Eric Martin and Daniel N. Osherson present a theory of inductive logic built on model theory. Their aim is to extend the mathematics of Formal Learning Theory to a more general setting and to provide a more accurate image of empirical inquiry. The formal results of their study illuminate aspects of scientific inquiry that are not covered by the commonly applied Bayesian approach.

This paper shows how late 1960’s student protests influenced the thought of Imre Lakatos and Paul Feyerabend. I argue that student movements shaped their work from this period, specifically Lakatos’s “Falsification and the Methodology of Scientific Research Programmes” and Feyerabend’s Against Method. Archival evidence shows that their political environments at London and Berkeley inflected their writing on scientific method, entrenching Lakatos’s search for a rationalist account of scientific development, and encouraging Feyerabend’s ‘anarchistic’ theory of knowledge. I document this influence and (...) draw two broader lessons: one lesson about the role of biography in philosophy and the other about the historiography of student protests. (shrink)

This paper shows how late 1960’s student protests influenced the thought of Imre Lakatos and Paul Feyerabend. I argue that student movements shaped their work from this period, specifically Lakatos’s “Falsification and the Methodology of Scientific Research Programmes” and Feyerabend’s Against Method. Archival evidence shows that their political environments at London and Berkeley inflected their writing on scientific method, entrenching Lakatos’s search for a rationalist account of scientific development, and encouraging Feyerabend’s ‘anarchistic’ theory of knowledge. I document this influence and (...) draw two broader lessons: one lesson about the role of biography in philosophy and the other about the historiography of student protests. (shrink)

Scientific inquiry is represented as a process of rational hypothesis revision in the face of data. For the concept of rationality, we rely on the theory of belief dynamics as developed in [5, 9]. Among other things, it is shown that if belief states are left unclosed under deductive logic then scientific theories can be expanded in a uniform, consistent fashion that allows inquiry to proceed by any method of hypothesis revision based on "kernel" contraction. In contrast, if belief states (...) are closed under logic, then no such expansion is possible. (shrink)

Scientific inquiry is represented as a process of rational hypothesis revision in the face of data. For the concept of rationality, we rely on the theory of belief dynamics as developed in [5, 9]. Among other things, it is shown that if belief states are left unclosed under deductive logic then scientific theories can be expanded in a uniform, consistent fashion that allows inquiry to proceed by any method of hypothesis revision based on “kernel” contraction. In contrast, if belief states (...) are closed under logic, then no such expansion is possible. (shrink)

Science and Ideology This article illustrates some of the relationships between science and ideologies. It discusses how science has been enlisted to support particular ideologies and how ideologies have influenced the processes and interpretations of scientific inquiry. An example from the biological sciences illustrates this. In the early 20th century, evolutionary theory was used to … Continue reading Science and Ideology →.

A model of inductive inquiry is defined within a first-order context. Intuitively, the model pictures inquiry as a game between Nature and a scientist. To begin the game, a nonlogical vocabulary is agreed upon by the two players along with a partition of a class of structures for that vocabulary. Next, Nature secretly chooses one structure ("the real world") from some cell of the partition. She then presents the scientist with a sequence of atomic facts about the chosen structure. With (...) each new datum the scientist announces a guess about the cell to which the chosen structure belongs. To succeed in his inquiry, the scientist's successive conjectures must be correct all but finitely often, that is, the conjectures must converge in the limit to the correct cell. A special kind of scientist selects his hypotheses on the basis of a belief revision operator. We show that reliance on belief revision allows scientists to solve a wide class of problems. (shrink)

In Minker and Rajasekar (J Log Program 9(1):45–74, 1990), Minker proposed a semantics for negation-free disjunctive logic programs that offers a natural generalisation of the fixed point semantics for definite logic programs. We show that this semantics can be further generalised for disjunctive logic programs with classical negation, in a constructive modal-theoretic framework where rules are built from _claims_ and _hypotheses_, namely, formulas of the form \(\Box \varphi \) and \(\Diamond \Box \varphi \) where \(\varphi \) is a literal, respectively, (...) yielding a “base semantics” for general disjunctive logic programs. Model-theoretically, this base semantics is expressed in terms of a classical notion of logical consequence. It has a complete proof procedure based on a general form of the cut rule. Usually, alternative semantics of logic programs amount to a particular interpretation of nonclassical negation as “failure to derive.” The counterpart in our framework is to complement the original program with a set of hypotheses required to satisfy specific conditions, and apply the base semantics to the resulting set. We demonstrate the approach for the answer set semantics. The proposed framework is purely classical in mainly three ways. First, it uses classical negation as unique form of negation. Second, it advocates the computation of logical consequences rather than of particular models. Third, it makes no reference to a notion of preferred or minimal interpretation. (shrink)

We present a model for first-order belief revision that is characterized by an underlying relevance-like relation and a background proof system. The model is extremely general in order to allow for a wide variety in these characterizing parameters. It allows some weakenings of beliefs which were initially implicit to become explicit and survive the revision process. The effects of revision are localized to the part of the theory that is influenced by the the new information. Iterated revision in this model (...) is handled trivially since the revision operator is constructive by definition. The usage of deductively limited proof systems permits an inconsistency tolerant model. The notion of a part of a theory capable of being influenced by new information (designed to accomodate the specific character of first-order languages) is shown to satisfy some intuitive and desirable properties. We show that for particular parametrizations, standard revision schemes can be embedded into our paradigm. (shrink)

The inductive logic developed in the second and third essays is limited in important ways. For example: (a) the logic makes no provision for missing or misleading data; (b) it gives the scientist no control over the evidence reaching him; (c) revision-based scientist must work with theories written in the cramped idiom of firstorder logic; (d) the idea of efficient induction is only weakly expressed (in terms of “dominance”).

The following correction will help establish more clearly and fully the ways in which I drew on Matteo Collodel’s work in the composition of this paper. Given that we were researching similar topics and that I was acquainted with his achievements.

Parametric logic is a framework that generalises classical first-order logic. A generalised notion of logical consequence—a form of preferential entailment based on a closed world assumption—is defined as a function of some parameters. A concept of possible knowledge base—the counterpart to the consistent theories of first-order logic—is introduced. The notion of compactness is weakened. The degree of weakening is quantified by a nonnull ordinal—the larger the ordinal, the more significant the weakening. For every possible knowledge base T, a hierarchy of (...) sentences that are generalised logical consequences of T is built. The first layer of the hierarchies corresponds to sentences that can be obtained by a deductive inference, characterised by the compactness property. The second layer of the hierarchies corresponds to sentences that can be obtained by an inductive inference, characterised by the property of weak compactness quantified by 1. Weaker forms of compactness—quantified by nonnull ordinals—determine higher layers in the hierarchies, corresponding to more complex inferences. The naturalness of the hierarchies built over the possible knowledge bases is attested by fundamental connections with notions from Learning theory and from topology. The naturalness of the hierarchies built over the possible knowledge bases is attested by fundamental connections with notions from Learning theory—classification in the limit, with or without a bounded number of mind changes—and from topology—in reference to the Borel and the difference hierarchies. In this paper, we introduce the key model-theoretic aspects of Parametric logic, justify the concept of the knowledge base, define the hierarchies of generalised logical consequences and illustrate their relevance to Nonmonotonic reasoning. More specifically, we show that the degree of nonmonotonicity that is required to infer a sentence can be characterised by the least nonnull ordinal that quantifies the weakening of compactness used to locate the inferred sentence in the hierarchies. (shrink)

This paper reviews the natural philosophies of John Polkinghorne and Nancy Cartwright, with particular emphasis on the role of pluralism in their respective writings. While often motivated by distinct projects, their philosophies display some interesting and perhaps unexpected similarities. It is suggested that Polkinghorne's views are not far away from some of Cartwright's proposals, and further that certain debates about God's providential action could be helpfully reoriented if the insistence on natural laws as a centerpiece of scientific explanation were relinquished.

A model of inductive inquiry is defined within the context of first‐order logic. The model conceives of inquiry as a game between Nature and a scientist. To begin the game, a nonlogical vocabulary is agreed upon by the two players, along with a partition of a class of countable structures for that vocabulary. Next, Nature secretly chooses one structure from some cell of the partition. She then presents the scientist with a sequence of facts about the chosen structure. With each (...) new datum the scientist announces a guess about the cell to which the chosen structure belongs. To succeed in his or her inquiry, the scientist’s successive conjectures must be correct all but finitely often, that is, the conjectures must converge in the limit to the correct cell. Different kinds of scientists can be investigated within this framework. At opposite ends of the spectrum are dumb scientists that rely on the strategy of “induction by enumeration,” and smart scientists that rely on an operator of belief revision. We report some results about the scope and limits of these two inductive strategies. (shrink)

In the four papers available on our web site (of which this is the first), we propose to develop an inductive logic. By “inductive logic” we mean a set of principles that distinguish between successful and unsuccessful strategies for scientific inquiry. Our logic will have a technical character, since it is built from the concepts and terminology of (elementary) model theory. The reader may therefore wish to know something about the kind of results on offer before investing time in definitions (...) and notation. Providing such an informal overview is the purpose of the present essay. We begin with discussion of the central concept under investigation, namely, theory acceptance. (shrink)

A model of inductive inquiry is defined within the context of first‐order logic. The model conceives of inquiry as a game between Nature and a scientist. To begin the game, a nonlogical vocabulary is agreed upon by the two players, along with a partition of a class of countable structures for that vocabulary. Next, Nature secretly chooses one structure from some cell of the partition. She then presents the scientist with a sequence of facts about the chosen structure. With each (...) new datum the scientist announces a guess about the cell to which the chosen structure belongs. To succeed in his or her inquiry, the scientist’s successive conjectures must be correct all but finitely often, that is, the conjectures must converge in the limit to the correct cell. Different kinds of scientists can be investigated within this framework. At opposite ends of the spectrum are dumb scientists that rely on the strategy of “induction by enumeration,” and smart scientists that rely on an operator of belief revision. We report some results about the scope and limits of these two inductive strategies. (shrink)

We present a framework that provides a logic for science by generalizing the notion of logical (Tarskian) consequence. This framework will introduce hierarchies of logical consequences, the first level of each of which is identified with deduction. We argue for identification of the second level of the hierarchies with inductive inference. The notion of induction presented here has some resonance with Popper's notion of scientific discovery by refutation. Our framework rests on the assumption of a restricted class of structures in (...) contrast to the permissibility of classical first-order logic. We make a distinction between deductive and inductive inference via the notions of compactness and weak compactness. Connections with the arithmetical hierarchy and formal learning theory are explored. For the latter, we argue against the identification of inductive inference with the notion of learnable in the limit. Several results highlighting desirable properties of these hierarchies of generalized logical consequence are also presented. (shrink)