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 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)

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.

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)

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)

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)

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)

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)

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.

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)

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)

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.

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)

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)

In the future, when we compare the welfare of a being of one substrate (say, a human) with the welfare of another (say, an artificial intelligence system), we will be making an intersubstrate welfare comparison. In this paper, we argue that intersubstrate welfare comparisons are important, difficult, and potentially tractable. The world might soon contain a vast number of sentient or otherwise significant beings of different substrates, and moral agents will need to be able to compare their welfare levels. However, (...) this work will be difficult, because we lack the same kinds of commonalities across substrates that we have within them. Fortunately, we might be able to make at least some intersubstrate welfare comparisons responsibly in spite of these issues. We make the case for cautious optimism and call for more research. (shrink)

According to many philosophers, belief is a settling state. On this view, someone who believes p is disposed to take p for granted in practical and theoretical reasoning. This paper presents a simple objection to this settling conception of belief: it conflicts with our ordinary patterns of belief ascription. I show that ascriptions of unsettled beliefs are commonplace, and that they pose problems for all of the most promising ways of developing the settling conception. I proceed to explore the implications (...) of my argument for the relation between belief and credence, and for the relative importance of belief in psychological explanation. (shrink)

"The Symons Medal—one of Canada's most prestigious honours—recognizes an individual who has made an exceptional contribution to Canadian life. The 2020 Symons Medal was awarded to Mr. Bob Rae, P.C., C.C., O.Ont, Q.C. Mr. Rae is the 20th Medallist in this series, following a formidable line of recipients. Hon. Rae's lecture is Learning from The Past, Imagining the Future: Reflections from a Political Life. Throughout the address, published in a bilingual book format, he explores such themes as Canada's improbable origins (...) as a nation; post-war emergence onto the global stage; active membership within the United Nations; and the significance of the ever-evolving Canadian constitution—a "living tree document." Welcoming remarks were made by the Honourable Premier Dennis King, while the official response was delivered by the Honourable Roy Romanow, PC, OC, SOM, QC. Hon. Rae is the 2020 recipient in part for his many years of work on humanitarian issues, most recently the Rohingya Refugee crisis, as well as his time focusing on indigenous issues and his decades of public service and teaching. He has always been deeply involved in the political life of Canada—as MP, former premier of Ontario, and interim leader of the federal Liberal Party. The Symons Medal and Lecture Series is named in honour of Professor Thomas H.B. Symons, a long-time supporter of Confederation Centre and a Board Governor. Held each fall, the medal ceremony and its associated lecture offer a national platform for an eminent Canadian to discuss the nation’s current state, shared histories, and future prospects using themes related to their professional pursuits. The award is presented at Confederation Centre of the Arts—Canada’s living monument to Canadian Confederation—in honour of the meetings of the Fathers of Confederation, who first gathered at Province House National Historic Site in the fall of 1864."--. (shrink)

"We often find that the state of not-knowing can be a precursor to moments of rich discovery which possess a dynamic, transformative power that exceeds any prior expectation." From the Introduction In daily life, when we see, hear or touch something that we don't recognise, we are instantly at our most alert. In that condition of 'not-knowing' we are in a state of alive, lithe awareness: asking questions, inviting input, open to learning, looking for significance and meaning... These essays, most (...) by practising psychotherapists, some of them Buddhists, take as their starting point the idea that not-knowing is fundamental to conscious reflection and the desire to know must always arise in the first instance from the self-awareness of not-knowing. Read in any order, on waking, before sleeping or when in doubt... they will be of interest to anyone who has ever sat in either chair in a psychotherapist's or counsellor's room - as well as anyone troubled or intrigued by what it takes to live comfortably and well in a condition of not knowing all (or even any) of the answers. (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)

Arguments for veganism don’t make many vegans, or even many who think they ought to be vegans, at least when they’re written by philosophers. Others — such as the one by Jonathan Safran Foer — seem to do a bit better. Why? To answer this question, I sketch a theory of ordinary moral argumentation that highlights the importance of meaning-based considerations in arguing that people ought to act in ways that deviate from normal expectations for behaviour. In particular, I outline (...) an eclectic theory, where we draw on a variety of moral frameworks and don’t assume that morality is generally overriding. I suggest that meaning-based considerations help us sort through the array of reasons available to us, as well as explain why, in a particular case, what we ought to do morally is what we ought to do all things considered. (shrink)

In this collection of closely interrelated essays, Robert Young emphasizes the scope of the nineteenth-century debate on 'man's place in nature' at the same time as he engages with the approaches of scholars who write about it. He is critical of the separation of the writing of history from writing about history, historiography, and of the separation of history from politics and ideology, then or now. Dr Young challenges fellow historians for reimposing the very disciplinary boundaries that the nineteenth-century debate (...) showed were in the service of ideological forces in that culture. Rather, he proposes that the full weight of the contending forces should be made apparent and debated openly so that neither nineteenth-century nor contemporary issues about the role of science in culture should be treated in a narrow perspective. (shrink)

One leading account of justification comes from the evidentialist tradition. According to evidentialists, whether a doxastic attitude is justified depends on whether that attitude is supported by the believer’s evidence. This chapter assesses the prospects for evidentialism, focusing on the question of whether evidentialists can provide a satisfactory account of their key notions – evidence possession and evidential support – without helping themselves to the notion of justification.

In this wide-ranging interview, Bob Jessop discusses the development of, and many of the main themes in, his work over the last fifty years. He explains how he became interested in realism and Marx...

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 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)

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)

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)

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