Formal learning theory is the mathematical embodiment of a normative epistemology. It deals with the question of how an agent should use observations about her environment to arrive at correct and informative conclusions. Philosophers such as Putnam, Glymour and Kelly have developed learning theory as a normative framework for scientific reasoning and inductive inference.

Formal learning theory is an approach to the study of inductive inference that has been developed by computer scientists. In this paper, I discuss the relevance of formal learning theory to such standard topics in the philosophy of science as underdetermination, realism, scientific progress, methodology, bounded rationality, the problem of induction, the logic of discovery, the theory of knowledge, the philosophy of artificial intelligence, and the philosophy of psychology.

One version of the problem of induction is how to justify hypotheses in the face of data. Why advance hypothesis A rather than B — or in a probabilistic context, why attach greater probability to A than B? If the data arrive as a stream of observations (distributed through time) then the problem is to justify the associated stream of hypotheses. Several perspectives on this problem have been developed including Bayesianism (Howson and Urbach, 1993) and belief-updating (Hansson, 1999). These are (...) broad families of approaches; the citations are meant just as portals. (shrink)

This article reviews a number of different areas in the foundations of formal learning theory. After outlining the general framework for formal models of learning, the Bayesian approach to learning is summarized. This leads to a discussion of Solomonoff's Universal Prior Distribution for Bayesian learning. Gold's model of identification in the limit is also outlined. We next discuss a number of aspects of learning theory raised in contributed papers, related to both (...) computational and representational complexity. The article concludes with a description of how semi-supervised learning can be applied to the study of cognitive learning models. Throughout this overview, the specific points raised by our contributing authors are connected to the models and methods under review. (shrink)

This article argues that a successful answer to Hume's problem of induction can be developed from a sub-genre of philosophy of science known as formal learning theory. One of the central concepts of formal learning theory is logical reliability: roughly, a method is logically reliable when it is assured of eventually settling on the truth for every sequence of data that is possible given what we know. I show that the principle of induction (PI) (...) is necessary and sufficient for logical reliability in what I call simple enumerative induction. This answer to Hume's problem rests on interpreting PI as a normative claim justified by a non-empirical epistemic means-ends argument. In such an argument, a rule of inference is shown by mathematical or logical proof to promote a specified epistemic end. Since the proof concerning PI and logical reliability is not based on inductive reasoning, this argument avoids the circularity that Hume argued was inherent in any attempt to justify PI. (shrink)

This paper places formal learning theory in a broader philosophical context and provides a glimpse of what the philosophy of induction looks like from a learning-theoretic point of view. Formal learning theory is compared with other standard approaches to the philosophy of induction. Thereafter, we present some results and examples indicating its unique character and philosophical interest, with special attention to its unified perspective on inductive uncertainty and uncomputability.

Nelson Goodman's new riddle of induction forcefully illustrates a challenge that must be confronted by any adequate theory of inductive inference: provide some basis for choosing among alternative hypotheses that fit past data but make divergent predictions. One response to this challenge is to distinguish among alternatives by means of some epistemically significant characteristic beyond fit with the data. Statistical learning theory takes this approach by showing how a concept similar to Popper's notion of degrees of testability (...) is linked to minimizing expected predictive error. In contrast, formal learning theory appeals to Ockham's razor, which it justifies by reference to the goal of enhancing efficient convergence to the truth. In this essay, I show that, despite their differences, statistical and formal learning theory yield precisely the same result for a class of inductive problems that I call strongly VC ordered, of which Goodman's riddle is just one example. (shrink)

Nelson Goodman’s new riddle of induction forcefully illustrates a challenge that must be confronted by any adequate theory of inductive inference: provide some basis for choosing among alternative hypotheses that fit past data but make divergent predictions. One response to this challenge is to distinguish among alternatives by means of some epistemically significant characteristic beyond fit with the data. Statistical learning theory takes this approach by showing how a concept similar to Popper’s notion of degrees of testability (...) is linked to minimizing expected predictive error. In contrast, formal learning theory appeals to Ockham’s razor, which it justifies by reference to the goal of enhancing efficient convergence to the truth. In this essay, I show that, despite their differences, statistical and formal learning theory yield precisely the same result for a class of inductive problems that I call strongly VC ordered , of which Goodman’s riddle is just one example. (shrink)

This essay demonstrates a previously unnoticed connection between formal and statistical learning theory with regard to Nelson Goodman’s new riddle of induction. Discussions of Goodman’s riddle in formal learning theory explain how conjecturing “all green” before “all grue” can enhance efficient convergence to the truth, where efficiency is understood in terms of minimizing the maximum number of retractions or “mind changes.” Vapnik-Chervonenkis (VC) dimension is a central concept in statistical learning theory and (...) is similar to Popper’s notion of degrees of testability. I show that for a class inductive problems of which Goodman’s riddle is one example, a reliable inductive method minimizes the maximum number of mind changes exactly if it always conjectures the hypothesis from the set with lowest VC dimension consistent with the data. I also discuss the relevance of these results to language invariance and curve fitting. (shrink)

Computational learning theory explores the limits of learnability. Studying language acquisition from this perspective involves identifying classes of languages that are learnable from the available data, within the limits of time and computational resources available to the learner. Different models of learning can yield radically different learnability results, where these depend on the assumptions of the model about the nature of the learning process, and the data, time, and resources that learners have access to. To the (...) extent that such assumptions accurately reflect human language learning, a model that invokes them can offer important insights into the formal properties of natural languages, and the way in which their representations might be efficiently acquired. In this chapter we consider several computational learning models that have been applied to the language learning task. Some of these have yielded results that suggest that the class of natural languages cannot be efficiently learned from the primary linguistic data (PLD) available to children, through.. (shrink)

An interpretation of memory from the perspective of logical learning theory is presented. In contrast to traditional associationistic theories of learning and memory, which rest on mediation modeling, LLT rests on a predication model. Predication draws on formal and final causation whereas mediation is limited to material and efficient causation. It is held in LLT that memory begins in predicate organization, where framing meanings are logically extended to targets. Passage of time is irrelevant in this meaning (...) extension. The effectiveness of memory depends on the cohesiveness or "tightness" of meaningful organization as framed by a relevant context. Well-organized contexts facilitate memory and poorly organized contexts are detrimental to memory. Possible reasons for good or poor organization in memory are discussed, and relevant research findings cited. The paper closes with a separate definition of memory as a content and as a process. (shrink)

This Special Issue addresses a range of educational issues linked to main themes from our 2019 book The Emergence of Complexity: Rethinking Education as a Social Science. This book elaborated two major theses that raise fundamental questions for philosophy of education. First, that learning by groups is typically a distinctive kind of learning that is not reducible to learning by individuals. Second, that a degree of holism, as against a focus on individuals, is essential for achieving a (...) convincing understanding of this distinctive type of group learning. These two theses are of direct interest to philosophy of education since they challenges the resources of received theories of learning. Yet this kind of group learning characterises the vast bulk of human learning situations that occur outside of formal education systems. In this Special Issue introductory article we, firstly, introduce some basic principles of complexity theory, and demonstrate that, together with the concept of a ‘co-present group’ (between 2 and about 12 individuals), these ideas offer develop novel understandings of the distinctive learning that occurs within such groups. Secondly, we outline and illustrate with further examples the main features of co-present groups. Thirdly, we consolidate our conceptual work by showing five ways in which our account of group learning serves to refurbish the concept of ‘learning’. (shrink)

In "Learning at Not-School," Julian Sefton-Green explores studies and scholarly research on out-of-school learning, investigating just what it is that is distinctive about the quality of learning in these "not-school" settings.

We propose a formal framework for interpretable machine learning. Combining elements from statistical learning, causal interventionism, and decision theory, we design an idealised explanation game in which players collaborate to find the best explanation for a given algorithmic prediction. Through an iterative procedure of questions and answers, the players establish a three-dimensional Pareto frontier that describes the optimal trade-offs between explanatory accuracy, simplicity, and relevance. Multiple rounds are played at different levels of abstraction, allowing the players (...) to explore overlapping causal patterns of variable granularity and scope. We characterise the conditions under which such a game is almost surely guaranteed to converge on a optimal explanation surface in polynomial time, and highlight obstacles that will tend to prevent the players from advancing beyond certain explanatory thresholds. The game serves a descriptive and a normative function, establishing a conceptual space in which to analyse and compare existing proposals, as well as design new and improved solutions. (shrink)

Youth is a historical construction and an answer to a specific challenge of individualisation in biography. And, as a historical and social construction, youth has to be learned. This article focuses on youth development from an action or activity theory perspective and as a learning process. It demonstrates how different youth problems and forms of youth differentiation follow forms of youth learning. Moreover, it shows how late modern development creates the demand for a new non-formal (...) class='Hi'>learning perspective to secure the development of new forms of competence. Based on Danish research concerning peer learning as a non-formal learning context, some perspectives of peer-learning competence are discussed. (shrink)

We propose a formal framework for interpretable machine learning. Combining elements from statistical learning, causal interventionism, and decision theory, we design an idealisedexplanation gamein which players collaborate to find the best explanation(s) for a given algorithmic prediction. Through an iterative procedure of questions and answers, the players establish a three-dimensional Pareto frontier that describes the optimal trade-offs between explanatory accuracy, simplicity, and relevance. Multiple rounds are played at different levels of abstraction, allowing the players to explore (...) overlapping causal patterns of variable granularity and scope. We characterise the conditions under which such a game is almost surely guaranteed to converge on a (conditionally) optimal explanation surface in polynomial time, and highlight obstacles that will tend to prevent the players from advancing beyond certain explanatory thresholds. The game serves a descriptive and a normative function, establishing a conceptual space in which to analyse and compare existing proposals, as well as design new and improved solutions. (shrink)

This book analyzes iconic British children's literature through the lens of formal educational theory, policy, and practice. Examining themes like growth mindset and project-based learning alongside educational philosophers like Plato, Rousseau, and Dewey, the author sheds new light on children’s classics from Alice in Wonderland to Harry Potter.

[full article, abstract in English; abstract in Lithuanian] Many countries are adopting computing in schools, for pupils from 5 years of age. Educational philosophies that such curricula might be based on are not clear in curriculum documentation. Many Western countries’ curricula are based on developmental concepts of cognitive constructivism, with activities progressing through sensorimotor, preoperational, concrete operational, and formal operational stages. Social constructivism and constructionism add new dimensions to this learning framework, both fundamentally important for developing computing practices. (...) We review selected learning theories, and investigate features that should underpin computing curricula if practices and outcomes are to develop computing practitioner competencies of a software developer. (shrink)

We propose a formal framework for interpretable machine learning. Combining elements from statistical learning, causal interventionism, and decision theory, we design an idealised explanation game in which players collaborate to find the best explanation for a given algorithmic prediction. Through an iterative procedure of questions and answers, the players establish a three-dimensional Pareto frontier that describes the optimal trade-offs between explanatory accuracy, simplicity, and relevance. Multiple rounds are played at different levels of abstraction, allowing the players (...) to explore overlapping causal patterns of variable granularity and scope. We characterise the conditions under which such a game is almost surely guaranteed to converge on a optimal explanation surface in polynomial time, and highlight obstacles that will tend to prevent the players from advancing beyond certain explanatory thresholds. The game serves a descriptive and a normative function, establishing a conceptual space in which to analyse and compare existing proposals, as well as design new and improved solutions. (shrink)

Numerous learning tasks can be described as the process of extrapolating patterns from observed data. One of the driving intuitions behind the theory of algorithmic randomness is that randomness amounts to the absence of any effectively detectable patterns: it is thus natural to regard randomness as antithetical to inductive learning. Osherson and Weinstein [11] draw upon the identification of randomness with unlearnability to introduce a learning-theoretic framework (in the spirit of formal learning theory) (...) for modelling algorithmic randomness. They define two success criteria—specifying under what conditions a pattern may be said to have been detected by a computable learning function—and prove that the collections of data sequences on which these criteria cannot be satisfied respectively correspond to the set of weak 1-randoms and the set of weak 2-randoms. This learning-theoretic approach affords an intuitive perspective on algorithmic randomness, and it invites the question of whether restricting attention to learning-theoretic success criteria comes at an expressivity cost. In other words, is the framework expressive enough to capture most core algorithmic randomness notions and, in particular, Martin-Löf randomness—arguably, the most prominent algorithmic randomness notion in the literature? In this paper, we answer the latter question in the affirmative by providing a learning-theoretic characterisation of Martin-Löf randomness. We then show that Schnorr randomness, another central algorithmic randomness notion, also admits a learning-theoretic characterisation in this setting. (shrink)

In formal semantics, structure is treated as the essential ingredient in the creation of sentence meaning from individual word meaning. This book introduces some of the foundational concepts, principles and techniques in the formal semantics of natural language and outlines the mathematical principles that underlie linguistics meaning. Using English examples, Yoad Winter presents the most useful tools and concepts of formal semantics in an accessible style and includes a variety of practical exercises so that readers can learn (...) to utilize these tools effectively. For readers with an elementary background in set theory and linguistics or with an interest in mathematical modelling, this fascinating study is an ideal introduction to natural language semantics. Designed as a quick yet thorough introduction to one of the most vibrant areas of research in modern linguistics today this volume reveals the beauty and elegance of the mathematical study of meaning. (shrink)

The present study used social learning theory and situated learning theory as a way to examine secondary social studies teacher participants' formal and informal professional learning. Existing literature is just beginning to attend to the potential of informal professional learning and to distinguish between formal and informal professional learning, so this exploratory study used observations of scheduled and spontaneous professional learning experiences, semi-structured interviews with 12 secondary social studies teachers, and (...) relevant documents to consider the following research question: In what ways do high school social studies teacher participants engage in formal and informal professional learning? Results indicate that while secondary social studies teachers engage in a variety of formal and informal professional learning experiences, their informal learning experiences are the most valuable and significant for their classroom practice. While informal professional learning is not a replacement for traditional formal professional learning, lessons learned here can inform ways to improve formal professional learning and to capitalize upon teachers’ existing informal professional learning experiences. This study provides insight into the ways in which social studies teachers support their growth outside of formal professional development. (shrink)

Like other health care practices, the increasing complexity in dentistry signals the need for a reconceptualisation of dentist professional learning. Professional dental bodies, at large, still privilege formal continuing professional development (CPD) provisions focusing on off-the-job activities despite growing evidence that much invaluable learning occurs through and at work. In exploring the two common dentist CPD approaches, this article critiques the narrow conceptions of learning inscribed in these frameworks, which are individualistic and acquisition oriented. Drawing on (...) a vignette of dentists’ professional practice at work, this paper argues for a shift in discussion from an emphasis on which CPD models work best to what counts as professional learning for dentists. To flesh out these arguments, the paper proposes using an innovative conceptual approach through the lens of ‘complexity thinking’ and the concept of ‘co-present group’. Through this lens, the reframing of thinking brings out two key features of learning: Emergent learning cannot be specified in advance and much significant learning is typically beyond an individual’s learning. Given the learning potentials of group practice and group learning, the paper concludes with suggestions to support dentists’ lifelong learning at work. (shrink)

In this article we discuss what are the implications for improving the design of corporate ethics programs, if we focus on the moral motivation accounts offered by main ethical theories. Virtue ethics, deontological ethics and utilitarianism offer different criteria of judgment to face moral dilemmas: Aristotle's virtues of character, Kant's categorical imperative, and Mill's greatest happiness principle are, respectively, their criteria to answer the question "What is the right thing to do?" We look at ethical theories from a different perspective: (...) the question we ask is "Why should I do the right thing?" In other words, we deal with the problem of moral motivation, and we examine the different rationale the main ethical theories provide. We then point out the relation between moral motivation and the concept of rationality in the different approaches - is acting morally seen as an expression of rational behavior? Our analysis of moral motivation provides a useful framework to improve the understanding of the relationships between formal and informal elements of corporate ethics programs, emphasizing the importance of the latter, often overlooked in compliance-focused programs. We conclude by suggesting that the concept of moral imagination can provide a unifying approach to enhance the effectiveness of corporate ethics programs, by providing an intangible asset that supports the implementation of their formal components into management decision making. (shrink)

This book provides philosophers and logicians with a broad spectrum of views on contemporary research on the problem of deduction, its justification and explanation. The variety of distinct approaches exemplified by the single chapters allows for a dialogue between perspectives that, usually, barely communicate with each other. The contributions concern (in a possibly intertwined way) three major perspectives in logic: philosophical, historical, formal. The philosophical perspective has to do with the relationship between deductive validity and truth, and questions the (...) alleged conclusiveness of deduction and its epistemic contribution. It also discusses the role of linguistic acts in deductive practice, and provides a cognitive-didactic contribution on how we may learn through deduction. In the historical perspective, the contributions discuss the ideas of some major historical figures, such as Bolzano, Girard, Gödel, and Peano. Finally, in the formal perspective, the mathematics of deduction is dealt with mainly from an intuitionistic-constructivist or proof-theoretic point of view, with focus on “ecumenic” or internalistic approaches to logical validity, on the nature and identity of proofs, and on dialogical setups. Chapter [14] is available open access under a Creative Commons Attribution 4.0 International License via link.springer.com. (shrink)

In this paper, we present an intelligent tutoring system developed to help students in learning Computer Theory. The Intelligent tutoring system was built using ITSB authoring tool. The system helps students to learn finite automata, pushdown automata, Turing machines and examines the relationship between these automata and formal languages, deterministic and nondeterministic machines, regular expressions, context free grammars, undecidability, and complexity. During the process the intelligent tutoring system gives assistance and feedback of many types in an intelligent (...) manner according to the behavior of the student. An evaluation of the intelligent tutoring system has revealed reasonably acceptable results in terms of its usability and learning abilities are concerned. (shrink)

We examine the prospects for finding “best possible” or “ideal” computing machines for various learning tasks. For this purpose, several precise senses of “ideal machine” are considered within the context of formal learning theory. Generally negative results are provided concerning the existence of ideal learning‐machines in the senses considered.

In this paper we analyse the conditions for attributing to AI autonomous systems the ontological status of “artificial moral agents”, in the context of the “distributed responsibility” between humans and machines in Machine Ethics (ME). In order to address the fundamental issue in ME of the unavoidable “opacity” of their decisions with ethical/legal relevance, we start from the neuroethical evidence in cognitive science. In humans, the “transparency” and then the “ethical accountability” of their actions as responsible moral agents is not (...) in contradiction with the unavoidable “opacity” (unawareness) of the brain process by which they perform their moral judgements on the right action to execute. In fact, the moral accountability of our actions depends on what is immediately before and after our “moral judgements” on the right action to execute (formally, deontic first order logic (FOL) decisions). I.e., our moral accountability depends on the “ethical constraints” we imposed to our judgement before performing it in an opaque way. Anyway, our moral accountability depends overall on the “ethical assessment” or explicit “moral reasoning” after and over the moral judgement before executing our actions (deontic higher order logic (HOL) assessment). In this way, in the light of the AI “imitation game”, the consistent attribution of the status of ethically accountable artificial moral agents to autonomous AI systems depends on two similar conditions. Firstly, it depends on the presence of “ethical constraints” to be satisfied in their Machine Learning (ML) supervised optimization algorithm during its training phase, to give the system ethical skills (“competences”) in its decisions. Secondly – and definitely—, it depends on the presence in an AI autonomous system of a deontic HOL “ethical reasoner” to perform an automatic, and fully transparent assessment (metalogical deontic valuation) about the decisions taken by the ethically skilled ML algorithm about the right action to execute, before executing it. Finally, we show that the proper deontic FOL and HOL for this class of artificial moral agents is Kripke’s modal relational logic, in its algebraic topological formalization. This is naturally implemented in the dissipative QFT unsupervised deep learning of our brains, based on the “doubling of the degrees of freedom” (DDF), and then in the so-called “deep-belief” artificial neural networks for the statistical data pre-processing. This unsupervised learning procedure is also compliant with the usage of the “maximin fairness principle”, used as a balancing aggregation principle of the statistical variables in Sen’s formal theory of fairness. (shrink)

Formal learning theory formalizes the process of inferring a general result from examples, as in the case of inferring grammars from sentences when learning a language. In this work, we develop a general framework—the supervised learning game—to investigate the interaction between Teacher and Learner. In particular, our proposal highlights several interesting features of the agents: on the one hand, Learner may make mistakes in the learning process, and she may also ignore the potential relation (...) between different hypotheses; on the other hand, Teacher is able to correct Learner’s mistakes, eliminate potential mistakes and point out the facts ignored by Learner. To reason about strategies in this game, we develop a modal logic of supervised learning and study its properties. Broadly, this work takes a small step towards studying the interaction between graph games, logics and formal learning theory. (shrink)

Understanding causal structure is a central task of human cognition. Causal learning underpins the development of our concepts and categories, our intuitive theories, and our capacities for planning, imagination and inference. During the last few years, there has been an interdisciplinary revolution in our understanding of learning and reasoning: Researchers in philosophy, psychology, and computation have discovered new mechanisms for learning the causal structure of the world. This new work provides a rigorous, formal basis for (...) class='Hi'>theory theories of concepts and cognitive development, and moreover, the causal learning mechanisms it has uncovered go dramatically beyond the traditional mechanisms of both nativist theories, such as modularity theories, and empiricist ones, such as association or connectionism. (shrink)

This paper examines the learning and performance of the air traffic control (ATC) work domain. This domain was chosen because it embodies features that represent future work for many other industries (e.g., information service provision mediated by information technologies; a high reliance on communication skills and collaborative work; increasing complexity and intensity of the work activity), within an organisational context undergoing considerable change. In ATC work learning occurs formally as part of accredited training and informally, as part of (...) everyday practice. In this way learning and experience of ATC work shapes – and is shaped by – the way work is organised, which includes the divisions of labour and technological artefacts. ATC work is experienced temporally, complexly, through affect and socially. Controllers learn to use a variety of socio-material resources which include embodiment and artefacts to undertake the work. In so doing the article demonstrates the ways in which learning and performance is not just a cognitive undertaking, but one that draws on embodiment to achieve what here is called “interdependent sentience”. This involves using all their senses to gain an awareness of problems as they emerge to collectively accomplish a seamless air traffic flow in a complex interdependent socio-technical system. (shrink)

This book adds to the theoretical development of the emerging fields of experiential learning and outdoor education by examining the central concept, 'experience', and interrogating a central claim of experiential learning: whether, and if so how, a short-term singular experience can transform a participant's life as a whole and in a permanent way. While such a possibility has been corroborated by the personal testimonies of participants, and the activities of instructors over many years, the book argues that we (...) must go beyond this kind of 'evidence'. In comparing Anglophone and continental approaches and drawing on the work of Dewey, Dilthey and Merleau-Ponty in the philosophy of experience, Experiential Learning and Outdoor Education presents the first detailed review of the concept of 'experience' in European philosophy, as applied to outdoor experiential learning. A vital insight into the field, this is important reading for students and researchers working in the philosophy of sport, and pedagogical theory, especially in areas relating to the outdoors, but also to experiential education more generally."--Provided by publisher. (shrink)

There is currently no viable alternative to the Bayesian analysis of scientific inference, yet the available versions of Bayesianism fail to do justice to several aspects of the testing and confirmation of scientific hypotheses. Bayes or Bust? provides the first balanced treatment of the complex set of issues involved in this nagging conundrum in the philosophy of science. Both Bayesians and anti-Bayesians will find a wealth of new insights on topics ranging from Bayes's original paper to contemporary formal (...) class='Hi'>learning theory. In a paper published posthumously in 1763, the Reverend Thomas Bayes made a seminal contribution to the understanding of "analogical or inductive reasoning." Building on his insights, modem Bayesians have developed an account of scientific inference that has attracted numerous champions as well as numerous detractors. Earman argues that Bayesianism provides the best hope for a comprehensive and unified account of scientific inference, yet the presently available versions of Bayesianisin fail to do justice to several aspects of the testing and confirming of scientific theories and hypotheses. By focusing on the need for a resolution to this impasse, Earman sharpens the issues on which a resolution turns. John Earman is Professor of History and Philosophy of Science at the University of Pittsburgh. (shrink)

There is currently no viable alternative to the Bayesian analysis of scientific inference, yet the available versions of Bayesianism fail to do justice to several aspects of the testing and confirmation of scientific hypotheses. Bayes or Bust? provides the first balanced treatment of the complex set of issues involved in this nagging conundrum in the philosophy of science. Both Bayesians and anti-Bayesians will find a wealth of new insights on topics ranging from Bayes’s original paper to contemporary formal (...) class='Hi'>learning theory.In a paper published posthumously in 1763, the Reverend Thomas Bayes made a seminal contribution to the understanding of "analogical or inductive reasoning." Building on his insights, modem Bayesians have developed an account of scientific inference that has attracted numerous champions as well as numerous detractors. Earman argues that Bayesianism provides the best hope for a comprehensive and unified account of scientific inference, yet the presently available versions of Bayesianisin fail to do justice to several aspects of the testing and confirming of scientific theories and hypotheses. By focusing on the need for a resolution to this impasse, Earman sharpens the issues on which a resolution turns. John Earman is Professor of History and Philosophy of Science at the University of Pittsburgh. (shrink)

Convergent realists desire scientific methods that converge reliably to informative, true theories over a wide range of theoretical possibilities. Much attention has been paid to the problem of induction from quantifier-free data. In this paper, we employ the techniques of formal learning theory and model theory to explore the reliable inference of theories from data containing alternating quantifiers. We obtain a hierarchy of inductive problems depending on the quantifier prefix complexity of the formulas that constitute the (...) data, and we provide bounds relating the quantifier prefix complexity of the data to the quantifier prefix complexity of the theories that can be reliably inferred from such data without background knowledge. We also examine the question whether there are theories with mixed quantifiers that can be reliably inferred with closed, universal formulas in the data, but not without. (shrink)

I. Prelude -- About this Book -- II. School Mathematics and Its Consequences -- The Foundations of Mathematical Thinking -- Compression, Connection and Blending of Mathematical Ideas -- Set-befores, Met-befores and Long-term Learning -- Mathematics and the Emotions -- The Three Worlds of Mathematics -- Journeys through Embodiment and Symbolism -- Problem-Solving and Proof -- III. Interlude -- The Historical Evolution of Mathematics -- IV. University Mathematics and Beyond -- The Transition to Formal Knowledge -- Blending Knowledge Structures (...) in the Calculus -- Expert Thinking and Structure Theorems -- Contemplating the Infinitely Large and the Infinitely Small -- Expanding the Frontiers through Mathematical Research -- Reflections -- Appendix: Where the Ideas Came From. (shrink)

There is an active debate within the field of phonology concerning the cognitive status of substantive phonetic factors such as ease of articulation and perceptual distinctiveness. A new framework is proposed in which substance acts as a bias, or prior, on phonological learning. Two experiments tested this framework with a method in which participants are first provided highly impoverished evidence of a new phonological pattern, and then tested on how they extend this pattern to novel contexts and novel sounds. (...) Participants were found to generalize velar palatalization (e.g., the change from [k] as in keep to [t⌢∫S] as in cheap) in a way that accords with linguistic typology, and that is predicted by a cognitive bias in favor of changes that relate perceptually similar sounds. Velar palatalization was extended from the mid front vowel context (i.e., before [e] as in cape) to the high front vowel context (i.e., before [i] as in keep), but not vice versa. The key explanatory notion of perceptual similarity is quantified with a psychological model of categorization, and the substantively biased framework is formalized as a conditional random field. Implications of these results for the debate on substance, theories of phonological generalization, and the formalization of similarity are discussed. (shrink)

Imprecise probabilities are an increasingly popular way of reasoning about rational credence. However they are subject to an apparent failure to display convincing inductive learning. This paper demonstrates that a small modification to the update rule for IP allows us to overcome this problem, albeit at the cost of satisfying only a weaker concept of coherence.

This paper starts by summarizing work that philosophers have done in the fields of inductive logic since 1950s and truth approximation since 1970s. It then proceeds to interpret and critically evaluate the studies on machine learning within artificial intelligence since 1980s. Parallels are drawn between identifiability results within formal learning theory and convergence results within Hintikka’s inductive logic. Another comparison is made between the PAC-learning of concepts and the notion of probable approximate truth.

What makes our utterances mean what they do? In this work I formulate and justify a structural constraint on possible answers to this key question in the philosophy of language, and I show that accepting this constraint leads naturally to the adoption of an algebraic formalization of truth-theoretic semantics. I develop such a formalization, and show that applying algebraic methodology to the theory of meaning yields important insights into the nature of language. ;The constraint I propose is, roughly, this: (...) the meaning of an utterance of a natural language sentence is derived from the place of that sentence within a system that includes other sentences, and is structured in virtue of primitive semantic facts about complete sentences. This constraint calls for a formal semantics program where the meaning of natural language sentences is represented by an appropriate assignment of the elements of an algebraic structure to sentences, an assignment that represents facts concerning complete sentences. I call the above constraint 'The Algebraic Thesis', and the corresponding formal semantics program 'Algebraic Semantics.' ;I introduce AT by carving it out of the holistic views of language of Quine and Davidson, and I show that it is independent of other important features of these philosophers' views. I then argue for AT, taking into account recent work by Brandom, Dummett, Fodor and Lepore, Gaifman and Perry. In developing Algebraic Semantics I present structures of two basic types: Boolean algebras and cylindric algebras; these structures are construed in the context of AT in a novel way: they are treated as applying to classes of sentences as numbers apply to physical objects in measurement. Applying algebraic methodology to formal semantics and to the theory of meaning, I develop these results: ; An account of the semantics of necessity and possibility that avoids both Quine's dismissal of these notions and their reduction by David Lewis to quantification over possible worlds; ; The application of the model-theoretic notion of expansion to capture the process of our learning progressively more of the meaning of expressions in our first languages. (shrink)

Developmental psychologists have long recognized the extraordinary influence of action on learning (Held & Hein, 1963; Piaget, 1952). Action experiences begin to shape our perception of the world during infancy (e.g., as infants gain an understanding of others’ goal-directed actions; Woodward, 2009) and these effects persist into adulthood (e.g., as adults learn about complex concepts in the physical sciences; Kontra, Lyons, Fischer, & Beilock, 2012). Theories of embodied cognition provide a structure within which we can investigate the mechanisms underlying (...) action’s impact on thinking and reasoning. We argue that theories of embodiment can shed light on the role of action experience in early learning contexts, and further that these theories hold promise for using action to scaffold learning in more formal educational settings later in development. (shrink)

Holistic theories of meaning have, at least since Dummett’s Frege: The Philosophy of language, been assumed to be problematic from the perspective of the incremental nature of natural language learning. In this essay I argue that the general relationship between holism and language learning is in fact the opposite of that claimed by Dummett. It is only given a particular form of language learning, and a particular form of holism, that there is a problem at all; in (...) general, for all forms of holism, and irrespective of how language learning is understood, semantic holism is conducive to language learning. The paper has three main parts. In the first, I demonstrate with the use of a simple formal system, that the form of holism that generates the problem that Dummett draws attention to is really decomposable into three distinct components, each of which is necessary for the problem to arise. In the second part, I demonstrate that even Dummett’s strong form of holism is compatible with one natural way in which to understand the incremental nature of language learning. In the third part, I outline the reasons why all forms of holism are conducive to language learning and offer two ways in which this general fact can be spelled out precisely. I end the paper by addressing some possible objections, and in doing so I draw attention to some affinities between semantic holism and the principle of compositionality, a semantic principle which has long been assumed to be conducive to language learning. (shrink)

Linguists intuitions about language change can be captured by adynamical systems model derived from the dynamics of language acquisition.Rather than having to posit a separate model for diachronic change, as hassometimes been done by drawing on assumptions from population biology (cf.Cavalli-Sforza and Feldman, 1973; 1981; Kroch, 1990), this new modeldispenses with these independent assumptions by showing how the behavior ofindividual language learners leads to emergent, global populationcharacteristics of linguistic communities over several generations. As thesimplest case, we formalize the example of (...) two grammars and show that eventhis situation leads directly to a nonlinear (quadratic) dynamical system.We study this one parameter model in a variety of situations for differentkinds of acquisition algorithms and maturational times, showing howdifferent learning theories can have very different evolutionaryconsequences. This allows us to formulate an evolutionary criterion for theadequacy of grammatical and learning theories. An application of thecomputational model to the historical loss of Verb Second from Old French toModern French is described showing how otherwise adequate grammaticaltheories might fail the evolutionary criterion. (shrink)

The impetus for this volume developed from the 1982 University of Western Ontario Learnability Workshop, which was organized by the editors and sponsored by that University's Department of Philosophy and the Centre for Cognitive Science. The volume e~plores the import of learnability theory for contemporary linguistic theory, focusing on foundational learning-theoretic issues associated with the parametrized Government-Binding framework. Written by prominent re searchers in the field, all but two of the eight contributions are pre viously unpublished. The (...) editor's introduction provides an overview that interrelates the separate papers and elucidates the foundational issues addressed by the volume. Osherson, Stob, and Weinstein's "Learning Theory and Natural Language" first appeared in Cognition ; Matthews's "The Plausi bility of Rationalism" was published in the Journal of Philosophy. The editors would like to thank the publishers for permission to reprint these papers. Mr. Marin Marinov assisted with the preparation of the indices for the volume. VB ROBERT 1. MATTHEWS INTRODUCTION: LEARNABILITY AND LINGUISTIC THEORY 1. INTRODUCTION Formal learning theory, as the name suggests, studies the learnability of different classes of formal objects under different formal models of learning. The specification of such a model, which specifies a learning environment, a learn ing strategy, and a criterion for successful learning, determines a class of formal objects, namely, the class that can be acquired to the level of the specified success criterion by a learner implementing the specified strategy in the specified enviroment. (shrink)

Complex constraints like conditionals ('If A, then B') and probabilistic constraints ('The probability that A is p') pose problems for Bayesian theories of learning. Since these propositions do not express constraints on outcomes, agents cannot simply conditionalize on the new information. Furthermore, a natural extension of conditionalization, relative information minimization, leads to many counterintuitive predictions, evidenced by the sundowners problem and the Judy Benjamin problem. Building on the notion of a `paradigm shift' and empirical research in psychology and economics, (...) I argue that the model of hypothesis testing can explain how people learn complex, theory-laden propositions like conditionals and probability constraints. Theories are formalized as probability distributions over a set of possible outcomes and theory change is triggered by a constraint which is incompatible with the initial theory. This leads agents to consult a higher order probability function, or a 'prior over priors,' to choose the most likely alternative theory which satisfies the constraint. The hypothesis testing model is applied to three examples: learning a simple probabilistic constraint involving coin bias, the sundowners problem for conditional learning, and the Judy Benjamin problem for learning conditional probability constraints. (shrink)

Game theory is a formal approach to behavior that focuses on the strategic aspect of situations. The game theoretic approach originates in economics but has been embraced by scholars across disciplines, including many philosophers and biologists. This approach has an important weakness: the strategic aspect of a situation, which is its defining quality in game theory, is often not its most salient quality in human cognition. Evidence from a wide range of experiments highlights this shortcoming. Previous theoretical (...) and empirical work has sought to address this weakness by considering learning across an ensemble of multiple games simultaneously. Here we extend this framework, incorporating artificial neural networks, to allow for an investigation of the interaction between the perceptual and functional similarity of the games composing the larger ensemble. Using this framework, we conduct a theoretical investigation of a population that encounters both stag hunts and prisoner’s dilemmas, two situations that are strategically different but which may or may not be perceptually similar. (shrink)

In this article we present symmetric diffusion networks, a family of networks that instantiate the principles of continuous, stochastic, adaptive and interactive propagation of information. Using methods of Markovion diffusion theory, we formalize the activation dynamics of these networks and then show that they can be trained to reproduce entire multivariate probability distributions on their outputs using the contrastive Hebbion learning rule (CHL). We show that CHL performs gradient descent on an error function that captures differences between desired (...) and obtained continuous multivariate probability distributions. This allows the learning algorithm to go beyond expected values of output units and to approximate complete probability distributions on continuous multivariate activation spaces. We argue that learning continuous distributions is an important task underlying a variety of real‐life situations that were beyond the scope of previous connectionist networks. Deterministic networks, like back propagation, cannot learn this task because they are limited to learning average values of independent output units. Previous stochastic connectionist networks could learn probability distributions but they were limited to discrete variables. Simulations show that symmetric diffusion networks can be trained with the CHL rule to approximate discrete and continuous probability distributions of various types. (shrink)

Formal epistemology is the study of crucial concepts in general or main- stream epistemology including knowledge, belief , certainty, ra- tionality, reasoning, decision, justi cation, learning, agent interaction and information processing using a spread of di¤erent formal tools. These formal tools may be drawn from elds such as logic, probability theory, game theory, decision theory, formal learning theory, and distributed com- puting –such variety is typical in formal epistemology, a (...) eld in which interaction with topics outside of philosophy proper is the rule rather than the exception. Practitioners of formal epistemology include philosophers, computer scientists, social scientists, cognitive psychologists, theoretical economists, mathematicians, and theoretical linguists. The interdiscipli- nary nature of formal epistemology can make it di¢ cult for those new to the eld to have a sense of some of its basic agendas, actors, and issues. What follows is a breezy overview of formal epistemology as organized around notions of agency and interaction. (shrink)