In previous work we gave a mathematical foundation, referred to as DisCoCat, for how words interact in a sentence in order to produce the meaning of that sentence. To do so, we exploited the perfect structural match of grammar and categories of meaning spaces. Here, we give a mathematical foundation, referred to as DisCoCirc, for how sentences interact in texts in order to produce the meaning of that text. First we revisit DisCoCat. While in DisCoCat all meanings are fixed as (...) states, in DisCoCirc word meanings correspond to a type, or system, and the states of this system can evolve. Sentences are gates within a circuit which update the variable meanings of those words. Like in DisCoCat, word meanings can live in a variety of spaces e.g. propositional, vectorial, or cognitive. The compositional structure are string diagrams representing information flows, and an entire text yields a single string diagram in which word meanings lift to the meaning of the entire text. While the developments in this paper are independent of a physical embodiment, both the compositional formalism and suggested meaning model are highly quantum-inspired, and implementation on a quantum computer would come with a range of benefits. We also praise Jim Lambek for his role in mathematical linguistics in general, and the development of the DisCo program more specifically. (shrink)

It will be shown that the probability calculus of a quantum mechanical entity can be obtained in a deterministic framework, embedded in a real space, by introducing a lack of knowledge in the measurements on that entity. For all n ∃ ℕ we propose an explicit model in $\mathbb{R}^{n^2 } $ , which entails a representation for a quantum entity described by an n-dimensional complex Hilbert space þn, namely, the “þn,Euclidean hidden measurement representation.” This Euclidean hidden measurement representation is also (...) in a more general sense equivalent with the orthodox Hilbert space formulation of quantum mechanics, since every mathematical ingredient of ordinary quantum mechanics can easily be translated into the framework of these representations. (shrink)

The Distributional Compositional Categorical model is a mathematical framework that provides compositional semantics for meanings of natural language sentences. It consists of a computational procedure for constructing meanings of sentences, given their grammatical structure in terms of compositional type-logic, and given the empirically derived meanings of their words. For the particular case that the meaning of words is modelled within a distributional vector space model, its experimental predictions, derived from real large scale data, have outperformed other empirically validated methods that (...) could build vectors for a full sentence. This success can be attributed to a conceptually motivated mathematical underpinning, something which the other methods lack, by integrating qualitative compositional type-logic and quantitative modelling of meaning within a category-theoretic mathematical framework. The type-logic used in the DisCoCat model is Lambekʼs pregroup grammar. Pregroup types form a posetal compact closed category, which can be passed, in a functorial manner, on to the compact closed structure of vector spaces, linear maps and tensor product. The diagrammatic versions of the equational reasoning in compact closed categories can be interpreted as the flow of word meanings within sentences. Pregroups simplify Lambekʼs previous type-logic, the Lambek calculus. The latter and its extensions have been extensively used to formalise and reason about various linguistic phenomena. Hence, the apparent reliance of the DisCoCat on pregroups has been seen as a shortcoming. This paper addresses this concern, by pointing out that one may as well realise a functorial passage from the original type-logic of Lambek, a monoidal bi-closed category, to vector spaces, or to any other model of meaning organised within a monoidal bi-closed category. The corresponding string diagram calculus, due to Baez and Stay, now depicts the flow of word meanings, and also reflects the structure of the parse trees of the Lambek calculus. (shrink)

In their seminal paper Birkhoff and von Neumann revealed the following dilemma:[ ] whereas for logicians the orthocomplementation properties of negation were the ones least able to withstand a critical analysis, the study of mechanics points to the distributive identities as the weakest link in the algebra of logic.

In Coecke (2002) we proposed the intuitionistic or disjunctive representation of quantum logic, i.e., a representation of the property lattice of physical systems as a complete Heyting algebra of logical propositions on these properties, where this complete Heyting algebra goes equipped with an additional operation, the operational resolution, which identifies the properties within the logic of propositions. This representation has an important application towards dynamic quantum logic, namely in describing the temporal indeterministic propagation of actual properties of physical systems. (...) This paper can as such by conceived as an addendum to Quantum Logic in Intuitionistic Perspective that discusses spin-off and thus provides an additional motivation. We derive a quantaloidal semantics for dynamic disjunctive quantum logic and illustrate it for the particular case of a perfect (quantum) measurement. (shrink)

We generalize the results of Ref. 7 for the coherent states of a spin-1 entity to spin-S entities with S > 1 and to noncoherent spin states: through the introduction of “hidden correlations” (see Ref. 8) we introduce a representation for a spin-S entity as a compound system consisting of 2S “individual” spin-1/2 entities, each of them represented by a “proper state,” and such that we are able to consider a measurement on the spin-S entity as a measurement on each (...) of the individual spin1/2 entities. If the spin-S entity is in a maximal spin state, the 2S individual spin1/2 entities behave as a collection of indistinguishable but separated entities. If not so, we have to introduce the same kind of hidden correlations as required for a hidden correlation representation of a compound quantum system described by a symmetrical superposition. Moreover, by applying the Majorana representation of Ref. 11 and Aerts' representation for a spin-1/2 entity of Ref. 3, this hidden correlation representation yields a classical mechanistic representation of a spin-S entity in $ \mathbb{R}^{\text{3}} $. (shrink)

We introduce a graphical framework for Bayesian inference that is sufficiently general to accommodate not just the standard case but also recent proposals for a theory of quantum Bayesian inference wherein one considers density operators rather than probability distributions as representative of degrees of belief. The diagrammatic framework is stated in the graphical language of symmetric monoidal categories and of compact structures and Frobenius structures therein, in which Bayesian inversion boils down to transposition with respect to an appropriate compact structure. (...) We characterize classical Bayesian inference in terms of a graphical property and demonstrate that our approach eliminates some purely conventional elements that appear in common representations thereof, such as whether degrees of belief are represented by probabilities or entropie quantities. We also introduce a quantum-like calculus wherein the Frobenius structure is noncommutative and show that it can accommodate Leifer's calculus of 'conditional density operators'. The notion of conditional independence is also generalized to our graphical setting and we make some preliminary connections to the theory of Bayesian networks. Finally, we demonstrate how to construct a graphical Bayesian calculus within any dagger compact category. (shrink)

In Coecke (2002) we proposed the intuitionistic or disjunctive representation of quantum logic, i.e., a representation of the property lattice of physical systems as a complete Heyting algebra of logical propositions on these properties, where this complete Heyting algebra goes equipped with an additional operation, the operational resolution, which identifies the properties within the logic of propositions. This representation has an important application “towards dynamic quantum logic”, namely in describing the temporal indeterministic propagation of actual properties of physical systems. (...) This paper can as such by conceived as an addendum to “Quantum Logic in Intuitionistic Perspective” that discusses spin-off and thus provides an additional motivation. We derive a quantaloidal semantics for dynamic disjunctive quantum logic and illustrate it for the particular case of a perfect (quantum) measurement. (shrink)

It will be shown in this article that an ontological approach for some problems related to the interpretation of Quantum Mechanics could emerge from a re-evaluation of the main paradox of early Greek thought: the paradox of Being and non-Being, and the solutions presented to it by Plato and Aristotle. More well known are the derivative paradoxes of Zeno: the paradox of motion and the paradox of the One and the Many. They stem from what was perceived by classical philosophy (...) to be the fundamental enigma for thinking about the world: the seemingly contradictory results that followed from the co-incidence of being and non-being in the world of change and motion as we experience it, and the experience of absolute existence here and now. The most clear expression of both stances can be found, again following classical thought, in the thinking of Heraclitus of Ephesus and Parmenides of Elea. The problem put forward by these paradoxes reduces for both Plato and Aristotle to the possibility of the existence of stable objects as a necessary condition for knowledge. Hence the primarily ontological nature of the solutions they proposed: Plato's Theory of Forms and Aristotle's metaphysics and logic. Plato's and Aristotle's systems are argued here to do on the ontological level essentially the same: to introduce stability in the world by introducing the notion of a separable, stable object, for which a principle of contradiction is valid: an object cannot be and not-be at the same place at the same time. So it becomes possible to forbid contradiction on an epistemological level, and thus to guarantee the certainty of knowledge that seemed to be threatened before. After leaving Aristotelian metaphysics, early modern science had to cope with these problems: it did so by introducing "space" as the seat of stability, and "time" as the theater of motion. But the ontological structure present in this solution remained the same. Therefore the fundamental notion `separable system', related to the notions observation and measurement, themselves related to the modern concepts of space and time, appears to be intrinsically problematic, because it is inextricably connected to classical logic on the ontological level. We see therefore the problems dealt with by quantum logic not as merely formal, and the problem of `non-locality' as related to it, indicating the need to re-think the notions `system', `entity', as well as the implications of the operation `measurement', which is seen here as an application of classical logic on the material world. (shrink)

The creation discovery view and together with it its technically underlying hidden measurement formalism has been elaborated from the early eighties on, and many aspects of it have been exposed in different places [6, 7, 12, 13, 15, 16, 19, 20, 22, 23, 30–37]. In this paper we give an overview of the most important of these aspects.

We introduce an explicit definition for “hidden correlations” on individual entities in a compound system: when one individual entity is measured, this induces a well-defined transition of the “proper state” of the other individual entities. We prove that every compound quantum system described in the tensor product of a finite number of Hilbert spaces can be uniquely represented as a collection of individual entities between which there exist such hidden correlations. We investigate the significance of these hidden correlation representations within (...) Aerts' creation-discovery approach and in particular their compatibility with the hidden measurement formalism. This leads us to the introduction of the notions of “soft” and “hard” “acts of creation” and to the observation that our approach can be seen as a theory of individuals when it is compared to the standard quantum theory. (shrink)

For more than 60 years, Jim Lambek has been a profoundly inspirational mathematician, with groundbreaking contributions to algebra, category theory, linguistics, theoretical physics, logic and proof theory. This Festschrift was put together on the occasion of his 90th birthday. The papers in it give a good picture of the multiple research areas where the impact of Jim Lambek's work can be felt. The volume includes contributions by prominent researchers and by their students, showing how Jim Lambek's ideas keep inspiring upcoming (...) generations of scholars. (shrink)

A symmetric monoidal category naturally arises as the mathematical structure that organizes physical systems, processes, and composition thereof, both sequentially and in parallel. This structure admits a purely graphical calculus. This paper is concerned with the encoding of a fixed causal structure within a symmetric monoidal category: causal dependencies will correspond to topological connectedness in the graphical language. We show that correlations, either classical or quantum, force terminality of the tensor unit. We also show that well-definedness of the concept of (...) a global state forces the monoidal product to be only partially defined, which in turn results in a relativistic covariance theorem. Except for these assumptions, at no stage do we assume anything more than purely compositional symmetric-monoidal categorical structure. We cast these two structural results in terms of a mathematical entity, which we call a causal category . We provide methods of constructing causal categories, and we study the consequences of these methods for the general framework of categorical quantum mechanics. (shrink)

We show that the physical principle, “the adjoint associates to each state a ‘test’ for that state”, fully characterises the Hermitian adjoint for pure quantum theory, therefore providing the adjoint with operational meaning beyond its standard mathematical definition. Moreover, we demonstrate that for general process theories, which all admit a diagrammatic representation, this physical principle induces a diagrammatic reflection operation.

The book, which stepwise develops successively more powerful logical and grammatical systems, covers an unusually broad range of material.

This paper considers how 5- to 11-year-olds’ verbal reasoning about the causality underlying extended, dynamic natural processes links to various facets of their statistical thinking. Such continuous processes typically do not provide perceptually distinct causes and effect, and previous work suggests that spatial–temporal analysis, the ability to analyze spatial configurations that change over time, is a crucial predictor of reasoning about causal mechanism in such situations. Work in the Humean tradition to causality has long emphasized on the importance of statistical (...) thinking for inferring causal links between distinct cause and effect events, but here we assess whether this is also viable for causal thinking about continuous processes. Controlling for verbal and non-verbal ability, two studies administered a battery of covariation, probability, spatial–temporal, and causal measures. Results indicated that spatial–temporal analysis was the best predictor of causal thinking across both studies, but statistical thinking supported and informed spatial–temporal analysis: covariation assessment potentially assists with the identification of variables, while simple probability judgment potentially assists with thinking about unseen mechanisms. We conclude that the ability to find out patterns in data is even more widely important for causal analysis than commonly assumed, from childhood, having a role to play not just when causally linking already distinct events but also when analyzing the causal process underlying extended dynamic events without perceptually distinct components. (shrink)

According to expressivists, normative language expresses desire‐like states of mind. According to noncognitivists, normative beliefs have a desire‐like functional role. What is the relation between these two doctrines? It is widely assumed that expressivism commits you to noncognitivism, and vice versa. This paper opposes that assumption. I advance a view that combines a noncognitivist psychology with a descriptivist semantics for normative language. While this might seem like an ungainly hybrid, I argue that it has important advantages over more familiar metaethical (...) positions. The noncognitivist aspect of the theory captures all of the explanatory benefits standardly associated with expressivism. At the same time, the descriptivist element allows us to avoid the semantic headaches for expressivism. (shrink)

According to a rich tradition in philosophy of action, intentional action requires practical knowledge: someone who acts intentionally knows what they are doing while they are doing it. Piñeros Glasscock argues that an anti-luminosity argument, of the sort developed in Williamson, can be readily adapted to provide a reductio of an epistemic condition on intentional action. This paper undertakes a rescue mission on behalf of an epistemic condition on intentional action. We formulate and defend a version of an epistemic condition (...) that is free from any luminosity commitments. While this version of an epistemic condition escapes reductio, it comes with substantive commitments of its own. In particular, we will see that it forces us to deny the existence of any essentially intentional actions. We go on to argue that this consequence should be embraced. On the resulting picture, intentional action is not luminous. But it still entails practical knowledge. (shrink)

Relativism and expressivism offer two different semantic frameworks for grappling with a similar cluster of issues. What is the difference between these two frameworks? Should they be viewed as rivals? If so, how should we choose between them? This chapter sheds light on these questions. After providing an overview of relativism and expressivism, I discuss three potential choice points: their relation to truth conditional semantics, their pictures of belief and communication, and their explanations of disagreement.

This paper argues that we should assign certainty a central place in epistemology. While epistemic certainty played an important role in the history of epistemology, recent epistemology has tended to dismiss certainty as an unattainable ideal, focusing its attention on knowledge instead. I argue that this is a mistake. Attending to certainty attributions in the wild suggests that much of our everyday knowledge qualifies, in appropriate contexts, as certain. After developing a semantics for certainty ascriptions, I put certainty to explanatory (...) work. Specifically, I argue that by taking certainty as our central epistemic notion, we can shed light on a variety of important topics, including evidence and evidential probability, epistemic modals, and the normative constraints on credence and assertion. (shrink)

Here, Bob Hale and Crispin Wright assemble the key writings that lead to their distinctive neo-Fregean approach to the philosophy of mathematics. In addition to fourteen previously published papers, the volume features a new paper on the Julius Caesar problem; a substantial new introduction mapping out the program and the contributions made to it by the various papers; a section explaining which issues most require further attention; and bibliographies of references and further useful sources. It will be recognized as the (...) most powerful presentation yet of a neo-Fregean program. (shrink)

One attractive feature of process reliabilism is its reductive potential: it promises to explain justification in entirely non-epistemic terms. In this paper, I argue that the phenomenon of epistemic defeat poses a serious challenge for process reliabilism’s reductive ambitions. The standard process reliabilist analysis of defeat is the ‘Alternative Reliable Process Account’ (ARP). According to ARP, whether S’s belief is defeated depends on whether S has certain reliable processes available to her which, if they had been used, would have resulted (...) in S not holding the belief in question. Unfortunately, ARP proves untenable. I show, by way of counterexample, that ARP fails to articulate either necessary or sufficient conditions on defeat. Process reliabilists must either provide an alternative reductive account of defeat or renounce their reductive aspirations. (shrink)

This essay defends a novel form of virtue epistemology: Modal Virtue Epistemology. It borrows from traditional virtue epistemology the idea that knowledge is a type of skillful performance. But it goes on to understand skillfulness in purely modal terms — that is, in terms of success across a range of counterfactual scenarios. We argue that this approach offers a promising way of synthesizing virtue epistemology with a modal account of knowledge, according to which knowledge is safe belief. In particular, we (...) argue that the modal elements of the view offer important advantages over traditional virtue epistemology, which assigns a central explanatory role to aptness. At the same time, the virtue epistemological elements of the theory reveal new avenues for overcoming obstacles to traditional modal accounts of knowledge. Finally, we highlight the advantages of Modal Virtue Epistemology over alternative syntheses, such as Pritchard's (2012) hybrid approach. (shrink)

This paper develops a question-sensitive theory of intention. We show that this theory explains some puzzling closure properties of intention. In particular, it can be used to explain why one is rationally required to intend the means to one’s ends, even though one is not rationally required to intend all the foreseen consequences of one’s intended actions. It also explains why rational intention is not always closed under logical implication, and why one can only intend outcomes that one believes to (...) be under one’s control. (shrink)

Neoliberalization is a distinctive economic, political, and social project that promotes profit-oriented, market-mediated accumulation as the primary axis of societalization. This might suggest that neoliberalism promotes the primacy of the economic but, since its extension and reproduction require continuing state support and, indeed, involve what Weber called political capitalism, one might also argue that it entails a primacy of the political. To address this paradox, my article offers a baseline definition of neoliberalism and identifies four ideal-typical historical forms thereof; relates (...) neoliberalism to the world market, geopolitics and global governance; disambiguates the primacy of the economic; and addresses the role of the political in promoting neoliberalism and handling its contradictions and crisis-tendencies. It illustrates this exercise in critical theory from the North Atlantic Financial Crisis and how its management has strengthened the neoliberal project, enabled its main promoters and beneficiaries to escape the need to learn from their mistakes, and even enabled them to further enrich themselves. (shrink)

To what extent general intelligence mechanisms are associated with causal thinking is unclear. There has been little work done experimentally to determine which developing cognitive capacities help to integrate causal knowledge into explicit systems. To investigate this neglected aspect of development, 138 children aged 5–11 studying at mainstream primary schools completed a battery of three intelligence tests: one investigating verbal ability, another looking at verbal analogical, and a third assessing non-verbal/fluid reasoning. Children were also interviewed over the course of three (...) causal tasks, with the results showing that the developmental paths exhibited uneven profiles across the three causal phenomena. Children consistently found that explaining solution, where substances disappeared toward the end of the process, was more challenging. The confirmatory factor analyses suggested that the impact of cognitive ability factor in explicitly identifying causal relations was large. The proportion of the direct effect of general intelligence was 66% and it subsumed the variances of both verbal measures. Of this, 37% was the indirect effect of age. Fluid reasoning explained a further 28% of the variance, playing a unique role in causal explanation. The results suggested that, overall, cognitive abilities are substantially related to causal reasoning, but not entirely due to developmental differences in “g” during the age periods studied. (shrink)

The past decade has seen a protracted debate over the semantics of epistemic modals. According to contextualists, epistemic modals quantify over the possibilities compatible with some contextually determined group’s information. Relativists often object that contextualism fails to do justice to the way we assess utterances containing epistemic modals for truth or falsity. However, recent empirical work seems to cast doubt on the relativist’s claim, suggesting that ordinary speakers’ judgments about epistemic modals are more closely in line with contextualism than relativism (...) (Knobe & Yalcin 2014; Khoo 2015). This paper furthers the debate by reporting new empirical research revealing a previously overlooked dimension of speakers’ truth-value judgments concerning epistemic modals. Our results show that these judgments vary systematically with the question under discussion in the conversational context in which the utterance is being assessed. We argue that this ‘QUD effect’ is difficult to explain if contextualism is true, but is readily explained by a suitably flexible form of relativism. (shrink)

We often claim to know what might be—or probably is—the case. Modal knowledge along these lines creates a puzzle for information-sensitive semantics for epistemic modals. This paper develops a solution. We start with the idea that knowledge requires safe belief: a belief amounts to knowledge only if it could not easily have been held falsely. We then develop an interpretation of the modal operator in safety that allows it to non-trivially embed information-sensitive contents. The resulting theory avoids various paradoxes that (...) arise from other accounts of modal knowledge. It also delivers plausible predictions about modal Gettier cases. (shrink)

This volume provides a survey of contemporary philosophy of language. As well as providing a synoptic view of the key issues, figures, concepts and debates, each essay makes new and original contributions to ongoing debate.

One leading approach to justification comes from the reliabilist tradition, which maintains that a belief is justified provided that it is reliably formed. Another comes from the ‘Reasons First’ tradition, which claims that a belief is justified provided that it is based on reasons that support it. These two approaches are typically developed in isolation from each other; this essay motivates and defends a synthesis. On the view proposed here, justification is understood in terms of an agent’s reasons for belief, (...) which are in turn analyzed along reliabilist lines: an agent's reasons for belief are the states that serve as inputs to their reliable processes. I show that this synthesis allows each tradition to profit from the other's explanatory resources. In particular, it enables reliabilists to explain epistemic defeat without abandoning their naturalistic ambitions. I go on to compare my proposed synthesis with other hybrid versions of reliabilism that have been proposed in the literature. (shrink)

This chapter first talks about LEGO modal epistemology. Modal epistemology has the two parts. Some of it is the study of how one knows that some things are contingent and others necessary. The other part of modal epistemology concerns how much one know about what is contingent and necessary. The chapter then talks about what went wrong with the imagination‐based story. Whatever the story about how one knows what he/she can build, it had better be one that factors in his/her (...) background knowledge, since that is what makes the imagination useful. With working knowledge of those little pieces, one can make judgments about what can and can not be done with them. Crucially, there are various ways to make those judgments. The chapter further focuses on a segment of the LEGO community that might seem a bit fussy: purists. (shrink)

Quasi-realists face the challenge of providing a plausible analysis of acknowledgments of moral fallibility. This paper devel...

This book is an eye-opening account of transnational advocacy, not by environmental and rights groups, but by conservative activists. Mobilizing around diverse issues, these networks challenge progressive foes across borders and within institutions. In these globalized battles, opponents struggle as much to advance their own causes as to destroy their rivals. Deploying exclusionary strategies, negative tactics and dissuasive ideas, they aim both to make and unmake policy. In this work, Clifford Bob chronicles combat over homosexuality and gun control in the (...) UN, the Americas, Europe and elsewhere. He investigates the 'Baptist-burqa' network of conservative believers attacking gay rights, and the global gun coalition blasting efforts to control firearms. Bob draws critical conclusions about norms, activists and institutions, and his broad findings extend beyond the culture wars. They will change how campaigners fight, scholars study policy wars, and all of us think about global politics. (shrink)

Uses interviews with persons who have changed from belief to nonbelief or vice versa, and discusses what comfort people receive from religion.

What is it to believe something might be the case? We develop a puzzle that creates difficulties for standard answers to this question. We go on to propose our own solution, which integrates a Bayesian approach to belief with a dynamic semantics for epistemic modals. After showing how our account solves the puzzle, we explore a surprising consequence: virtually all of our beliefs about what might be the case provide counterexamples to the view that rational belief is closed under logical (...) implication. (shrink)

When is it permissible to rely on a proposition in practical reasoning? Standard answers to this question face serious challenges. This paper uses these challenges to motivate a certainty norm of practical reasoning. This norm holds that one is permitted to rely on p in practical reasoning if and only if p is epistemically certain. After developing and defending this norm, I consider its broader implications. Taking a certainty norm seriously calls into question traditional assumptions about the importance of belief (...) and knowledge. In particular, it raises the possibility that many epistemological jobs that are usually assigned to belief and knowledge should be reallocated to two related but importantly different states: psychological and epistemic certainty. (shrink)

Why engage in inquiry? According to many philosophers, the goal of inquiring into some question is to come to know its answer. While this view holds considerable appeal, this paper argues that it stands in tension with another highly attractive thesis: knowledge does not require absolute certainty. Forced to choose between these two theses, I argue that we should reject the idea that inquiry aims at knowledge. I go on to develop an alternative view, according to which inquiry aims at (...) maximizing the epistemic value of our credences. This alternative view makes room for knowledge that falls shy of certainty, and it coheres nicely with a rich body of work in epistemic decision theory. I proceed to highlight the implications of this replacement for some important topics in epistemology, including the dogmatism paradox, the nature of interrogative attitudes, and the norm of practical reasoning. (shrink)