and only if some of its subformulas is consistent. In addition, Ciore is algebraizable in the sense of (...) Blok and Pigozzi. On the other hand, the logic LFI2* satisfies a somewhat counter-intuitive property: the universal and the existential quantifier are inter-definable by means of the paraconsistent negation, as it happens in classical first-order logic with respect to the classical negation. This feature seems to be unnatural, given that both quantifiers have the classical meaning in LFI2*, and that this logic does not satisfy the De Morgan laws with respect to its paraconsistent negation. The first goal of the present article is to introduce a first-order version of Ciore (which we call QCiore) preserving the spirit of Ciore, that is, without introducing unexpected relationships between the quantifiers. The second goal of the article is to adapt to QCiore the partial structures semantics for the first-order paraconsistent logic LPT1 introduced by Coniglio and Silvestrini, which generalizes the semantic notion of quasi-truth considered by Mikeberg, da Costa and Chuaqui. Finally, some important results of classical Model Theory are obtained for this logic, such as Robinson’s joint consistency theorem, amalgamation and interpolation. Although we focus on QCiore, this framework can be adapted to other 3-valued first-order LFIs. (shrink)

'Mass terms', words like water, rice and traffic, have proved very difficult to accommodate in any theory of meaning since, unlike count nouns such as house or dog, they cannot be viewed as part of a logical set and differ in their grammatical properties. In this study, motivated by the need to design a computer program for understanding natural language utterances incorporating mass terms, Harry Bunt provides a thorough analysis of the problem and offers an original and detailed solution. An (...) extension of classical set theory, Ensemble Theory, is defined, and this provides the conceptual basis of a framework for the analysis of natural language meaning which Dr Bunt calls Two-level model-theoretic semantics. The validity of the framework is convincingly demonstrated by the formal analysis of a fragment of English including sentences with quantified and modified mass terms. Separate chapters of the book are devoted to an axiomatic definition of Ensemble Theory and a detailed discussion of its status as a mathematical formalism. (shrink)

The first section of the paper gives a very condensed history of the evolution of the author’s views on realism and anti-realism. It emphasizes that his previously accepted form of anti-realism was abandoned not because of the alleged fallacies in the model-theoretic argument against metaphysical realism, but due to his rejection of some of the assumptions on which it rests - assumptions which have been almost universal in philosophy after Descartes. The second section discusses and defends the part of the (...) model-theoretic argument that many critics have almost universally found fallacious, namely the „just more theory” move. In the appendix to the paper the author elaborates what does his return to common sense realism mean. He points out that his position involves a negative element, the rejection of the idea that truth cannot outrun verifiability, and two positive elements: returning (as close as possible) to „naive realism” with respect to perception, and a disquotational account of truth similar to one endorsed by Wittgenstein. (shrink)