A reasoned argument or tarka is essential for a wholesome vāda that aims at establishing the truth. A strong tarka constitutes of a number of elements including an anumāna based on a valid hetu. Several scholars, such as Dharmakīrti, Vasubandhu and Dignāga, have worked on theories for the establishment of a valid hetu to distinguish it from an invalid one. This paper aims to interpret Dignāga’s hetu-cakra, called the wheel of grounds, from a modern philosophical perspective by deconstructing it (...) into a simple probabilistic mathematical model. The objective is to understand how and why a vāda based on a probabilistically weaker hetu can degrade into a Jalpa or vitaṇḍā. To do so, the paper maps the concept of ‘Bounded Rationality’ onto the hetu-cakra. Bounded Rationality, an idea coined by the management thinker Herbert Simon, is often employed in understanding decision-making processes of rational agents. In the context of this paper, the concept would state that the prativādin and ālocaka (debater) may not hold unbounded information to back their pratijñā (proposition). The paper argues that within the probabilistically deconstructed hetu-cakra model, most people argue in the ‘Zone of Bounded Rationality’, and thus, the probability of a debate degrading into Jalpa or vitaṇḍā is high. -/- . (shrink)

To explore the relation between mathematical models and reality, four different domains of reality are distinguished: observer-independent reality, personal reality, social reality and mathematical/formal reality. The concepts of personal and social reality are strongly inspired by constructivist ideas. Mathematical reality is social as well, but constructed as an autonomous system in order to make absolute agreement possible. The essential problem of mathematical modelling is that within mathematics there is agreement about ‘truth’, but the assignment (...) of mathematics to informal reality is not itself formally analysable, and it is dependent on social and personal construction processes. On these levels, absolute agreement cannot be expected. Starting from this point of view, repercussion of mathematical on social and personal reality, the historical development of mathematical modelling, and the role, use and interpretation of mathematical models in scientific practice are discussed. (shrink)

The aim of this paper is to show that it’s not a good idea to have a theory of truth that is consistent but ω-inconsistent. In order to bring out this point, it is useful to consider a particular case: Yablo’s Paradox. In theories of truth without standard models, the introduction of the truth-predicate to a first order theory does not maintain the standard ontology. Firstly, I exhibit some conceptual problems that follow from so introducing it. (...) Secondly, I show that in second order theories with standard semantics the same procedure yields a theory that doesn’t have models. So, while having an ω- inconsistent theory is a bad thing, having an unsatisfiable theory of truth is actually worse. This casts doubts on whether the predicate in question is, after all, a truthpredicate for that language. Finally, I present some alternatives to prove an inconsistency adding plausible principles to certain theories of truth. (shrink)

Evolutionary anthropologists and archaeologists have been considerably successful in modelling the cumulative evolution of culture, of technological skills and knowledge in particular. Recently, one of these models has been introduced in the philosophy of science by De Cruz and De Smedt (Philos Stud 157:411–429, 2012), in an attempt to demonstrate that scientists may collectively come to hold more truth-approximating beliefs, despite the cognitive biases which they individually are known to be subject to. Here we identify a major shortcoming (...) in that attempt: De Cruz & De Smedt’s mathematical model makes one particularly strong tractability assumption that causes the model to largely miss its target (namely, truth accumulation in science), and that moreover conflicts with empirical observations. The second, more constructive part of the paper presents an alternative, agent-based model, which allows one to much better examine the conditions for scientific progress and decline. (shrink)

1. Setting off: An introduction. 1.1. A measure of motivation. 1.2. Computable mathematics. 1.3. Reverse mathematics. 1.4. An overview. 1.5. Further reading -- 2. Gathering our tools: Basic concepts and notation. 2.1. Computability theory. 2.2. Computability theoretic reductions. 2.3. Forcing -- 3. Finding our path: Konig's lemma and computability. 3.1. II[symbol] classes, basis theorems, and PA degrees. 3.2. Versions of Konig's lemma -- 4. Gauging our strength: Reverse mathematics. 4.1. RCA[symbol]. 4.2. Working in RCA[symbol]. 4.3. ACA[symbol]. 4.4. WKL[symbol]. 4.5. [symbol]- (...) class='Hi'>models. 4.6. First order axioms. 4.7. Further remarks -- 5. In defense of disarray -- 6. Achieving consensus: Ramsey's theorem. 6.1. Three proofs of Ramsey's theorem. 6.2. Ramsey's theorem and the arithmetic hierarchy. 6.3. RT, ACA[symbol], and the Paris-Harrington theorem. 6.4. Stability and cohesiveness. 6.5. Mathias forcing and cohesive sets. 6.6. Mathias forcing and stable colorings. 6.7. Seetapun's theorem and its extensions. 6.8. Ramsey's theorem and first order axioms. 6.9. Uniformity -- 7. Preserving our power: Conservativity. 7.1. Conservativity over first order systems. 7.2. WKL[symbol] and II[symbol]-conservativity. 7.3. COH and r-II[symbol]-conservativity -- 8. Drawing a map: Five diagrams -- 9. Exploring our surroundings: The world below RT[symbol]. 9.1. Ascending and descending sequences. 9.2. Other combinatorial principles provable from RT[symbol]. 9.3. Atomic models and omitting types -- 10. Charging ahead: Further topics. 10.1. The Dushnik-Miller theorem. 10.2. Linearizing well-founded partial orders. 10.3. The world above ACA[symbol]. 10.4. Still further topics, and a final exercise. (shrink)

Analysing several characteristic mathematical models: natural and real numbers, Euclidean geometry, group theory, and set theory, I argue that a mathematical model in its final form is a junction of a set of axioms and an internal partial interpretation of the corresponding language. It follows from the analysis that (i) mathematical objects do not exist in the external world: they are our internally imagined objects, some of which, at least approximately, we can realize or represent; (ii) (...) mathematical truths are not truths about the external world but specifications (formulations) of mathematical conceptions; (iii) mathematics is first and foremost our imagined tool by which, with certain assumptions about its applicability, we explore nature and synthesize our rational cognition of it. (shrink)

In the following paper we propose a model-theoretical way of comparing the “strength” of various truth theories which are conservative over PA\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ PA $$\end{document}. Let Th\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak {Th}}$$\end{document} denote the class of models of PA\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ PA $$\end{document} which admit an expansion to a model of theory Th\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} (...) \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${ Th}$$\end{document}. We show that PA⊃TB⊃RS⊃UTB⊇CT-,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {{\mathfrak {PA}}}\supset {\mathfrak {TB}}\supset {{\mathfrak {RS}}}\supset {\mathfrak {UTB}}\supseteq \mathfrak {CT^-}, \end{aligned}$$\end{document}where PA\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak {PA}}$$\end{document} denotes simply the class of all models of PA\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ PA $$\end{document} and RS\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak {RS}}$$\end{document} denotes the class of recursively saturated models of PA\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ PA $$\end{document}. Our main original result is that every model of PA\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ PA $$\end{document} which admits an expansion to a model of CT-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ CT ^-$$\end{document}, admits also an expansion to a model of UTB\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ UTB $$\end{document}. Moreover, as a corollary to one of the results we conclude that UTB\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ UTB $$\end{document} is not relatively interpretable in TB\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ TB $$\end{document}, thus answering the question from Fujimoto. (shrink)

This introduction to mathematical logic starts with propositional calculus and first-order logic. Topics covered include syntax, semantics, soundness, completeness, independence, normal forms, vertical paths through negation normal formulas, compactness, Smullyan's Unifying Principle, natural deduction, cut-elimination, semantic tableaux, Skolemization, Herbrand's Theorem, unification, duality, interpolation, and definability. The last three chapters of the book provide an introduction to type theory (higher-order logic). It is shown how various mathematical concepts can be formalized in this very expressive formal language. This expressive notation (...) facilitates proofs of the classical incompleteness and undecidability theorems which are very elegant and easy to understand. The discussion of semantics makes clear the important distinction between standard and nonstandard models which is so important in understanding puzzling phenomena such as the incompleteness theorems and Skolem's Paradox about countable models of set theory. Some of the numerous exercises require giving formal proofs. A computer program called ETPS which is available from the web facilitates doing and checking such exercises. Audience: This volume will be of interest to mathematicians, computer scientists, and philosophers in universities, as well as to computer scientists in industry who wish to use higher-order logic for hardware and software specification and verification. (shrink)

The paper argues that there is a fruitful analogy to be made between classic pre-analytic Euclidean geometry and a certain kind of mechanism models, called ideal mechanisms. Both supply necessary truths. Bunge is of the opinion that pure mathematics is about fictions, but that mathematics nonetheless is useful in science and technology because we can go “to reality through fictions.” Similarly, the paper claims that ideal mechanisms are useful because we can go to real mechanisms through the fictions of (...) ideal mechanisms. The view put forward takes it for granted that two important distinctions concerned with the classification of fictions can be made. One is between ideal and non-ideal fictions, and the other between social and non-social fictions. Pure numbers, purely geometric figures, and ideal mechanisms are claimed to be ideal and social fictions. (shrink)

It is argued that the truth status of emergent properties of complex adaptive systems models should be based on an epistemology of proof by constructive verification and therefore on the ontological axioms of a non-realist logical system such as constructivism or intuitionism. ‘Emergent’ properties of complex adaptive systems models create particular epistemological and ontological challenges. These challenges bear directly on current debates in the philosophy of mathematics and in theoretical computer science. CAS research, with its emphasis on (...) computer simulation, is heavily reliant on models which explore the entailments of Formal Axiomatic Systems. The incompleteness results of G¨odel, the incomputability results of Turing, and the Algorithmic Information Theory results of Chaitin, undermine a realist truth model of emergent properties. These same findings support the hegemony of epistemology over ontology and point to alternative truth models such as intuitionism, constructivism and quasi-empiricism. (shrink)

This papers deals with the class of axiomatic theories of truth for semantically closed languages, where the theories do not allow for standard models; i.e., those theories cannot be interpreted as referring to the natural number codes of sentences only (for an overview of axiomatic theories of truth in general, see Halbach[6]). We are going to give new proofs for two well-known results in this area, and we also prove a new theorem on the nonstandardness of a (...) certain theory of truth. The results indicate that the proof strategies for all the theorems on the nonstandardness of such theories are "essentially" of the same kind of structure. (shrink)

This book examines the role of acts of choice in classical and intuitionistic mathematics. Featuring fifteen papers - both new and previously published - it offers a fresh analysis of concepts developed by the mathematician and philosopher L.E.J. Brouwer, the founder of intuitionism. The author explores Brouwer's idealization of the creative subject as the basis for intuitionistic truth, and in the process he also discusses an important, related question: to what extent does the intuitionistic perspective succeed in avoiding the (...) classical realistic notion of truth? The papers detail realistic aspects in the idealization of the creative subject and investigate the hidden role of choice even in classical logic and mathematics, covering such topics as bar theorem, type theory, inductive evidence, Beth models, fallible models, and more. In addition, the author offers a critical analysis of the response of key mathematicians and philosophers to Brouwer's work. These figures include Michael Dummett, Saul Kripke, Per Martin-Löf, and Arend Heyting. This book appeals to researchers and graduate students with an interest in philosophy of mathematics, linguistics, and mathematics. (shrink)

Where does math come from? From a textbook? From rules? From deduction? From logic? Not really, Eugenia Cheng writes in Is Math Real?: it comes from curiosity, from instinctive human curiosity, "from people not being satisfied with answers and always wanting to understand more." And most importantly, she says, "it comes from questions": not from answering them, but from posing them. Nothing could seem more at odds from the way most of us were taught math: a rigid and autocratic model (...) which taught us to follow specific steps to reach specific answers. Instead of encouraging a child who asks why 1+1 is 2, our methods of education force them to accept it. Instead of exploring why we multiply before we add, a textbook says, just to get on with the order of operations. Indeed, the point is usually just about getting the right answer, and those that are good at that, become "good at math" while those who question, are not. And that's terrible: These very same questions, as Cheng shows, aren't simply annoying questions coming from people who just don't "get it" and so can't do math. Rather, they are what drives mathematical research and push the boundaries in our understanding of all things. Legitimizing those questions, she invites everyone in, whether they think they are good at math or not. And by highlighting the development of mathematics outside Europe, Cheng shows that-western chauvinism notwithstanding--that math can be for anyone who wishes to do it, and how much we gain when anyone can. (shrink)

This article investigates models of axiomatizations related to the semantic conception of truth presented by Kripke (J Philos 72(19):690–716, 1975), the so-called _fixed-point semantics_. Among the various proof systems devised as a proof-theoretic characterization of the fixed-point semantics, in recent years two alternatives have received particular attention: _classical systems_ (i.e., systems based on classical logic) and _nonclassical systems_ (i.e., systems based on some nonclassical logic). The present article, building on Halbach and Nicolai (J Philos Log 47(2):227–257, 2018), shows (...) that there is a sense in which classical and nonclassical theories (in suitable variants) have the same models. (shrink)

Definitional and axiomatic theories of truth -- Objects of truth -- Tarski -- Truth and set theory -- Technical preliminaries -- Comparing axiomatic theories of truth -- Disquotation -- Classical compositional truth -- Hierarchies -- Typed and type-free theories of truth -- Reasons against typing -- Axioms and rules -- Axioms for type-free truth -- Classical symmetric truth -- Kripke-Feferman -- Axiomatizing Kripke's theory in partial logic -- Grounded truth -- Alternative (...) evaluation schemata -- Disquotation -- Classical logic -- Deflationism -- Reflection -- Ontological reduction -- Applying theories of truth. (shrink)

In the epistemological tradition, there are two main interpretations of the semantic relation that an empirical theory may bear to the real world. According to realism, the theory-world relationship should be conceived as truth; according to instrumentalism, instead, it should be limited to empirical adequacy. Then, depending on how empirical theories are conceived, either syntactically as a class of sentences, or semantically as a class of models, the concepts of truth and empirical adequacy assume different and specific (...) forms. In this paper, we review two main conceptions of truth (one sentence-based and one model-based) and two of empirical adequacy (one sentence-based and one model-based), we point out their respective difficulties, and we give a first formulation of a new general view of the theory-world relationship, which we call Methodological Constructive Realism (MCR). We then show how the content of MCR can be further specified and expressed in a definite and precise form. The bulk of the paper shows in detail how it is possible to accomplish this goal for the special case of deterministic dynamical phenomena and their correlated deterministic models. This special version of MCR is formulated as an axiomatic extension of set theory, whose specific axioms constitute a formal ontology that provides an adequate framework for analyzing the two semantic relations of truth and empirical correctness, as well as their connections. (shrink)

We present a construction of a truth class (an interpretation of a compositional truth predicate) in an arbitrary countable recursively saturated model of first-order arithmetic. The construction is fully classical in that it employs nothing more than the classical techniques of formal proof theory.

This article surveys recent literature by Parsons, McGee, Shapiro and others on the significance of categoricity arguments in the philosophy of mathematics. After discussing whether categoricity arguments are sufficient to secure reference to mathematical structures up to isomorphism, we assess what exactly is achieved by recent ‘internal’ renditions of the famous categoricity arguments for arithmetic and set theory.

The notion of truth is a central subject both in Philosophy and Mathematical Logic. The logical approach on the one side and the philosophical one on the other, however, mostly deal with problems which, apparently, require different tools to be tackled. In this paper I argue that such a separation can and should be overcome, and, in order to build a bridge, I focus on the philosophical issue of the insubstantiality of truth, which is a crucial topic (...) to distinguish inflationist from deflationist proposals. Elaborating on the interpretation of insubstantiality in terms of the sparse/abundant classification of properties, I put forward a refined version in which certain flaws afflicting other formulations are solved. Then, I show how, using this improved variant, the philosophical notion of abundance can be fruitfully related to the formal notion of expandability of models, if a logical framework is adopted. Among other virtues, the obtained link can shed new light on the debate of deflationism and conservativity. (shrink)

In the paper the problem of definability and undefinability of the concept of satisfaction and truth is considered. Connections between satisfaction and truth on the one hand and consistency of certain systems of omega-logic and transfinite induction on the other are indicated.

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

Undergraduate students with no prior classroom instruction in mathematical logic will benefit from this evenhanded multipart text by one of the centuries greatest authorities on the subject. Part I offers an elementary but thorough overview of mathematical logic of first order. The treatment does not stop with a single method of formulating logic; students receive instruction in a variety of techniques, first learning model theory (truth tables), then Hilbert-type proof theory, and proof theory handled through derived rules. (...) Part II supplements the material covered in Part I and introduces some of the newer ideas and the more profound results of logical research in the twentieth century. Subsequent chapters introduce the study of formal number theory, with surveys of the famous incompleteness and undecidability results of Godel, Church, Turing, and others. The emphasis in the final chapter reverts to logic, with examinations of Godel's completeness theorem, Gentzen's theorem, Skolem's paradox and nonstandard models of arithmetic, and other theorems. Unabridged republication of the edition published by John Wiley & Sons, Inc. New York, 1967. Preface. Bibliography. Theorem and Lemma Numbers: Pages. List of Postulates. Symbols and Notations. Index. (shrink)

In recent years, several remarkable results have shown that certain theorems of finite combinatorics are unprovable in certain logical systems. These developments have been instrumental in stimulating research in both areas, with the interface between logic and combinatorics being especially important because of its relation to crucial issues in the foundations of mathematics which were raised by the work of Kurt Godel. Because of the diversity of the lines of research that have begun to shed light on these issues, there (...) was a need for a comprehensive overview which would tie the lines together. This volume fills that need by presenting a balanced mixture of high quality expository and research articles that were presented at the August 1985 AMS-IMS-SIAM Joint Summer Research Conference, held at Humboldt State University in Arcata, California.With an introductory survey to put the works into an appropriate context, the collection consists of papers dealing with various aspects of 'unprovable theorems and fast-growing functions'. Among the topics addressed are: ordinal notations, the dynamical systems approach to Ramsey theory, Hindman's finite sums theorem and related ultrafilters, well quasiordering theory, uncountable combinatorics, nonstandard models of set theory, and a length-of-proof analysis of Godel's incompleteness theorem. Many of the articles bring the reader to the frontiers of research in this area, and most assume familiarity with combinatorics and/or mathematical logic only at the senior undergraduate or first-year graduate level. (shrink)

I begin by distinguishing two notions of model, the notion of a truth-making structure and the notion of a mathematical model (in one specific sense). I then argue that although the models of the semantic view have often been taken to be both truth-making structures and mathematical models, this is in part due to a failure to distinguish between two ways of truth-making; in fact, the talk of truth-making is best excised from (...) the view altogether. The result is a version of the semantic view which is better supported by the direct evidence offered for it, better equipped to achieve its avowed aims, and, I think, closer to the intentions of the original proponents of the view in many ways, despite some of their own declarations to the contrary. (shrink)

How do we explain the truth of true propositions? Truthmaker theory is the branch of metaphysics that explores the relationships between what is true and what exists. It plays an important role in contemporary debates about the nature of metaphysics and metaphysical enquiry. -/- In this book Jonathan Tallant argues, controversially, that we should reject truthmaker theory. In its place he argues for an 'explanationist' approach. Drawing on a deflationary theory of truth he shows that it allows us (...) to explain the truth of true propositions and respond to recent arguments that purport to show otherwise. He augments this with a distinction between internally and externally quantified claims: externally quantified claims are claims that quantify over elements of our ontology that play an indispensable explanatory role; internally quantified claims do not. He deploys this union of deflationism and a distinction between kinds of quantification to pursue metaphysical inquiry, sketching the implications for a number of first-order debates, including those in the philosophy of time, modality and mathematics, and also shows how this explanationist model can be used to solve the key problems that afflicted truthmaker theory. -/- Truth and the World is an important contribution to debates about truth and truthmaker theory as well as metametaphysics, the metaphysics of time and the metaphysics of mathematics, and is essential reading for students and scholars engaged in the study of these topics. (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)

Modeling is an important scientific practice, yet it raises significant philosophical puzzles. Models are typically idealized, and they are often explored via imaginative engagement and at a certain “distance” from empirical reality. These features raise questions such as what models are and how they relate to the world. Recent years have seen a growing discussion of these issues, including a number of views that treat modeling in terms of indirect representation and analysis. Indirect views treat the model as (...) a bona fide object, specified by the modeler and used to represent and reason about some portion of the concrete empirical world. On some indirect views, model systems are abstract entities, such as mathematical structures, while on other views they are concrete hypothetical things. Here I assess these views and offer a novel account of models. I argue that regarding models as abstracta results in some significant tensions with the practice of modeling, especially in areas where non-mathematical models are common. Furthermore, viewing models as concrete hypotheticals raises difficult questions about model-world relations. The view I argue for treats models as direct, albeit simplified, representations of targets in the world. I close by suggesting a treatment of model-world relations that draws on a recent work by Stephen Yablo concerning the notion of partial truth. (shrink)

SummaryThe platonist, in affirming the principle of bivalence for sentences for which there is no decision procedure, disconnects their truth‐conditions from conditions that would enable us to prove them ‐ as if Goldbach's conjecture, say, might just happen to be true. According to Dummett, what has gone wrong here is that the meaning of the relevant sentences has been conceived so as to go beyond what could be learned in learning to use them, or displayed in using them competently. (...) Dummett draws the general conclusion that accounts of meaning must traffic only in decidable circumstances. I suggest that Dummett can be right about platonism but wrong in this general conclusion: the centrality of decidable circumstances in competent use of language is a special feature of mathematical language. This means that someone who recoils from the anti‐realism constituted by Dummett's generalized anti‐platonism, in the case of, say, statements about other minds, need not be recoiling into a close analogue of platonism, as Dummett suggests. We can reinstate the intuitive idea that platonism goes wrong by inappropriately modelling the epistemology and metaphysics of mathematics on the epistemology and metaphysics of the natural world. And we make room for the suggestion that anti‐realism makes a converse mistake; in this vein, I propose a picture of Dummettian anti‐realism as a novel expression of familiar and suspect epistemological and metaphysical thoughts. (shrink)

On a traditional view, the primary role of a mathematical proof is to warrant the truth of the resulting theorem. This view fails to explain why it is very often the case that a new proof of a theorem is deemed important. Three case studies from elementary arithmetic show, informally, that there are many criteria by which ordinary proofs are valued. I argue that at least some of these criteria depend on the methods of inference the proofs employ, (...) and that standard models of formal deduction are not well-equipped to support such evaluations. I discuss a model of proof that is used in the automated deduction community, and show that this model does better in that respect. (shrink)

After a brief survey of the different meanings of consistency, the study is restricted to consistency understood as non-contradiction of sets of sentences. The philosophical reasons for this requirement are discussed, both in relation to the problem of sense and the problem of truth. The issue of mathematical truth is then addressed, and the different conceptions of it are put in relation with consistency. The formal treatment of consistency and truth in mathematical logic is then (...) considered, with particular attention paid to the relation between syntactic and semantic properties of sets and calculi. After the crisis of mathematical intuition and the dominance of the formalistic view, it seemed that consistency could totally replace the requirement of truth in mathematics, also in the sense that the existence of "objects" of axiomatic systems could be granted by their consistency. A rejection of this claim is presented, whose central point is a detailed analysis of the theorem that any consistent set S of sentences of first order logic has a model. A critical scrutiny shows that this model is very peculiar, being offered by the elements of the same language that is being interpreted, and the satisfiability conditions for any sentence being constituted by the mere fact of belonging to S. Though not being insignificant from a metatheoretical point of view, this theorem fails to endow consistency with an "ontological creativity", that is, with the capability of providing a model ontologically distinct from the language itself. A final discussion regarding the different "ontological regions" and the referential nature of truth clarifies the different aspects of the whole issue discussed. (shrink)

This volume presents the proceedings from the Eleventh Brazilian Logic Conference on Mathematical Logic held by the Brazilian Logic Society in Salvador, Bahia, Brazil. The conference and the volume are dedicated to the memory of professor Mario Tourasse Teixeira, an educator and researcher who contributed to the formation of several generations of Brazilian logicians. Contributions were made from leading Brazilian logicians and their Latin-American and European colleagues. All papers were selected by a careful refereeing processs and were revised and (...) updated by their authors for publication in this volume. There are three sections: Advances in Logic, Advances in Theoretical Computer Science, and Advances in Philosophical Logic. Well-known specialists present original research on several aspects of model theory, proof theory, algebraic logic, category theory, connections between logic and computer science, and topics of philosophical logic of current interest. Topics interweave proof-theoretical, semantical, foundational, and philosophical aspects with algorithmic and algebraic views, offering lively high-level research results. (shrink)

We study, from a classical point of view, how the truth of a statement about higher type functionals depends on the underlying model. The models considered are the classical set-theoretic finite type hierarchy and the constructively more meaningful models of continuous functionals, hereditarily effective operations, as well as the closed term model of Gödel's system T. The main results are characterisations of prenex classes for which truth in the full set-theoretic model transfers to truth in (...) the other models. As a corollary we obtain that the axiom of choice is not conservative over Gödel's system T with classical logic for closed ∃2-formulas. We hope that this study will contribute to a clearer delineation and perception of constructive mathematics from a classical perspective. (shrink)

§1. The mission of axiomatic set theory. What is set theory needed for in the foundations of mathematics? Why cannot we transact whatever foundational business we have to transact in terms of our ordinary logic without resorting to set theory? There are many possible answers, but most of them are likely to be variations of the same theme. The core area of ordinary logic is by a fairly common consent the received first-order logic. Why cannot it take care of itself? (...) What is it that it cannot do? A large part of every answer is probably that first-order logic cannot handle its own model theory and other metatheory. For instance, a first-order language does not allow the codification of the most important semantical concept, viz. the notion of truth, for that language in that language itself, as shown already in Tarski. In view of such negative results it is generally thought that one of the most important missions of set theory is to provide the wherewithal for a model theory of logic. For instance Gregory H. Moore asserts in his encyclopedia article “Logic and set theory” thatSet theory influenced logic, both through its semantics, by expanding the possible models of various theories and by the formal definition of a model; and through its syntax, by allowing for logical languages in which formulas can be infinite in length or in which the number of symbols is uncountable. (shrink)

There is a long line of thinkers in the philosophy of mathematics who have sought to base an account of proof on what might be called a 'metaphysical ordering' of the truths of mathematics. Use the term 'metaphysical' to describe these orderings is intended to call attention to the fact that they are regarded as objective and not subjective and that they are conceived primarily as orderings of truths and only secondarily as orderings of beliefs. -/- I describe and consider (...) two models for such orderings, (i) an implicational model and something I call (ii) a mosaic model. -/- Both models fail to do what such orderings should do. Moreover, in both cases, the trouble arises because the class of truths to be ordered is taken to be closed under various weakening implications. This is a result of their employing a global conception of logic. (shrink)

Well-written undergraduate-level introduction begins with symbolic logic and set theory, followed by presentation of statement calculus and predicate calculus. First-order theories are discussed in some detail, with special emphasis on number theory. After a discussion of truth and models, the completeness theorem is proved. "...an excellent text."—Mathematical Reviews. Exercises. Bibliography.

In this article, we study logics of unknown truths and false beliefs under neighborhood semantics. We compare the relative expressivity of the two logics. It turns out that they are incomparable over various classes of neighborhood models, and the combination of the two logics are equally expressive as standard modal logic over any class of neighborhood models. We propose morphisms for each logic, which can help us explore the frame definability problem, show a general soundness and completeness result, (...) and generalize some results in the literature. We axiomatize the two logics over various classes of neighborhood frames. Most importantly, by adopting the intersection semantics and the subset semantics in the literature, we extend the results to the case of public announcements, which gives us the desired reduction axioms and has good applications related to Moore sentences, successful formulas and self-refuting formulas. Also, we can say something about the comparative merits of the intersection semantics and the subset semantics. (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)

The prime purpose of this paper is, first, to restore to discourse-bound occasion sentences their rightful central place in semantics and secondly, taking these as the basic propositional elements in the logical analysis of language, to contribute to the development of an adequate logic of occasion sentences and a mathematical foundation for such a logic, thus preparing the ground for more adequate semantic, logical and mathematical foundations of the study of natural language. Some of the insights elaborated in (...) this paper have appeared in the literature over the past thirty years, and a number of new developments have resulted from them. The present paper aims atproviding an integrated conceptual basis for this new development in semantics. In Section 1 it is argued that the reduction by translation of occasion sentences to eternal sentences, as proposed by Russell and Quine, is semantically and thus logically inadequate. Natural language is a system of occasion sentences, eternal sentences being merely boundary cases. The logic hasfewer tasks than is standardly assumed, as it excludes semantic calculi, which depend crucially on information supplied by cognition and context and thus belong to cognitive psychology rather than to logic. For sentences to express a proposition and thus be interpretable and informative, they must first be properly anchored in context. A proposition has a truth value when it is, moreover, properly keyed in the world, i.e. is about a situation in the world. Section 2 deals with the logical properties of natural language. It argues that presuppositional phenomena require trivalence and presents the trivalent logic PPC3, with two kinds of falsity and two negations. It introduces the notion of -space for a sentence A as the basis of logical model theory, and the notion of P A /, functioning as a `private' subuniverse for A/A. The trivalent Kleene calculus is reinterpreted as a logical account of vagueness, rather than of presupposition. PPC3 and the Kleene calculus are refinements of standard bivalent logic and can be combined into one logical system. In Section 3 the adequacy of PPC3 as a truth-functional model of presupposition is considered more closely and given a Boolean foundation. In a noncompositional extended Boolean algebra, three operators are defined: 1a for the conjoined presuppositions of a, ã for the complement of a within 1a, and â for the complement of 1a within Boolean 1. The logical properties of this extended Boolean algebra are axiomatically defined and proved for all possible models. Proofs are provided of the consistency and the completeness of the system. Section 4 is a provisional exploration of the possibility of using the results obtained for a new discourse-dependent account of the logic of modalities in natural language. The overall result is a modified and refined logical and model-theoretic machinery, which takes into account both the discourse-dependency of natural language sentences and the necessity of selecting a key in the world before a truth value can be assigned. (shrink)

The prime purpose of this paper is, first, to restore to discourse-bound occasion sentences their rightful central place in semantics and secondly, taking these as the basic propositional elements in the logical analysis of language, to contribute to the development of an adequate logic of occasion sentences and a mathematical (Boolean) foundation for such a logic, thus preparing the ground for more adequate semantic, logical and mathematical foundations of the study of natural language. Some of the insights elaborated (...) in this paper have appeared in the literature over the past thirty years, and a number of new developments have resulted from them. The present paper aims at providing an integrated conceptual basis for this new development in semantics. In Section 1 it is argued that the reduction by translation of occasion sentences to eternal sentences, as proposed by Russell and Quine, is semantically and thus logically inadequate. Natural language is a system of occasion sentences, eternal sentences being merely boundary cases. The logic has fewer tasks than is standardly assumed, as it excludes semantic calculi, which depend crucially on information supplied by cognition and context and thus belong to cognitive psychology rather than to logic. For sentences to express a proposition and thus be interpretable and informative, they must first be properly anchored in context. A proposition has a truth value when it is, moreover, properly keyed in the world, i.e. is about a situation in the world. Section 2 deals with the logical properties of natural language. It argues that presuppositional phenomena require trivalence and presents the trivalent logic ${\rm PPC}_{3}$ , with two kinds of falsity and two negations. It introduces the notion of Σ-space for a sentence A (or / A /, the set of situations in which A is true) as the basis of logical model theory, and the notion of / ${\bf P}^{A}$ / (the Σ-space of the presuppositions of A), functioning as a 'private' subuniverse for / A /. The trivalent Kleene calculus is reinterpreted as a logical account of vagueness, rather than of presupposition. ${\rm PPC}_{3}$ and the Kleene calculus are refinements of standard bivalent logic and can be combined into one logical system. In Section 3 the adequacy of ${\rm PPC}_{3}$ as a truth-functional model of presupposition is considered more closely and given a Boolean foundation. In a noncompositional extended Boolean algebra, three operators are defined: $1_{a}$ for the conjoined presuppositions of a, ã for the complement of a within $1_{a}$ , and â for the complement of $1_{a}$ within Boolean 1. The logical properties of this extended Boolean algebra are axiomatically defined and proved for all possible models. Proofs are provided of the consistency and the completeness of the system. Section 4 is a provisional exploration of the possibility of using the results obtained for a new discourse-dependent account of the logic of modalities in natural language. The overall result is a modified and refined logical and model-theoretic machinery, which takes into account both the discourse-dependency of natural language sentences and the necessity of selecting a key in the world before a truth value can be assigned. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Journal of the History of Philosophy 41.3 (2003) 426-427 [Access article in PDF] Timothy Smiley, editor. Mathematics and Necessity: Essays in the History of Philosophy. New York: Oxford University Press, 2000. Pp. ix + 166. Cloth, $35.00.Mathematics and Necessity contains essays by M. F. Burnyeat, Ian Hacking, and Jonathan Bennett based on lectures given to the British Academy in 1998. All concern the history of the philosophical treatment of (...) mathematics, necessity, or both, although there is no single thread running through all three essays.Burnyeat focuses on Plato's understanding of mathematics in the Republic, making use of the Timeaus in particular to defend his interpretation. Plato claimed that in the ideal society mathematics should be studied for a full ten years, which many have thought excessive. To explain it, Burnyeat argues that the role of mathematics is pervasive: ethics and politics are mathematical, because in Plato's view mathematics is a constitutive part of ethical understanding. On Burnyeat's interpretation, goodness is part of the objective world, and mathematics is a science of values. The link between mathematics and goodness in the objective world is forged by the concord, harmony, and unity of proportions, which express the goodness of the Divine Craftsman's beneficent design. Proportions are both intrinsically good and the proper study of mathematics.Burnyeat explains why Plato thinks abstract kinematics (the pure non-sensible counterpart to astronomy) and abstract harmonics lead to knowledge of the good. Abstract kinematics concerns the "harmony of the spheres" described in the Republic. Drawing heavily on the Timeaus, Burnyeat argues that the circular motions of the spheres are motions in the intelligence of the world soul and that there are similar motions in human souls. The world soul and human souls are spatial and have circular motions (invisibility and intangibility make them incorporeal). Souls are the most important subject matter of abstract harmonics—the study of good proportions. A soul that understands the various mathematical disciplines and their relations takes on the harmony and hence goodness of the world soul. Such a soul serves as a model for organizing the social world, a model which properly trained rulers can use to shape a community. Burnyeat's well-argued essay is compelling and exciting. [End Page 427]Ian Hacking's views provide a great contrast to Burnyeat's Plato. Hacking conducts a phenomenological inquiry into why some philosophers have become obsessed by mathematics as a source of philosophical inspiration. He holds that mathematics concerns a relatively minor part of human culture and that the influence of mathematics on philosophy is pernicious; he even describes it as an infection.Hacking examines the views of six philosophers: Plato, Leibniz, Mill, Descartes, Lakatos, and Wittgenstein. Hacking's assessment of the six philosophers is at times disputable, but it is also interesting and stimulating, and serves his phenomenological inquiry.Hacking traces philosophy's infection by mathematics to two factors. A mathematical proof can be perspicuous; that is, a proof can be "grasped," which requires seeing directly why something must be true and being able to recapitulate it. A mathematical proof can also anticipate facts about the world. He thinks that these two factors affect us in two ways: they give us the idea of a priori knowledge, and they dazzle us into unreasoned conviction in proofs. One might respond that, setting aside the feeling proofs give us, we are still justified in being impressed by them. Hacking argues that much of what impresses is based on a misunderstanding of their nature, which he attempts to correct.According to Hacking, a mathematical demonstration involves shifts of meaning that analytically bind the proof and the theorem it establishes. They thereby mutually support each other and provide what he calls self-authentication. The correctness of the proof is supported by the "analytified" truth of the theorem and the theorem is considered true because proven.This bootstrapping feature of mathematical proofs supports Hacking's deflationary account of necessity. Hacking suggests that the "must" of logical necessity rests upon our unwillingness to accept empirical counter-examples to the... (shrink)

In recent years, semantical partiality has emerged as an important explanatory concept in philosophical logic as well as in the study of natural language semantics. Despite the many applications, however, a number of mathematically intriguing questions associated with this concept have received only very limited attention. ;The present dissertation aims to present a systematic study of certain types of partiality in the area of basic model theory. Two types of issues are given special attention: Introducing partially defined models, there (...) are many ways to generalize the classical truth definition for sentences of a simple first order language relative to standard, complete models. Different interpretations of the formal framework motivate conflicting truth definitions between language and partial models: A partial model can be taken to represent a part of the world, or a partial information set. The truth of a sentence can be supported directly by a part of the world, but can also follow indirectly from an information set. These notions are related, and the relation motivates a comparison between various weaker and stronger alternative truth definitions. Results are obtained about the extent to which these truth definitions differ, and a number of characterization results are deduced. ;Among other conditions that are not expressible in the framework of standard, complete model theory, a condition of monotonicity or persistence of truth relative to partial models is argued to follow under both the given interpretations of the formal framework. The final chapter investigates the relation between such conditions and expressibility properties in general. These discussions culminate with a combined Lindstrom and persistence characterization theorem. (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)

The model theoretic concept of truth has thus far been applied mainly to mathematical languages and theories. The paper presents an attempt to apply it to languages of empirical theories. Such an application must do justice to some characteristic features of empirical discourse. The paper outlines the main problems which a model theoretic theory of truth for empirical languages is bound to face and suggests some solutions to those problems.

The model theoretic concept of truth has thus far been applied mainly to mathematical languages and theories. The paper presents an attempt to apply it to languages of empirical theories. Such an application must do justice to some characteristic features of empirical discourse. The paper outlines the main problems which a model theoretic theory of truth for empirical languages is bound to face and suggests some solutions to those problems.

Different formal tools are useful for different purposes. For example, when it comes to modelling degrees of belief, probability theory is a better tool than classical logic; when it comes to modelling the truth of mathematical claims, classical logic is a better tool than probability theory. In this paper I focus on a widely used formal tool and argue that it does not provide a good model of a phenomenon of which many think it does provide a good (...) model: I shall argue that while supervaluationism may provide a model of probability of truth, or of assertability, it cannot provide a good model of truth -- supertruth cannot be truth. The core of the argument is that an adequate model of truth must render certain connectives truth-functional (at least in certain circumstances) -- and supervaluationism does not do so (in those circumstances). (shrink)

The relation between theoretical models and metaphors has been studied since at least the 1950s. The relation between metaphors and mathematical modelling is less well researched. This article takes the media coverage of the foot and mouth modelling exercise in 2001 as an occasion to examine the metaphors of mathematical modelling that were proposed by the UK press during that time to make sense of this new scientific policy tool. One can detect a gradual change in metaphor (...) use by the newspapers from conceptualizing modellers as detectives and models as mapping tools to modellers as soldiers and heroes, to modellers as liars and models as tools to distort the truth. This seems to indicate a shift in reporting from seeing models as a legitimate and "objective" basis used by decision makers to pursue science-based policies towards seeing models as tools used to legitimize increasingly difficult political decisions. (shrink)