Carnap's early system of inductive logic make degrees of confirmation depend on the languages in which they are expressed. They are sensitive to which predicates are, in the language, taken as primitive. Hence they fail to be ‘linguistically invariant’. His later systems, in which prior probabilities are assigned to elements of a model rather than sentences of a language, are sensitive to which properties in the model are called primitive. Critics have often protested against these features (...) of his work. This paper shows how to make his systems independent of any choice of primitive predicates or primitive properties.The solution is related to another criticism of inductive logic. It has been noticed that Carnap's systems are too all-embracing. Hisc(h, e) is defined for all sentencesh ande. Yet for manyh ande, the evidencee does not warrant any assessment of the probability ofh. We need an inductive logic in whichc(h, e) is defined only whene really does bear onh. This paper sketches the measure theory of such a logic, and, within this measure theory, provides ‘relativized’ versions of Carnap's systems which are linguistically invariant. (shrink)

We give a brief account of some de Finetti style representation theorems for probability functions satisfying Spectrum Exchangeability in Polyadic Inductive Logic, together with applications to Non-splitting, Language Invariance, extensions with Equality and Instantial Relevance.

In Pure Inductive Logic, the rational principle of Predicate Exchangeability states that permuting the predicates in a given language L and replacing each occurrence of a predicate in an L-sentence phi according to this permutation should not change our belief in the truth of phi. In this paper we study when a prior probability function w on a purely unary language L satisfying Predicate Exchangeability also satisfies the principle of Unary Language Invariance.

The structure of Reichenbach's pragmatic vindication of induction is analysed in detail. The argument is seen to proceed in two stages, the first being a pragmatic justification of the frequency interpretation of probability which is taken as a license for considering the aim of induction to be the discovery of limiting relative frequencies, and the second being the pragmatic justification of induction itself. Both justifications are found to contain flaws, and the arguments used to support Reichenbach's definition of the aim (...) of induction presuppose the availability of a type of predicate which generates paradoxes closely related to Goodman's "grue-bleen" paradox. Next, Salmon's "Criterion of Linguistic Invariance" is evaluated as a canon of inductive logic, which singles out the "straight rule" from an infinite class of convergent inductive rules. Upon close examination, its credentials are seen to be unimpressive. In connection with Salmon's work on linguistic invariance, we take a closer look at the impact of the Goodman paradox on probability estimator rules like Reichenbach's "straight rule." We find that its undesirable consequences for such systems of inductive logic cannot be escaped by Salmon's solution, or indeed by any solution whose motivating idea is to rule "queer" predicates out of court. Finally, the discussion of the Goodman paradox leads us to a modest methodological proposal for systems of inductive logic which incorperate probability estimator rules of the type at issue. (shrink)

Although the invariance criterion of logicality first emerged as a criterion of a purely mathematical interest, it has developed into a criterion of considerable linguistic and philosophical interest. In this paper I compare two different perspectives on this criterion. The first is the perspective of natural language. Here, the invariance criterion is measured by its success in capturing our linguistic intuitions about logicality and explaining our logical behavior in natural-linguistic settings. The second perspective is more theoretical. Here, the invariance criterion (...) is used as a tool for developing a theoretical foundation of logic, focused on a critical examination, explanation, and justification of its veridicality and modal force. (shrink)

A necessary and sufficient condition in terms of a de Finetti style representation is given for a probability function in Polyadic Inductive Logic to satisfy being part of a Language Invariant family satisfying Spectrum Exchangeability. This theorem is then considered in relation to the unary Carnap and Nix–Paris Continua.

According to a popular interpretation, Carnap’s interpretation of probability had evolved from a logical towards a subjective conception. However Carnap himself insisted that his basic philosophical view of probability was always the same. I address this apparent clash between Carnap's self-identification and the subsequent interpretations of his work. Following its original intentions, I reconstruct inductive logic as an explication. The emerging picture is of a versatile linguistic framework, whose main function is not the discovery of objective logical relations (...) in the object language, but the stipulation of conceptual possibilities. Within this representation, I map out the changes that the project went through. Seen from such an explication-based perspective, inductive logic becomes quite hard to categorize using the standard labels. (shrink)

A family of symmetries of polyadic inductive logic are described which in turn give rise to the purportedly rational Permutation Invariance Principle stating that a rational assignment of probabilities should respect these symmetries. An equivalent, and more practical, version of this principle is then derived.

We give a unified account of some results in the development of Polyadic Inductive Logic in the last decade with particular reference to the Principle of Spectrum Exchangeability, its consequences for Instantial Relevance, Language Invariance and Johnson's Sufficientness Principle, and the corresponding de Finetti style representation theorems.

We investigate the relative probabilistic support afforded by the combination of two analogies based on possibly different, structural similarity (as opposed to e.g. shared predicates) within the context of Pure Inductive Logic and under the assumption of Language Invariance. We show that whilst repeated analogies grounded on the same structural similarity only strengthen the probabilistic support this need not be the case when combining analogies based on different structural similarities. That is, two analogies may provide less support than (...) each would individually. (shrink)

We consider two formalizations of the notion of irrelevance as a rationality principle within the framework of (Carnapian) Inductive Logic: Johnson's Sufficientness Principle, JSP, which is classically important because it leads to Carnap's influential Continuum of Inductive Methods and the recently proposed Weak Irrelevance Principle, WIP. We give a complete characterization of the language invariant probability functions satisfying WIP which generalizes the Nix-Paris Continuum. We argue that the derivation of two very disparate families of inductive (...) methods from alternative perceptions of 'irrelevance' is an indication that this notion is imperfectly understood at present. (shrink)

According to the objective Bayesian approach to inductive logic, premisses inductively entail a conclusion just when every probability function with maximal entropy, from all those that satisfy the premisses, satisfies the conclusion. When premisses and conclusion are constraints on probabilities of sentences of a first-order predicate language, however, it is by no means obvious how to determine these maximal entropy functions. This paper makes progress on the problem in the following ways. Firstly, we introduce the concept of a (...) limit in entropy and show that, if the set of probability functions satisfying the premisses contains a limit in entropy, then this limit point is unique and is the maximal entropy probability function. Next, we turn to the special case in which the premisses are categorical sentences of the logical language. We show that if the uniform probability function gives the premisses positive probability, then the maximal entropy function can be found by simply conditionalising this uniform prior on the premisses. We generalise our results to demonstrate agreement between the maximal entropy approach and Jeffrey conditionalisation in the case in which there is a single premiss that specifies the probability of a sentence of the language. We show that, after learning such a premiss, certain inferences are preserved, namely inferences to inductive tautologies. Finally, we consider potential pathologies of the approach: we explore the extent to which the maximal entropy approach is invariant under permutations of the constants of the language, and we discuss some cases in which there is no maximal entropy probability function. (shrink)

We discuss the principles for a primitive, object-linguistic notion of consequence proposed by ) that yield a version of Curry’s paradox. We propose and study several strategies to weaken these principles and overcome paradox: all these strategies are based on the intuition that the object-linguistic consequence predicate internalizes whichever meta-linguistic notion of consequence we accept in the first place. To these solutions will correspond different conceptions of consequence. In one possible reading of these principles, they give rise to a notion (...) of logical consequence: we study the corresponding theory of validity by showing that it is conservative over a wide range of base theories: this result is achieved via a well-behaved form of local reduction. The theory of logical consequence is based on a restriction of the introduction rule for the consequence predicate. To unrestrictedly maintain this principle, we develop a conception of object-linguistic consequence, which we call grounded consequence, that displays a restriction of the structural rule of reflexivity. This construction is obtained by generalizing Saul Kripke’s inductive theory of truth. Grounded validity will be shown to satisfy several desirable principles for a naïve, self-applicable notion of consequence. (shrink)

This collection of new essays presents cutting-edge research on the semantic conception of logic, the invariance criteria of logicality, grammaticality, and logical truth. Contributors explore the history of the semantic tradition, starting with Tarski, and its historical applications, while central criticisms of the tradition, and especially the use of invariance criteria to explain logicality, are revisited by the original participants in that debate. Other essays discuss more recent criticism of the approach, and researchers from mathematics and linguistics weigh in (...) on the role of the semantic tradition in their disciplines. This book will be invaluable to philosophers and logicians alike. (shrink)

This volume comprises a selection of papers that were contributed to the 7th International Congress of Logic, Methodology and Philosophy of Science, which was held in Salzburg from the 11th - 16th July, 1983. There were 14 sections in this congress: 1. proof theory and foundations of mathematics 2. model theory and its applica ti on 3. recursion theory and theory of computation 4. axiomatic set theory 5. philosophical logic 6. general methodology of science 7. foundations of probability (...) and induction 8. foundations and philosophy of the physical sciences 9. foundati ons and phi 1 osophy of biology 10. foundations and philosophy of psychology foundations and philosophy 11. of the social sciences 12. foundati ons and philosophy of linguistics 13. history of logic, methodology and philosophy of science 14. fundamental principles of the ethics of science In each section, three or four invited addresses were given, which will be published in the Congress Proceedings (Ruth Barcan Marcus, Georg J. W. Dorn and Paul Weingartner, eds. : Logic, Metho dology and Philosophy of Science VII. Proceedings of the Seventh International Congress of Logic, Methodology and Philosophy of v PREFACE Science, Salzburg, 1983. - Amsterdam, New York, Oxford: North-Holland Publishing 'Company, 1985. ) Every section with the exception of section 14 also contained contributed papers. (shrink)

There were in the past, just as there are in the present, several diverse attempts to establish a unique theory capable of identifying in all natural languages a similar, invariable basic structure of a logical nature. If such a theory exists, then there must be principles that rule the functioning of these languages and they must have a logical origin. Based on a work by the French linguist, Oswald Ducrot, entitled D’un mauvais usage de la logique , this paper aims (...) to present in a concise manner two of the above mentioned attempts. They were elaborated in diverse epochs and different arguments were put forward to support them. The first attempt was in XVII century France and its theoretic basis was the renowned ‘Port-Royal Logic’. The second attempt is recent and its theoretic support comes from Contemporary Logic. DOI: 10.5007/1808-1711.2011v15n1p111. (shrink)

This paper deals with the problem of vindicating a particular type of inductive rule, a rule to govern inferences from observed frequencies to limits of relative frequencies. Reichenbach's rule of induction is defended. By application of two conditions, normalizing conditions and a criterion of linguistic invariance, it is argued that alternative rules lead to contradiction. It is then argued that the rule of induction does not lead to contradiction when suitable restrictions are placed upon the predicates admitted. Goodman's grue-bleen (...) paradox is considered, and an attempt to resolve it is offered. Finally, Reichenbach's pragmatic argument, hinging on convergence properties, is applied. (shrink)

There were in the past, just as there are in the present, several diverse attempts to establish a unique theory capable of identifying in all natural languages a similar, invariable basic structure of a logical nature. If such a theory exists, then there must be principles that rule the functioning of these languages and they must have a logical origin. Based on a work by the French linguist, Oswald Ducrot, entitled D’un mauvais usage de la logique, this paper aims to (...) present in a concise manner two of the above mentioned attempts. They were elaborated in diverse epochs and different arguments were put forward to support them. The first attempt was in XVII century France and its theoretic basis was the renowned ‘Port-Royal Logic’. The second attempt is recent and its theoretic support comes from Contemporary Logic. (shrink)

In his 1974 paper “Popper’s qualitative theory of verisimilitude” published in the British Journal for the Philosophy of Science David Miller gave his so called ‘Weather Example’ to argue that the Hamming distance between constituents is flawed as a measure of proximity to truth since the former is not, unlike the latter, translation invariant. In this present paper we generalise David Miller’s Weather Example in both the unary and polyadic cases, characterising precisely which permutations of constituents/atoms can be effected (...) by translations. In turn this suggests a meta-principle of the rational assignment of subjective probabilities, that rational principles should be preserved under translations, which we formalise and give a particular characterisation of in the context of Unary Pure Inductive Logic. (shrink)

A computational theory of induction must be able to identify the projectible predicates, that is to distinguish between which predicates can be used in inductive inferences and which cannot. The problems of projectibility are introduced by reviewing some of the stumbling blocks for the theory of induction that was developed by the logical empiricists. My diagnosis of these problems is that the traditional theory of induction, which started from a given (observational) language in relation to which all inductive (...) rules are formulated, does not go deep enough in representing the kind of information used in inductive inferences. As an interlude, I argue that the problem of induction, like so many other problems within AI, is a problem of knowledge representation. To the extent that AI-systems are based on linguistic representations of knowledge, these systems will face basically the same problems as did the logical empiricists over induction. In a more constructive mode, I then outline a non-linguistic knowledge representation based on conceptual spaces. The fundamental units of these spaces are "quality dimensions". In relation to such a representation it is possible to define "natural" properties which can be used for inductive projections. I argue that this approach evades most of the traditional problems. (shrink)

This is part II in a series of papers outlining Abstraction Theory, a theory that I propose provides a solution to the characterisation or epistemological problem of induction. Logic is built from first principles severed from language such that there is one universal logic independent of specific logical languages. A theory of (non-linguistic) meaning is developed which provides the basis for the dissolution of the `grue' problem and problems of the non-uniqueness of probabilities in inductive logics. The (...) problem of counterfactual conditionals is generalised to a problem of truth conditions of hypotheses and this general problem is then solved by the notion of abstractions. The probability calculus is developed with examples given. In future parts of the series the full decision theory is developed and its properties explored. (shrink)

The Fifth International Congress of Logic, Methodology and Philosophy of Science was held at the University of Western Ontario, London, Canada, 27 August to 2 September 1975. The Congress was held under the auspices of the International Union of History and Philosophy of Science, Division of Logic, Methodology and Philosophy of Science, and was sponsored by the National Research Council of Canada and the University of Western Ontario. As those associated closely with the work of the Division over (...) the years know well, the work undertaken by its members varies greatly and spans a number of fields not always obviously related. In addition, the volume of work done by first rate scholars and scientists in the various fields of the Division has risen enormously. For these and related reasons it seemed to the editors chosen by the Divisional officers that the usual format of publishing the proceedings of the Congress be abandoned in favour of a somewhat more flexible, and hopefully acceptable, method of pre sentation. Accordingly, the work of the invited participants to the Congress has been divided into four volumes appearing in the University of Western Ontario Series in Philosophy of Science. The volumes are entitled, Logic, Foundations of Mathematics and Computability Theory, Foun dational Problems in the Special Sciences, Basic Problems in Methodol ogy and Linguistics, and Historical and Philosophical Dimensions of Logic, Methodology and Philosophy of Science. (shrink)

Gupta-Belnap-style circular definitions use all real numbers as possible starting points of revision sequences. In that sense they are boldface definitions. We discuss lightface versions of circular definitions and boldface versions of inductive definitions.

The new area of logic and computation is now undergoing rapid development. This has affected the social pattern of research in the area. A new topic may rise very quickly with a significant body of research around it. The community, however, cannot wait the traditional two years for a book to appear. This has given greater importance to thematic collections of papers, centred around a topic and addressing it from several points of view, usually as a result of a (...) workshop, summer school, or just a scientific initiative. Such a collection may not be as coherent as a book by one or two authors yet it is more focused than a collection of key papers on a certain topic. It is best thought of as a thematic collection, a study in the area of logic and computation. The new series Studies in Logic and Computation is intended to provide a home for such thematic collections. Substructural logics are nonclassical logics, which arose in response to problems in foundations of mathematics and logic, theoretical computer science, mathematical linguistics, and category theory. They include intuitionistic logic, relevant logic, BCK logic, linear logic, and Lambek's calculus of syntactic categories. Substructural logics differ from classical logics, and from each other, in their presuppositions about Gentzen's structural rules, although their presuppositions about the deductive role of logical constants are invariant. Substructural logics have been a subject of study for logicians during the last sixty years. Specialists have often worked in isolation, however, largely unaware of the contributions of others. This book brings together new papers by some of the most eminent authorities in these varioustraditions to produce a unified view of substructural logics. (shrink)

Logic, Methodology, and the Philosophy of Science, the Proceedings of the 1960 International Congress at Stanford, is heavily weighted towards technical problems of logic, foundations of mathematics, and the special sciences, especially psychology, economic models, and structural linguistics, with little discussion of general problems of the philosophy of science. Problems about the idealization involved in the relation of theories to the world become problems about probabilistic models at various levels of abstraction ; induction becomes a problem in decision (...) theory ; language becomes the subject-matter of formalized syntactics, and so on. There is nothing in the least undesirable about these inquiries in a Philosophy of Science Congress, but there is a danger that, if they are not balanced by more informal discussions, real points of dispute are missed, or decided over-hastily in the brief preliminaries of formalization. The philosopher's job ought sometimes to be the more irritating one of returning to these points of dispute; a job which is well done in some of the contributions to this volume, but one could wish to find it being done in more. In what follows I shall have to restrict myself to commenting on only a handful of these more philosophical papers. (shrink)

Gupta-Belnap-style circular definitions use all real numbers as possible starting points of revision sequences. In that sense they are boldface definitions. We discuss lightface versions of circular definitions and boldface versions of inductive definitions.

I present a solution to the epistemological or characterisation problem of induction. In part I, Bayesian Confirmation Theory (BCT) is discussed as a good contender for such a solution but with a fundamental explanatory gap (along with other well discussed problems); useful assigned probabilities like priors require substantive degrees of belief about the world. I assert that one does not have such substantive information about the world. Consequently, an explanation is needed for how one can be licensed to act as (...) if one has substantive information about the world when one does not. I sketch the outlines of a solution in part I, showing how it differs from others, with full details to follow in subsequent parts. The solution is pragmatic in sentiment (though differs in specifics to arguments from, for example, William James); the conceptions we use to guide our actions are and should be at least partly determined by preferences. This is cashed out in a reformulation of decision theory motivated by a non-reductive formulation of hypotheses and logic. A distinction emerges between initial assumptions--that can be non-dogmatic--and effective assumptions that can simultaneously be substantive. An explanation is provided for the plausibility arguments used to explain assigned probabilities in BCT. -/- In subsequent parts, logic is constructed from principles independent of language and mind. In particular, propositions are defined to not have form. Probabilities are logical and uniquely determined by assumptions. The problems considered fatal to logical probabilities--Goodman's `grue' problem and the uniqueness of priors problem are dissolved due to the particular formulation of logic used. Other problems such as the zero-prior problem are also solved. -/- A universal theory of (non-linguistic) meaning is developed. Problems with counterfactual conditionals are solved by developing concepts of abstractions and corresponding pictures that make up hypotheses. Spaces of hypotheses and the version of Bayes' theorem that utilises them emerge from first principles. -/- Theoretical virtues for hypotheses emerge from the theory. Explanatory force is explicated. The significance of effective assumptions is partly determined by combinatoric factors relating to the structure of hypotheses. I conjecture that this is the origin of simplicity. (shrink)

We study a simple game theoretic model of information transfer which we consider to be a baseline model for capturing strategic aspects of epistemological questions. In particular, we focus on the question whether simple learning rules lead to an efficient transfer of information. We find that reinforcement learning, which is based exclusively on payoff experiences, is inadequate to generate efficient networks of information transfer. Fictitious play, the game theoretic counterpart to Carnapian inductive logic and a more sophisticated kind (...) of learning, suffices to produce efficiency in information transfer. (shrink)

Linguistics as a science has rapidly changed during the course of a relatively short period. The mathematical foundations of the science, however, present a different story below the surface. In this paper, I argue that due to the former, the seismic shifts in theory over the past 80 years opens linguistics up to the problem of pessimistic meta-induction or radical theory change. I further argue that, due to the latter, one current solution to this problem in the philosophy of science, (...) namely structural realism :403–424, 1998; French in Proc Aristot Soc 106:167–185, 2006), should be viewed as especially enticing for linguists, as their field is a largely structural enterprise. I discuss particular historical instances of theory change in generative syntax before investigating two views on the nature of structural properties and eventually proposing an approach in terms of invariance :162–186, 2015) as a grounding for structural realism in the history and philosophy of linguistics. (shrink)

Frege proved an important result, concerning the relation of arithmetic to second-order logic, that bears on several issues in linguistics. Frege’s Theorem illustrates the logic of relations like PRECEDES(x, y) and TALLER(x, y), while raising doubts about the idea that we understand sentences like ‘Carl is taller than Al’ in terms of abstracta like heights and numbers. Abstract paraphrase can be useful—as when we say that Carl’s height exceeds Al’s—without reflecting semantic structure. Related points apply to causal relations, (...) and even grammatical relations like DOMINATES(x, y). Perhaps surprisingly, Frege provides the resources needed to recursively characterize labelled expressions without characterizing them as sets. His theorem may also bear on questions about the meaning and acquisition of number words. (shrink)

By formalizing recent syntactic theories for natural languages Stabler shows how their complexity can be handled without guesswork or oversimplification. By formalizing recent syntactic theories for natural languages in the tradition of Chomsky's Barriers, Stabler shows how their complexity can be handled without guesswork or oversimplification. He introduces logical representations of these theories together with special deductive techniques for exploring their consequences that will provide linguists with a valuable tool for deriving and testing theoretical predictions and for experimenting with alternative (...) formulations of grammatical principles. Stabler's novel approach allows results to be deduced with straightforward calculations and provides a systematic framework for tackling the problem of how speakers can infer the properties of an utterance from principles of the grammar. The special treatment of equality, induction principles, and inclusion of a general method for collecting structures from proofs means that sophisticated linguistic arguments can be carried out in detail, giving a rich perspective to issues in linguistic theory and parsing. (shrink)

Logic is a celebrated representation language because of its formal generality. But there are two senses in which a logic may be considered general, one that concerns a technical ability to discriminate between different types of individuals, and another that concerns constitutive norms for reasoning as such. This essay embraces the former, permutation-invariance conception of logic and rejects the latter, Fregean conception of logic. The question of how to apply logic under this pure invariantist view (...) is addressed, and a methodology is given. The pure invariantist view is contrasted with logical pluralism, and a methodology for applied logic is demonstrated in remarks on a variety of issues concerning non-monotonic logic and non-monotonic inference, including Charles Morgan’s impossibility results for non-monotonic logic, David Makinson’s normative constraints for non-monotonic inference, and Igor Douven and Timothy Williamson’s proposed formal constraints on rational acceptance. (shrink)

In his thesis 'Para uma Teoria Geral dos Homomorfismos' (1944) the Portuguese mathematician José Sebastião e Silva constructed an abstract or generalized Galois theory, that is intimately linked to F. Klein’s Erlangen Program and that foreshadows some notions and results of today’s model theory; an analogous theory was independently worked out by M. Krasner in 1938. In this paper, we present a version of the theory making use of tools which were not at Silva’s disposal. At the same time, we (...) tried to keep in mind, so much as possible, the gist of his standpoint. (shrink)

On the one hand, the absence of contraction is a safeguard against the logical (property theoretic) paradoxes; but on the other hand, it also disables inductive and recursive definitions, in its most basic form the definition of the series of natural numbers, for instance. The reason for this is simply that the effectiveness of a recursion clause depends on its being available after application, something that is usually assured by contraction. This paper presents a way of overcoming this problem (...) within the framework of a logic based on inclusion and unrestricted abstraction, without any form of extensionality. (shrink)

We revisit a debate initiated some 15 years ago by Ray Elugardo and Robert Stainton about the domain of arguments. Our main result is that arguments are not exclusively sets of linguistic expressions. Instead, as we put it, some non-linguistic items have ‘logical form’. The crucial examples are arguments, both deductive and inductive, made with unembedded words and phrases. … subsentential expressions such as singular terms and predicates… cannot serve as premises or conclusions in inferences.

Symbolic logic is sited at intersection of philosophy, mathematics, linguistics and computer science. It deals with the structure of reasoning, and the formal features of information. Work in symbolic logic has almost exclusively treated the deductive validity of arguments: those arguments for which it is impossible for the premises to be true and the conclusion false. However, techniques from twentieth-century logic have found a place in the study of inductive or probabilistic reasoning, in which premises need (...) not render their conclusions certain. (shrink)

Abstract.In this paper, we construct, by effective induction, a ∏11 equivalence relation E on ωω for which there is no largest E-thin E-invariant ∏11 subset of ωω by effective induction. This result answers a problem asked by Kechris in [1].

We present a method for integrating rippling-based rewriting into matrix-based theorem proving as a means for automating inductive specification proofs. The selection of connections in an inductive matrix proof is guided by symmetries between induction hypothesis and induction conclusion. Unification is extended by decision procedures and a rippling/reverse-rippling heuristic. Conditional substitutions are generated whenever a uniform substitution is impossible. We illustrate the integrated method by discussing several inductive proofs for the integer square root problem as well as (...) the algorithms extracted from these proofs. (shrink)

Designed for students with no prior training in logic, INTRODUCTION TO LOGIC AND CRITICAL THINKING offers an accessible treatment of logic that enhances understanding of reasoning in everyday life. The text begins with an introduction to arguments. After some linguistic preliminaries, the text presents a detailed analysis of inductive reasoning and associated fallacies. This order of presentation helps to motivate the use of formal methods in the subsequent sections on deductive logic and fallacies. Lively and (...) straightforward prose assists students in gaining facility with the sometimes challenging concepts of logic. By combining a sensitive treatment of ordinary language arguments with a simple but rigorous exposition of basic principles of logic, the text develops students' understanding of the relationships between logic and language, and strengthens their skills in critical thinking. Important Notice: Media content referenced within the product description or the product text may not be available in the ebook version. (shrink)

Vagueness is epistemic, according to some. Vagueness is ontological, according to others. This article deploys what I take to be a compromise position. Predicates are coined in specific contexts for specific purposes, but these limited practices do not automatically fix the extensions of predicates over the domain of all objects. The linguistic community using the predicate has rarely considered, much less decided, all questions that might arise about the predicate’s extension. To this extent, the ontological view is correct. But a (...) predicate that applies in some contexts can be reasonably extended to other contexts where it is initially vague. This process of development approximates the cognitive remedy for vagueness that the epistemic view prescribes. The process is piecemeal and inductive, akin to what von Wright described as the molding of concepts. Vagueness cannot be understood apart from the backdrop of classification, for vagueness is classification gone awry. Hence these pages explore the classification of particulars, both its clear successes and vague failures. How we classify unique particulars is the theme of Sections 2 and 3, which are primarily descriptive. Section 2 identifies a way of classifying particulars that pervades discourse of all sorts, and Section 3 illustrates its use in a field notorious for vagueness: ethics. Why a certain particular should (or should not) be classified in a certain way is a normative question, however, and it occupies Sections 4 and 5. Section 4 proposes a norm for strong arguments by analogy, and Section 5 illustrates how the norm might resolve vagueness in one kind of ethical dispute. This norm, which has a strong probabilistic component, is one way of affirming that probability is a guide to life. (shrink)

The idea that the world is made of particles — little discrete, interacting objects that compose the material bodies of everyday experience — is a durable one. Following the advent of quantum theory, the idea was revised but not abandoned. It remains manifest in the explanatory language of physics, chemistry, and molecular biology. Aside from its durability, there is good reason for the scientific realist to embrace the particle interpretation: such a view can account for the prominent epistemic fact that (...) only limited knowledge of a portion of the material universe is needed in order to make reliable predictions about that portion. Thus, particle interpretations can support an abductive argument from the epistemic facts in favor of a realist reading of physical theory. However, any particle interpretation with this property is untenable. The empirical adequacy of modern particle theories requires adoption of a postulate known as permutation invariance — the claim that interchanging the role of two particles of the same kind in a dynamical state description results in a description of the identical state. It is the central claim of this essay that PI is incompatible with any particle interpretation strong enough to account for the epistemic facts. This incompatibility extends across all physical theories. To frame and motivate the inconsistency argument, I begin by fixing the relevant notion of particle. To single out those accounts of greatest appeal to the realist, I develop the logically weakest particle ontology that entails the epistemic fact that the world is piecewise predictable, an ontology I call ‘minimal atomism’. The entire series of scientific conceptions of the particle, from Newton’s mechanically interacting corpuscles to the ‘centers of force’ in classical field theories, all comport with MA. As long as PI is left out, even quantum mechanics can be viewed this way. To assess the impact of PI on this picture, I present a framework for rigorously connecting interpretations to physical theories. In particular, I represent MA as a set of formal conditions on the models of physical theories, the mathematical structures taken to represent states of the world. I also formulate PI — originally introduced as a postulate of non-relativistic quantum mechanics — in theory independent terms. With all of these pieces in hand, I am then able to present a proof of the inconsistency of PI and MA. In the second part of the essay, I survey responses to the inconsistency result open to the scientific realist. The two most plausible approaches involve abandoning particles in one way or another. The first alternative interpretation considered takes the property bearing objects of the world to be regions of space rather than particles. In this view, the properties once attributed to particles in quantum states are attributed instead to one or more regions of space. PI no longer obtains in this case, at least not as a statement about the permutation symmetry of property bearers. Rather, the new interpretation naturally imposes an analogous constraint on quantum states. The second major approach to evading the inconsistency result is to dispense with objects altogether. This is the recommendation of so-called ‘Ontic Structural Realism’. The central OSR thesis is that structure rather than entities are the basic ontological components of the world. OSR is intended to embrace the ‘miracle’ argument in favor of scientific realism while avoiding the pessimistic meta-induction. I demonstrate that one principal motivation for OSR based on the under-determination of interpretations in QM is actually dissolved by the incompatibility result. At the same time, I suggest reasons to think that OSR fares no better with respect to the pessimistic meta-induction than traditional realism does. Thus, while PI and MA may be incompatible, object ontologies remain the best option for the realist. (shrink)