The purpose of this article is to explore the extent to which mathematics is subject to open texture and the extent to which mathematics resists open texture. The resistance is tied to the importance of proof and, in particular, rigor, in mathematics.

Many of the terms of our language, such as “jar”, are open-textured in the sense that their applicability to novel objects is not entirely determined by their past usage. Many others, such as the verbs “use” and “have”, are schematic in the sense that they have only a very general meaning although on any particular occasion of use they denote some more particular relation. The phenomena of open texture and schematicity constitute a sharp challenge to referential semantics, (...) which assumes that every non-logical term has a definite extension. A different, non-referential approach to formal semantics defines truth as relative to a context and defines contexts as built up from exclusively linguistic entities. For any given utterance of a sentence, there will be one of these contexts that pertains to it. In this framework, open texture and schematicity can be understood as consequences of the complex nature of the pertaining relation between contexts and utterances. (shrink)

Open texture is a kind of semantic indeterminacy first systematically studied by Waismann. In this paper, extant definitions of open texture will be compared and contrasted, with a view towards the consequences of open-textured concepts in mathematics. It has been suggested that these would threaten the traditional virtues of proof, primarily the certainty bestowed by proof-possession, and this suggestion will be critically investigated using recent work on informal proof. It will be argued that informal proofs (...) have virtues that mitigate the danger posed by open texture. Moreover, it will be argued that while rigor in the guise of formalisation and axiomatisation might banish open texture from mathematical theories through implicit definition, it can do so only at the cost of restricting the tamed concepts in certain ways. (shrink)

In this paper, we will survey the different uses of the term algorithm in contemporary legal practice. We will argue that the concept of algorithm currently exhibits a substantial degree of open texture, co-determined by the open texture of the concept of algorithm itself and by the open texture inherent to legal discourse. We will substantiate our argument by virtue of a case study, in which we analyze a recent jurisprudential case where the first (...) and second-degree judges have carved-out contrasting notions of algorithm. We will see that, thanks to our analysis of the open texture of the notion of algorithm in legal language, we can make sense of the different decisions taken by the judges as different contextually-determined sharpenings of the concept of algorithm. Finally, we will draw some general conclusions concerning the use of technical terms in legal instruments that address new technologies, such as the EU AI Act. (shrink)

The article presents a careful analysis of the idea of the “open texture” of empirical concepts and the problems of verification in the way that they were formulated by Friedrich Waismann. The idea of the “open texture” means for Waismann a certain type of a linguistic indeterminacy or a sort of lack of definition, which must be distinguished from, and linked to, another types like vagueness or ambiguity. It is shown that empirical statements are not conclusively (...) verifiable for two different reasons: the incompleteness of description of the material object and the open texture of the terms involved. We cannot conclusively verify statements in which the empirical concepts are used, because we cannot define these concepts in an exhaustive way because of their open texture. Thus, the definition of the concept will be incomplete. Waismann’s approach to definition plays here a key role, and it is directly related to the open texture of concepts. The author proposes interpreting the open texture as an immanent property of the concept, as something that is embedded in it a priori, and which can cause a vagueness. Nevertheless, an open texture must be distinguished form a vagueness. This leads to the conclusion that an open texture is a possibility of vagueness; vagueness can be remedied by giving more accurate rules, open texture cannot. In this sense, the “open texture” of a language allows for a more precise definition of concepts (by adjusting the definition) if appropriate circumstances arise. This justifies the thesis that the argument of the open texture is the ontological basis of the linguistic anti-reductionism of Friedrich Waismann. (shrink)

This edited collection covers Friedrich Waismann's most influential contributions to twentieth-century philosophy of language: his concepts of open texture and language strata, his early criticism of verificationism and the analytic-synthetic distinction, as well as their significance for experimental and legal philosophy. -/- In addition, Waismann's original papers in ethics, metaphysics, epistemology and the philosophy of mathematics are here evaluated. They introduce Waismann's theory of action along with his groundbreaking work on fiction, proper names and Kafka's Trial. -/- Waismann (...) is known as the voice of Ludwig Wittgenstein in the Vienna Circle. At the same time we find in his works a determined critic of logical positivism and ordinary language philosophy, who anticipated much later developments in the analytic tradition and devised his very own vision for its future. (shrink)

An important challenge when creating automatically processable laws concerns open-textured terms. The ability to measure open-texture can assist in determining the feasibility of encoding regulation and where additional legal information is required to properly assess a legal issue or dispute. In this article, we propose a novel conceptualisation of open-texture with the aim of determining the extent of open-textured terms in legal documents. We conceptualise open-texture as a lever whose state is impacted (...) by three types of forces: internal forces (the words within the text themselves), external forces (the resources brought to challenge the definition of words), and lateral forces (the merit of such challenges). We tested part of this conceptualisation with 26 participants by investigating agreement in paired annotators. Five key findings emerged. First, agreement on which words are open-texture within a legal text is possible and statistically significant. Second, agreement is even high at an average inter-rater reliability of 0.7 (Cohen’s kappa). Third, when there is agreement on the words, agreement on the Open-Texture Value is high. Fourth, there is a dependence between the Open-Texture Value and reasons invoked behind open-texture. Fifth, involving only four annotators can yield similar results compared to involving twenty more when it comes to only flagging clauses containing open-texture. We conclude the article by discussing limitations of our experiment and which remaining questions in real life cases are still outstanding. (shrink)

Just about every theorist holds that vague terms are context-sensitive to some extent. What counts as ?tall?, ?rich?, and ?bald? depends on the ambient comparison class, paradigm cases, and/or the like. To take a stock example, a given person might be tall with respect to European entrepreneurs and downright short with respect to professional basketball players. It is also generally agreed that vagueness remains even after comparison class, paradigm cases, etc. are fixed, and so this context sensitivity does not solve (...) the problems with vague terms. In this paper, I briefly sketch the main features of my broadly contextualist account of vagueness, that of the late Ruth Manor, and that of Agustín Rayo, showing how the three accounts relate to, and reinforce, each other, noting a seeming difference. A key item used to articulate my own view is David Lewis's notion of conversational score. This is used to track which borderline cases have been called in the course of a conversation. Manor alludes to Lewis's notion of accommodation, a key aspect of the kinematics of conversation, and Rayo invokes Stalnaker's notion of common ground. I'll show how the conversational score could be employed to develop and extend their views, showing how vague terms, as they construe them, are (or can be) deployed in conversation, consistent with both the underlying indeterminacy of the terms and normal communicative goals. To help develop all three views further, I invoke Craige Roberts's notion of Retrievability, a tool developed to show how definites, such as definite descriptions, singular pronouns, and proper names, are deployed in conversation, in a Lewis-style scorekeeping framework. This, I think, is exactly the right way to understand conversations in what Manor calls non-sorities situations and, to invoke my own view, in sorites situations as well. (shrink)

Kripke, argued like this: it seems possible that E; the appearance can't be explained away as really pertaining to a "presentation" of E; so, pending a different explanation, it is possible that E. Textbook Kripkeans see in the contrast between E and its presentation intimations of a quite general distinction between two sorts of meaning. E's secondary or a posteriori meaning is the set of all worlds w which E, as employed here, truly describes. Its primary or a priori meaning (...) is the set of all w such that if w is actual, then E is true. "Conceivability error" occurs when a primary possibility is mistaken for a secondary one. Textbook Kripkeanism is rejected on the grounds that it makes meaning too modal and modality too much a matter of meaning. (shrink)

This new addition to the series New Studies in Practical Philosophy edited by W. D. Hudson is a study of deontic moral judgment, in particular of moral concepts which embody standards for the assessment of claims to right or wrong actions. Three main theses are quite clearly stated. The first thesis concerns the distinctive character of the moral point of view which is irreducible to either logical or factual considerations. The second thesis is that moral judgments claim interpersonal validity in (...) the sense that the speaker makes the implicit claim that every reflective agent who considers the judgment "ought to agree" with the judgment as a correct one. The third thesis pertains to the moral disagreement. It is maintained that moral disagreements are "in principle" decidable, not by way of deductive subsumption under the criteria governing the uses of moral concepts, but by a method of inquiry analogous to that of scientific inquiry involving the use of hypothesis. The author acknowledges his debt to Arthur Prior, J. Kovesi, and M. Polanyi. The book is good in raising questions on the relation between moral inquiry and scientific inquiry, the significance of the open texture of moral terms and its relation to ethical argumentation, and the problem of reconciling differences of moral beliefs in the setting of their connection with nonmoral beliefs.—A. S. C. (shrink)

The appearance in 1945 of the idea of the open texture of empirical concepts, which anticipated Friedrich Waismann’s thesis of a many-level-structure of language, led to a re-evaluation of “context”. It widens the sense of context that we are accustomed to mentioning as being Wittgenstein’s conception of meaning in his later philosophy. The new idea Waismann brought into the landscape is how to “clarify the context”, which is in a way a very non-Wittgensteinian question as well as an (...) “explanation of context”, where open texture plays a key role. But despite the amount of literature about open texture, this idea of Waismann is still not properly understood. Open texture must be situated as an obstacle to exhaustiveness of definition for internal reasons not only because of unforeseen conditions that could always arise in the future, but also by virtue of an a priori aspect of the texture of concepts. Thus, the main goal of this paper is to propose an interpretation of open texture as an immanent property of a concept, that is, as something that is underlying its nature. (shrink)

In this paper we discuss a view of the Machine Learning technique called Explanation-Based Learning (EBL) or Explanation-Based Generalization (EBG) as a process for the interpretation of vague concepts in logic-based models of law.The open-textured nature of legal terms is a well-known open problem in the building of knowledge-based legal systems. EBG is a technique which creates generalizations of given examples on the basis of background domain knowledge. We relate these two topics by considering EBG''s domain knowledge as (...) corresponding to statute law rules, and EBG''s training example as corresponding to a precedent case. (shrink)

Planning problems arise in law when an individual (or corporation)wants to perform a sequence of actions that raises legal issues. Manylawyers make their living planning transactions, and a system thathelped them to solve these problems would be in demand.The designer of such a system in a common-law domain must addressseveral difficult issues, including the open-textured nature of legal rules,the relationship between legal rules and cases, the adversarial nature ofthe domain, and the role of argument. In addition, the system's design (...) isconstrained by the fact that the intended users are lawyers, and its operation and output must be convenient for lawyers to use.In this article, I describe a system called CHIRON that I have developed to explore solutions to these issues. This system develops simple plans from representations of statutes and cases in the domain of United States personal income tax planning. (shrink)

H. L. A. Hart and the "Open Texture" of Language tries to clarify the writings of both Hart and Friedrich Waismann on "open texture". In Waismann's work, "open texture" referred to the potential vagueness of words under extreme (hypothetical) circumstances. Hart's use of the term was quite different, and his work has been misunderstood because those differences were underestimated. Hart should not be read as basing his argument for judicial discretion on the nature of (...) language; primarily, he was putting forward a policy argument for why rules should be applied in a way which would require that discretion. (shrink)

It is a commonly raised argument against the family resemblance account of concepts that there is no limit to a concept's extension. An account of family resemblance which attempts to provide a solution to this problem by including both similarity among instances and dissimilarity to non-instances has been developed by the philosopher of science Thomas Kuhn. Similar solutions have been hinted at in the literature on family resemblance concepts, but the solution has never received a detailed investigation. I shall provide (...) a reconstruction of Kuhn's theory and argue that his solution necessitates a developmental perspective wchich builds on both the transmission of taxonomies between generations and a progressive development through history. (shrink)

This paper examines how the Barthesian difference between écrivains et écrivants can help analyse H.L.A. Hart’s Open Texture of the law. By comparing literary and judicial interpretation, one is able to better understand how lawyers, judges, and legislators can better interpret, use, and draft legislation and case law. In using concrete examples of legislation with regard to the phrase “cruel and usual”, this paper evaluates different interpretive techniques, uncovering key differences. In particular, it compares what implications the Death (...) of the Author would have on legislation that may be considered as vague, or open-ended. Could this mean the Death of the Legislator? What kind of impact may this have on a democracy’s authority? Ultimately, it is suggested that Barthe’s interpretive literary theories are undesirable when it comes to deciphering legislation. Indeed, more concrete and structured approaches are needed. (shrink)

It is a commonly raised argument against thefamily resemblance account of concepts that, on thisaccount, there is no limit to a concept's extension.An account of family resemblance which attempts toprovide a solution to this problem by including bothsimilarity among instances and dissimilarity tonon-instances has been developed by the philosopher ofscience Thomas Kuhn. Similar solutions have beenhinted at in the literature on family resemblanceconcepts, but the solution has never received adetailed investigation. I shall provide areconstruction of Kuhn's theory and argue that (...) hissolution necessitates a developmental perspective withbuilds on both the transmission of taxonomies betweengenerations and a progressive development throughhistory. (shrink)

This book offers a systematized overview of Ian Hacking's work. It presents Hacking’s oeuvre as a network made up of four interconnected key nodes: styles of scientific thinking & doing, probability, making up people, and experimentation and scientific realism. Its central claim is that Michel Foucault’s influence is the underlying thread that runs across the Canadian philosopher’s oeuvre. Foucault’s imprint on Hacking’s work is usually mentioned in relation to styles of scientific reasoning and the human sciences. This research shows that (...) Foucault’s influence can in fact be extended beyond these fields, insofar the underlying interest to the whole corpus of Hacking’s works, namely the analysis of conditions of possibility, is stimulated by the work of the French philosopher. Displacing scientific realism as the central focus of Ian Hacking’s oeuvre opens up a very different landscape, showing, behind the apparent dispersion of his works, the far-reaching interest that amalgamates them: to reveal the historical and situated conditions of possibility for the emergence of scientific objects and concepts. This book shows how Hacking’s deployment concepts such as looping effect, making up people, and interactive kinds, can complement Foucauldian analyses, offering an overarching perspective that can provide a better explanation of the objects of the human sciences and their behaviors. (shrink)

ABSTRACTIn this paper I investigate how conceptual engineering applies to mathematical concepts in particular. I begin with a discussion of Waismann’s notion of open texture, and compare it to Shapiro’s modern usage of the term. Next I set out the position taken by Lakatos which sees mathematical concepts as dynamic and open to improvement and development, arguing that Waismann’s open texture applies to mathematical concepts too. With the perspective of mathematics as open-textured, I make (...) the case that this allows us to deploy the tools of conceptual engineering in mathematics. I will examine Cappelen’s recent argument that there are no conceptual safe spaces and consider whether mathematics constitutes a counterexample. I argue that it does not, drawing on Haslanger’s distinction between manifest and operative concepts, and applying this in a novel way to set-theoretic foundations. I then set out some of the questions that need to be engaged with to establish mathematics as involving a kind of conceptual engineering. I finish with a case study of how the tools of conceptual engineering will give us a way to progress in the debate between advocates of the Universe view and the Multiverse view in set theory. (shrink)

By deepening Austin’s reflections on the ‘open texture’ of empirical concepts, Frederick L. Will defends an ‘externalist’ account of mental content: as human beings we could not think, were we not in fact cognizant of a natural world structured by events and objects with identifiable and repeatable similarities and differences. I explicate and defend Will’s insight by developing a parallel critique of Kant’s and Carnap’s rejections of realism, both of whom cannot account properly for the content of experience. (...) This critique shows that Will has identified a genuinely transcendental basis for defending common-sense realism, without appeal to any high-level theory of knowledge. I then show how Will’s realism undercuts Simon Blackburn’s quasi-realism, Michael Williams’ partial defense of skepticism, John Haldane’s attempt to rehabilitate Aquinas’ account of concepts, and how it augments Crispin Wright’s defense of realism. (shrink)

A representation methodology for knowledge allowing multiple interpretations is described. It is based on the following conception of legal knowledge and its open texture. Since indeterminate, legal knowledge must be adapted to fit the circumstances of the cases to which it is applied. Whether a certain adaptation is lawful or not is measured by metaknowledge. But as this too is indeterminate, its adaptation to the case must be measured by metametaknowledge, etc. This hierarchical model of law is quite (...) well-established and may serve well as a basis for a legal knowledge system. To account for the indeterminacy of law such a system should support the construction of different arguments for and against various interpretations of legal sources. However, automatizing this reasoning fully is unsound since it would imply a restriction to arguments defending interpretations anticipated at programming time. Therefore, the system must be interactive and the user''s knowledge be furnished in a principled way. Contrary to the widespread opinion that classical logic is inadequate for representing open-textured knowledge, the framework outlined herein is given a formalization in first order logic. (shrink)

Ben-Yami presents Wittgenstein’s explicit criticism of the Platonic identification of an explanation with a definition and the alternative forms of explanation he employed. He then discusses a few predecessors of Wittgenstein’s criticisms and the Fregean background against which he wrote. Next, the idea of family resemblance is introduced, and objections answered. Wittgenstein’s endorsement of vagueness and the indeterminacy of sense are presented, as well as the open texture of concepts. Common misunderstandings are addressed along the way. Wittgenstein’s ideas, (...) as is then shown, have far-reaching implications for knowledge of meaning and the nature of logic, and with them to the nature of the philosophical project and its possible achievements. (shrink)

In this article the focus is on the topic of indeterminacy of the sources of law. Indeterminacy in this domain may be a product of a lack of legislative regulation or open texture of law, this analysis of the causes of indeterminacy regarding the sources of law is applied to the Cuban legal system.Resumen:El tema de las fuentes del derecho sigue siendo de primer orden en la ciencia jurídica contemporánea, por lo cual resulta de gran utilidad para los (...) juristas abordar las problemáticas que se generan en torno al mismo. Las causas de indeterminación de las fuentes del derecho —determinadas en unos casos por la falta de regulación por parte del legislador y en otros por la textura abierta del derecho que regula el sistema de fuentes— en el sistema jurídico, es una de dichas problemáticas. El objetivo: analizar las causas de indeterminación de las fuentes del derecho y su expresión en contextos jurídicos como el cubano. (shrink)