This paper presents an extended version of the Quantified Argument Calculus (Quarc). Quarc is a logic comparable to the first-order predicate calculus. It employs several nonstandard syntactic and semantic devices, which bring it closer to natural language in several respects. Most notably, quantifiers in this logic are attached to one-place predicates; the resulting quantified constructions are then allowed to occupy the argument places of predicates. The version presented here is capable of straightforwardly translating natural-language sentences involving defining clauses. A three-valued, (...) model-theoretic semantics for Quarc is presented. Interpretations in this semantics are not equipped with domains of quantification: they are just interpretation functions. This reflects the analysis of natural-language quantification on which Quarc is based. A proof system is presented, and a completeness result is obtained. The logic presented here is capable of straightforward translation of the classical first-order predicate calculus, the translation preserving truth values as well as entailment. The first-order predicate calculus and its devices of quantification can be seen as resulting from Quarc on certain semantic and syntactic restrictions, akin to simplifying assumptions. An analogous, straightforward translation of Quarc into the first-order predicate calculus is impossible. (shrink)

The purpose of this paper is to communicate - as a proposal - a general method of assigning a 'truthmaker' to any 1st order sentence in each of its models. The respective construct is derived from the standard model theoretic (recursive) satisfaction definition for 1st order languages and is a conservative extension thereof. The heuristics of the proposal (which has been somewhat idiosyncratic from the current point of view) and some more technical detail of the construction may be found (...) in my article on part I of Spinoza's 'ethica, ordine geometrico demonstrata', which is the context within which I elaborated the assignment. But this context need not be repeated here, the presentation of the truthmaker assignment will be comprehensible to anybody with solid basic knowledge in 1st order model theory, anything used is standard and no advanced techniques are required. (shrink)

This bibliographical review of the modelling of the mitotic apparatus covers a period of one hundred and twenty years, from the discovery of the bipolar mitotic spindle up to the present day. Without attempting to be fully comprehensive, it will describe the evolution of the main ideas that have left their mark on a century of experimental and theoretical research. Fol and Bütschli's first writings date back to 1873, at a time when Schleiden and Schwann's cell theory was rapidly (...) gaining ground throughout Germany. Both mitosis and chromosomes were to be discovered within the space of thirty years, along with the two key events in the animal and plant reproductive cycle, namely fecondation and meiosis. The mitotic pole, a term still in use to this day, was employed to describe a morphological fact which was noted as early as 1876, namely that the lines and the dots of the karyokinetic figure, with its spindle and asters, looks remarkably like the lines of force around a bar magnet. This was to lead to models designed to explain the movements of chromosomes which take place when the cell nucleus appears to cease to exist as an organelle during mitosis. The nature of those mechanisms and the origin of the forces behind the chromosomes' ordered movements were central to the debate. Auguste Prenant, in a remarkable bibliographical synthesis published in 1910, summed up the opposing viewpoints of the vitalists, on the one hand, who favoured the theory of contractility or extensility in spindle fibres, and of those who believed in models based on physical phenomena, on the other. The latter subdivided into two groups: some, like Bütschli, Rhumbler or Leduc, referred to diffusion, osmosis and superficial tension, whilst the others, led by Gallardo and Hartog, focussed on the laws of electromagnetism. Lillie, Kuwada and Darlington followed up this line of research. The mid-20th century was a major turning point. Most of the modelling mentioned above was criticized and fell into disuse after disappearing from research publications and textbooks.This marked the onset of a new era, as electron microscopes made possible the materialization and detailed study of the macromolecular elements of the fibres, filaments and microtubules of the cytoskeleton. The successive phases of (a) de Harven and Bernhard's 1956 discovery of the centriole's ultrastructure, (b) its identification with the basal body of the cilia and flagella, confirming the theory set out by Henneguy and von Lenhossek (1898–99), (c) the universal presence of microtubules in animal, vegetal and eukaryotic protist cells, (d) the polymerization-depolymerization induced reversible transformations of the tubulin pool in mitosing cells (Inoue, 1960), (e) ultrastructural comparative studies of the mitotic apparatus of eukaryotes illustrating the Pickett-Heaps integrating concept of the MTOC (microtubule-organizing centre), (f) the possibility ofin vitro experiments on mtocs or on microtubules, brings us upon the present day, which has seen the focus placed on the concept of motor-proteins (kinesin, dynein) and on cell cycle models. The latter are based on a close coincidence between the observable modifications of the mitotic apparatus and the periodic variations in intracellular concentrations of calcium or of certain enzymes (cyclins, Cdc2) during the main transitions of the cell cycle. (shrink)

In the context of proliferation of many logical systems in the area of mathematical logic and computer science, we present a generalization of forcing in institution-independent model theory which is used to prove two abstract results: Downward Löwenheim-Skolem Theorem and Omitting Types Theorem . We instantiate these general results to many first-order logics, which are, roughly speaking, logics whose sentences can be constructed from atomic formulas by means of Boolean connectives and classical first-order quantifiers. These include first-order logic (...) , logic of order-sorted algebras , preorder algebras , as well as their infinitary variants \ , \ , \ . In addition to the first technique for proving OTT, we develop another one, in the spirit of institution-independent model theory, which consists of borrowing the Omitting Types Property from a simpler institution across an institution comorphism. As a result we export the OTP from FOL to partial algebras and higher-order logic with Henkin semantics , and from the institution of \ to \ and \ . The second technique successfully extends the domain of application of OTT to non conventional logical systems for which the standard methods may fail. (shrink)

We aim to compile some means for a rational reconstruction of a named part of the start-over of Baruch (Benedictus) de Spinoza's metaphysics in 'de deo' (which is 'pars prima' of the 'ethica, ordine geometrico demonstrata' ) in terms of 1st order model theory. In so far, as our approach will be judged successful, it may, besides providing some help in understanding Spinoza, also contribute to the discussion of some or other philosophical evergreen, e.g. 'ontological commitment'. For this (...) text we assume the reader familiar with 'de deo' as well as with some basic concepts and results of 1st order model theory. Before we start reconstruction, we will first revisit shortly the concept of 'attributum' (definitio IV) in it's setting in 'de deo' , next scan for formalizable aspects of 'in suo genere finita' ('de deo', definitio II), subsequently list the model theoretic constructs we will make use of. Then we begin reconstruction by stating "coordinative definitions" for the notions of 'attribute (of a substance)', 'modus (as conceived via an attribute)' and 'substance (as conceived via an attribute)', reasoning shortly for each of them. The "coordinative definitions", we will arrive at, must not be understood as literal translations of Spinoza's concepts - of course, there can't be such a thing as a literal translation - they are meant as formal analoga of these concepts, mapping some logical structure. But even with this caveat they may seem strange to the reader at this stage of discussion. Additional justification for them then should be found in our endeavour, to map some argumentation of Spinoza's proofs of some of his propositions from this starting point. -/- [ Version 1 of this paper originated from 2020-03-14. This is version 2 from 2021-09-12 with some readability amendments. ]. (shrink)

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

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.

Provability, Computability and Reflection.

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

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)

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)

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)

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)

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.

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 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 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)

My goal in this paper is, to tentatively sketch and try defend some observations regarding the ontological dignity of object references, as they may be used from within in a formalized language. -/- Hence I try to explore, what properties objects are presupposed to have, in order to enter the universe of discourse of an interpreted formalized language. -/- First I review Frege′s analysis of the logical structure of truth value definite sentences of scientific colloquial language, to draw suggestions from (...) his saturated vs. unsaturated sentence components paradigm. -/- Next try investigate, in how far reference to non pure math objects might allow for a role as argument of a truth value function. Object kinds to be considered are: common sense objects, technical objects, humanities object kinds (social, psychical, ...), objects of art, ... , be they abstract, concrete, or in this respect mixed objects. -/- Then have a comment on the just referenced label abstract objects. -/- Next try to get an idea wrt the ontological significance of the fact, that pure math objects and functions in some important classical cases can be uniquely defined by means of categorical theories. Here, in the course of my argument, I have a little corollary wrt the standard model of first order Peano arithmetic, reducing the epistemological significance of the existence of the non standard models. -/- Next, wrt a special concept of a formalized empirical theory, I care for whether the impure math objects and mixed objects described here, and complying best with truth value function mapping, are also reliable candidates for the ontological commitment of such theories: and discuss an alternative, which reduces the ontological significance of the universe of discourse of the theories intended models. -/- In the end I sum up what is - from my point of view - the status quo, according to my respective findings. (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)

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)

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)

Let G be a finite group. We explore the model-theoretic properties of the class of differential fields of characteristic zero in m commuting derivations equipped with a G-action by differential fie...

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)

The question “Are ‘biorobots' good models of biological behaviour?” can be seen as a specific instance of a more general question about the relation between computer programs and models, between models and theories, and between theories and reality. This commentary develops a personal view of these relations, from an antirealism perspective. Programs, models, theories and reality are separate and distinct entities which may converge in particular cases but should never be confused.

This paper presents a new theory of modal reasoning, i.e. reasoning about what may or may not be the case, and what must or must not be the case. It postulates that individuals construct models of the premises in which they make explicit only what is true. A conclusion is possible if it holds in at least one model, whereas it is necessary if it holds in all the models. The theory makes three predictions, which are corroborated (...) experimentally. First, conclusions correspond to the true, but not the false, components of possibilities. Second, there is a key interaction: it is easier to infer that a situation is possible as opposed to impossible, whereas it is easier to infer that a situation is not necessary as opposed to necessary. Third, individuals make systematic errors of omission and of commission. We contrast the theory with theories based on formal rules. (shrink)

We prove Łos conjecture = Morley theorem in ZF, with the same characterization, i.e., of first order countable theories categorical in $N_\alpha $ for some (equivalently for every ordinal) α > 0. Another central result here in this context is: the number of models of a countable first order T of cardinality $N_\alpha $ is either ≥ |α| for every α or it has a small upper bound (independent of α close to Ð₂).

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)

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)

We explore Shelah's model‐theoretic dividing line methodology. In particular, we discuss how problems in model theory motivated new techniques in model theory, for example classifying theories by their potential (consistently with Zermelo–Fraenkel set theory with the axiom of choice (ZFC)) spectrum of cardinals in which there is a universal model. Two other examples are the study (with Malliaris) of the Keisler order leading to a new ZFC result on cardinal invariants and attempts to (...) clarify the “main gap” by reducing the dependence of certain versions on (highly independent) cardinal arithmetic. (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)

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)

This paper develops the model theory of ordered structures that satisfy Keisler’s regularity scheme and its strengthening REF ${(\mathcal{L})}$ (the reflection scheme) which is an analogue of the reflection principle of Zermelo-Fraenkel set theory. Here ${\mathcal{L}}$ is a language with a distinguished linear order <, and REF ${(\mathcal {L})}$ consists of formulas of the form $$\exists x \forall y_{1} < x \ldots \forall y_{n} < x \varphi (y_{1},\ldots ,y_{n})\leftrightarrow \varphi^{ < x}(y_1, \ldots ,y_n),$$ where φ is an (...) ${\mathcal{L}}$ -formula, φ theory T in a countable language ${\mathcal{L}}$ with a distinguished linear order:Some model of T has an elementary end extension with a first new element.T ⊢ REF ${(\mathcal{L})}$ .T has an ω 1-like model that continuously embeds ω 1.For some regular uncountable cardinal κ, T has a κ-like model that continuously embeds a stationary subset of κ.For some regular uncountable cardinal κ, T has a κ-like model ${\mathfrak{M}}$ that has an elementary extension in which the supremum of M exists.Moreover, if κ is a regular cardinal satisfying κ = κ <κ , then each of the above conditions is equivalent to: T has a κ + -like model that continuously embeds a stationary subset of κ. (shrink)

Researchers currently working on relational reasoning typically argue that mental model theory (MMT) is a better account than the inference rule approach (IRA). They predict and observe that determinate (or one-model) problems are easier than indeterminate (or two-model) problems, whereas according to them, IRA should lead to the opposite prediction. However, the predictions attributed to IRA are based on a mistaken argument. The IRA is generally presented in such a way that inference rules only deal with (...) determinate relations and not with indeterminate ones. However, (a) there is no reason to presuppose that a rule-based procedure could not deal with indeterminate relations, and (b) applying a rule-based procedure to indeterminate relations should result in greater difficulty. Hence, none of the recent articles devoted to relational reasoning currently presents a conclusive case for discarding IRA by using the well-known determinate vs indeterminate problems comparison. (shrink)

One of the most debated questions in psychology and cognitive science is the nature and the functioning of the mental processes involved in deductive reasoning. However, all existing theories refer to a specific deductive domain, like syllogistic, propositional or relational reasoning.Our goal is to unify the main types of deductive reasoning into a single set of basic procedures. In particular, we bring together the microtheories developed from a mental models perspective in a single theory, for which we provide a (...) formal foundation. We validate the theory through a computational model (UNICORE) which allows fine‐grained predictions of subjects' performance in different reasoning domains.The performance of the model is tested against the performance of experimental subjects—as reported in the relevant literature—in the three areas of syllogistic, relational and propositional reasoning. The computational model proves to be a satisfactory artificial subject, reproducing both correct and erroneous performance of the human subjects. Moreover, we introduce a developmental trend in the program, in order to simulate the performance of subjects of different ages, ranging from children (3–6) to adolescents (8–12) to adults (>21). The simulation model performs similarly to the subjects of different ages.Our conclusion is that the validity of the mental model approach is confirmed for the deductive reasoning domain, and that it is possible to devise a unique mechanism able to deal with the specific subareas. The proposed computational model (UNICORE) represents such a unifying structure. (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)

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)

We study the model-theoretic aspects of a probability logic suited for talking about measure spaces. This nonclassical logic has a model theory rather different from that of classical predicate logic. In general, not every satisfiable set of sentences has a countable model, but we show that one can always build a model on the unit interval. Also, the probability logic under consideration is not compact. However, using ultraproducts we can prove a compactness theorem for a (...) certain class of weak models. (shrink)

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)

Abstract‘Skolem arithmetic’ is the complete theory T of the multiplicative monoid. We give a full characterization of the ‐definable stably embedded sets of T, showing in particular that, up to the relation of having the same definable closure, there is only one non‐trivial one: the set of squarefree elements. We then prove that T has weak elimination of imaginaries but not elimination of finite imaginaries.