This paper considers the extent to which the earliest stages of learning about systems of inscription requires not just individual mental effort, but effort that is distributed across a wide physical and intellectual environment. It is particularly concerned with how children under the age of three learn about notational systems, including writing, and examines parallels with the evolution of written systems. It considers the position that children gain knowledge incrementally over the early months and years of life, supported by a (...) commonly accepted view that learning is a mental activity in which knowledge accumulates hierarchically in the minds of individual children, starting with smallest parts of the system. The paper presents evidence to the contrary, suggesting that the early learning of inscriptional systems is associated not just with individual minds, but with social and cultural cognition that is dispersed across minds, bodies, tools, and material environments. It presents data and evidence from a small study of the sign-making of children under the age of three that indicates that children of this age already use notations purposefully in the construction of signs that are intentional, multimodal, and unbounded, and that already have features associated with conventional systems of inscription. In their early sign-making they use certain underlying principles of symbolic reference associated with conventional systems, including the use of ‘generic’ structures derived from social, bodily, and material experience. Central to this process are networks of interactions between co-participants, tools, materials, and the physical environment. (shrink)

Rousseau's understanding of legislation as the expression of the general will implies a constitutional principle of legislative supremacy. In turn, this should translate to a narrow, mechanical account of adjudication, lest creative judicial interpretation subvert the primacy of legislative power. Yet in his constitutional writings, Rousseau recommends open-textured and vague legislative codes, which he openly admits will require judicial development. Thus he apparently trusts a great deal in judicial discretion. Ostensibly, then, he overlooks the problem of how legislative indeterminacy—and correspondingly, (...) judicial discretion—may undermine the authority of the general will. However, I argue that Rousseau aims to check judicial subversion of legislative supremacy simply by extending his broader social politics—and specifically, his peculiar concept of republican virtue—to the domain of law. His main concern is that the law should not develop as a mystifying expert practice; therefore, he necessarily rejects any understanding of judicial virtue as lying in principled discourse. Instead, he envisages that judicial power will be checked by a more generic sense of republican virtue. In turn this echoes his apprehension of social differentiation and social complexity as sources of domination and hierarchy. (shrink)

This is the first English translation directly based on the original Polish ‘Zasada sprzeczności a logika symboliczna’, the appendix on symbolic logic of Jan Łukasiewicz's 1910 book O zasadzie sprzeczności u Arytotelesa (On the Principle of Contradiction in Aristotle).

A topological model of elementary semiotic schemes is presented. Implications are discussed with respect to the establishment of abstract terms and the search for ultimate meaning.

Since Sun-Joo Shin's groundbreaking study (2002), Peirce's existential graphs have attracted much attention as a way of writing logic that seems profoundly different from our usual logical calculi. In particular, Shin argued that existential graphs enjoy a distinctive property that marks them out as "diagrammatic": they are "multiply readable," in the sense that there are several di erent, equally legitimate ways to translate one and the same graph into a standard logical language. Stenning (2000) and Bellucci and Pietarinen (2016) have (...) retorted that similar phenomena of multiple readability can arise for sentential notations as well. Focusing on the simplest kinds of existential graphs, called alpha graphs (AGs), this paper argues that multiple readability does point to important features of AGs, but that both Shin and her critics have misdiagnosed its source. As a preliminary, and because the existing literature often glosses over such issues, we show that despite their non-linearity, AGs are uniquely parsable and allow for inductive de nitions. Extending earlier discussions, we then show that that in principle, all propositional calculi are multiply readable, just like AGs: contrary to what has been suggested in the literature, multiple readability is linked neither to non-linearity nor to AGs' dearth of connectives. However, we argue that in practice, AGs are more amenable to multiple readability than our usual notations, because the patterns that one needs to recognize to multiply translate an AG form what we call complex symbols, whose structural properties make it easy to perceive and process them as units. Nevertheless, we show that such complex symbols, though largely absent from our usual notations, are not inherently diagrammatic and can be found in seemingly sentential languages. Hence, while ultimately vindicating Shin's idea of multiple readability, our analysis traces it to a di erent source and thus severs its link with diagrammaticity. (shrink)

Understanding Ukrainian sacred art is impossible without understanding how ancient Ukrainians felt space and time, transformed and materialized this understanding in signs, the most ancient among which is the circle, square and cross. These symbols are universal spatial and temporal signs that play the role of archetypes and have deep pre-Christian roots and origins. Their original, cosmological essence of the understanding of nature, the desire to convey the divine essence through comprehension of space and time converges the sacred art of (...) the Christian, Hindu and Islamic worlds. (shrink)

Barash's work is both a philosophical work and a history of ideas. The one offers a conceptual account of collective memory, and the other a narrative of changing conceptions and ideological uses of “memory.” In both cases, he argues that careful attention to the border between memory and history is politically significant for criticising appeals to mythical bases of political unity.The main contention of the book is this: collective memory designates a restricted sphere of past references.

Prior to the twentieth century, theories of knowledge were inherently perceptual. Since then, developments in logic, statis- tics, and programming languages have inspired amodal theories that rest on principles fundamentally different from those underlying perception. In addition, perceptual approaches have become widely viewed as untenable because they are assumed to implement record- ing systems, not conceptual systems. A perceptual theory of knowledge is developed here in the context of current cognitive science and neuroscience. During perceptual experience, association areas in (...) the brain capture bottom-up patterns of activation in sensory-motor areas. Later, in a top-down manner, association areas partially reactivate sensory-motor areas to implement perceptual symbols. The stor- age and reactivation of perceptual symbols operates at the level of perceptual components – not at the level of holistic perceptual expe- riences. Through the use of selective attention, schematic representations of perceptual components are extracted from experience and stored in memory (e.g., individual memories of green, purr, hot). As memories of the same component become organized around a com- mon frame, they implement a simulator that produces limitless simulations of the component (e.g., simulations of purr). Not only do such simulators develop for aspects of sensory experience, they also develop for aspects of proprioception (e.g., lift, run) and introspec- tion (e.g., compare, memory, happy, hungry). Once established, these simulators implement a basic conceptual system that represents types, supports categorization, and produces categorical inferences. These simulators further support productivity, propositions, and ab- stract concepts, thereby implementing a fully functional conceptual system. Productivity results from integrating simulators combinato- rially and recursively to produce complex simulations. Propositions result from binding simulators to perceived individuals to represent type-token relations. Abstract concepts are grounded in complex simulations of combined physical and introspective events. Thus, a per- ceptual theory of knowledge can implement a fully functional conceptual system while avoiding problems associated with amodal sym- bol systems. Implications for cognition, neuroscience, evolution, development, and artificial intelligence are explored. (shrink)

This is a companion article to the translation of ‘Zasada sprzeczności a logika symboliczna’, the appendix on symbolic logic of Jan Łukasiewicz's 1910 book O zasadzie sprzeczności u Arytotelesa (On the Principle of Contradiction in Aristotle). While the appendix closely follows Couturat's 1905 book L'algebra de la logique (The Algebra of Logic), footnotes show that Łukasiewicz was aware of the work of Peirce, Huntington and Russell (before Principia Mathematica). This appendix was influential in the development of the Polish school (...) of logic, directly inspiring Stanisław Leśniewski and Leon Chwistek and more widely by serving as a text of the new symbolic logic. This appendix was an important source of the dominant algebraic logic in Poland, but also indicates that Łukasiewicz appreciated Russell's axiomatic approach to logic. (shrink)

Kant uses the concept of the symbol to show the complicated relationship between the autonomy of beauty and its systematic function as a transition from nature to freedom, which are the two most important topics in the third Critique. Beauty’s symbolism of morality lies in the analog between aesthetic reflection and moral disposition; concretely, it lies in the purity or disinterestedness and self-legislation as negative and positive freedom in both subjective states of mind. In this scenario, beauty’s symbolism does not (...) refer to aesthetic ideas that either involve intelligent interests (in the beauty of nature) or presuppose an end (in the beauty of art); it also cannot be grounded in the supersensible substrate, which is an elevated and metaphysical principle of the judgment of taste given in the Dialectic but not the original principle of subjective purposiveness in the Analytic. With this formal relationship, beauty and morality accelerate each other in the empirical-anthological sense—but they are also not a sufficient or necessary condition for each other. Furthermore, through symbolism, taste looks toward the intelligible and serves as a transition from nature to freedom from the transcendental perspective. (shrink)

We investigate the complexity of various combinatorial theorems about linear and partial orders, from the points of view of computability theory and reverse mathematics. We focus in particular on the principles ADS (Ascending or Descending Sequence), which states that every infinite linear order has either an infinite descending sequence or an infinite ascending sequence, and CAC (Chain-AntiChain), which states that every infinite partial order has either an infinite chain or an infinite antichain. It is well-known that Ramsey's Theorem for (...) pairs (RT₂²) splits into a stable version (SRT₂²) and a cohesive principle (COH). We show that the same is true of ADS and CAC, and that in their cases the stable versions are strictly weaker than the full ones (which is not known to be the case for RT₂² and SRT₂²). We also analyze the relationships between these principles and other systems and principles previously studied by reverse mathematics, such as WKL₀, DNR, and BΣ₂. We show, for instance, that WKL₀ is incomparable with all of the systems we study. We also prove computability-theoretic and conservation results for them. Among these results are a strengthening of the fact, proved by Cholak, Jockusch, and Slaman, that COH is Π₁¹-conservative over the base system RCA₀. We also prove that CAC does not imply DNR which, combined with a recent result of Hirschfeldt, Jockusch, Kjos-Hanssen, Lempp, and Slaman, shows that CAC does not imply SRT₂² (and so does not imply RT₂²). This answers a question of Cholak, Jockusch, and Slaman. Our proofs suggest that the essential distinction between ADS and CAC on the one hand and RT₂² on the other is that the colorings needed for our analysis are in some way transitive. We formalize this intuition as the notions of transitive and semitransitive colorings and show that the existence of homogeneous sets for such colorings is equivalent to ADS and CAC, respectively. We finish with several open questions. (shrink)

This ambitious work reclassifies and restructures the history of ideas and the philosophy of culture through a wide-ranging and novel use of the idea of the organon. It does so by radically revising standard interpretations and theories of all branches of philosophy, and by providing an intellectual and philosophical foundation for the new organon of the cultural sciences. Furthermore, the seeded idea that saw its growth in the form of this book is the unshakable conviction that the only way by (...) which a new apparatus of philosophy, an organon, could be created is by harking back to the vast sources of imagination, inspiration and mimÄ"sis. This entire study is based on the notion that metaphysics, insofar as it is concerned with the world in its entirety and with human beingâ (TM)s existence and thought, should provide the foundation for the organon of cultural sciences, based on symbolic forms. Given that the colossal amount of information and knowledge of philosophy, arts, humanities, logic, mathematics, social sciences and natural sciences cannot be comprised, analyzed and comprehended per se, it is the organonâ (TM)s objective to extract the main principles, ideas, postulates, theorems and theories of the cultural sciences, and, subsequently, to shape and restructure them as symbolic forms. Since all these principles are grounded on Becomingâ "which is not a stable or fixed entity such as Being, substance or thingâ "the symbolic forms preserve and change, elevate and further the organon of the cultural sciences, via a critical-dialectical process. (shrink)

This paper is concerned with the status of a symbol in Wittgenstein’s Tractatus. It is claimed in the first section that a Tractarian symbol, whilst essentially a syntactic entity to be distinguished from the mark or sound that is its sign, bears its semantic significance only inessentially. In the second and third sections I pursue this point of exegesis through the Tractarian discussions of nonsense and the context principle respectively. The final section of the paper places the forgoing work in (...) a secondary context, addressing in particular a debate regarding the realism of the Tractatus. (shrink)

Although symbolic logic has grown considerably in the subsequent decades, this book remains a classic.

As well as aiming to revive interest in Meinong's thought, this book challenges many of the most widespread assumptions of philosophical logic.

In the first section of the article, we examine some recent criticisms of the connectionist enterprise: first, that connectionist models are fundamentally behaviorist in nature (and, therefore, non-cognitive), and second that connectionist models are fundamentally associationist in nature (and, therefore, cognitively weak). We argue that, for a limited class of connectionist models (feed-forward, pattern-associator models), the first criticism is unavoidable. With respect to the second criticism, we propose that connectionist modelsare fundamentally associationist but that this is appropriate for building models (...) of human cognition. However, we do accept the point that there are cognitive capacities for which any purely associative model cannot provide a satisfactory account. The implication that we draw from is this is not that associationist models and mechanisms should be scrapped, but rather that they should be enhanced.In the next section of the article, we identify a set of connectionist approaches which are characterized by “active symbols” — recurrent circuits which are the basis of knowledge representation. We claim that such approaches avoid criticisms of behaviorism and are, in principle, capable of supporting full cognition. In the final section of the article, we speculate at some length about what we believe would be the characteristics of a fully realized active symbol system. This includes both potential problems and possible solutions (for example, mechanisms needed to control activity in a complex recurrent network) as well as the promise of such systems (in particular, the emergence of knowledge structures which would constitute genuine internal models). (shrink)

A large proportion of hospital inpatients are affected by cognitive impairment, posing challenges in the provision of their care in busy, fast-paced acute wards. Signs and symbols, known as visual identifiers, are employed in many U.K. hospitals with the intention of helping healthcare professionals identify and respond to the needs of these patients. Although widely considered useful, these tools are used inconsistently, have not been subject to full evaluation, and attract criticism for acting as a shorthand for a routinized response. (...) In order for visual identifiers to be used effectively in acute care settings, thorough consideration must be given to the ethical and legal issues that are engaged in this context, and their potential benefits and harms must be weighed and balanced. This paper proposes a set of legal and ethical principles that can be used to guide the implementation of visual identifiers. Together, these principles provide a framework applicable in the design and implementation phases to systematically identify relevant considerations arising from the use of these tools. We outline some tensions that arise between principles and conclude that selecting a preferred moral framework could help to guide decision-making, as does clarity around the purpose and objectives of the identifier. (shrink)

No one need deny the importance of language to thought and cognition. At the same time, there is a tendency in studies of mind and mental functioning to assume that properties and principles of linguistic, or language-like, forms of representation must hold of forms of thought and representation in general. Consideration of a wider range of symbol systems shows that this is not so. In turn, various claims and arguments in cognitive theory that depend on assumptions applicable only to (...) linguistic systems, do not go through or become difficult to state in a manner that makes them both interesting and plausible. (shrink)

Computationalist theories of mind require brain symbols, that is, neural events that represent kinds or instances of kinds. Standard models of computation require multiple inscriptions of symbols with the same representational content. The satisfaction of two conditions makes it easy to see how this requirement is met in computers, but we have no reason to think that these conditions are satisfied in the brain. Thus, if we wish to give computationalist explanations of human cognition, without committing ourselvesa priori to a (...) strong and unsupported claim in neuroscience, we must first either explain how we can provide multiple brain symbols with the same content, or explain how we can abandon standard models of computation. It is argued that both of these alternatives require us to explain the execution of complex tasks that have a cognition-like structure. Circularity or regress are thus threatened, unless noncomputationalist principles can provide the required explanations. But in the latter case, we do not know that noncomputationalist principles might not bear most of the weight of explaining cognition. Four possible types of computationalist theory are discussed; none appears to provide a promising solution to the problem. Thus, despite known difficulties in noncomputationalist investigations, we have every reason to pursue the search for noncomputationalist principles in cognitive theory. (shrink)

This article proposes a neurobiologically motivated theory of meaning as internal representation that holds on to the principle of compositionality, but negates the principle of semantic constituency. The approach builds on neurobiological findings regarding topologically structured cortical feature maps and the mechanism of object-related binding by neuronal synchronization. It incorporates the Gestalt principles of psychology and is implemented by recurrent neural networks. The semantics to be developed is structurally analogous to some variant of model-theoretical semantics. The semantics to be (...) developed is a neuro-emulative model-theoretical semantics of a first-order language. The model-theoretical semantics is merely denotational and does not imply anything about the structures of the mind or the underlying neural mechanisms that enable producing and comprehend meaningful expressions. The relation between a mental representation and its content is some form of causal-informational covariation. (shrink)

I argue that neural activity, strictly speaking, is not computation. This is because computation, strictly speaking, is the processing of strings of symbols, and neuroscience shows that there are no neural strings of symbols. This has two consequences. On the one hand, the following widely held consequences of computationalism must either be abandoned or supported on grounds independent of computationalism: (i) that in principle we can capture what is functionally relevant to neural processes in terms of some formalism taken from (...) computability theory (such as Turing Machines), (ii) that it is possible to design computer programs that are functionally equivalent to neural processes in the same sense in which it is possible to design computer programs that are functionally equivalent to each other, (iii) that the study of neural (or mental) computation is independent of the study of neural implementation, (iv) that the Church-Turing thesis applies to neural activity in the sense in which it applies to digital computers. On the other hand, we need to gradually reinterpret or replace computational theories in psychology in terms of theoretical constructs that can be realized by known neural processes, such as the spike trains of neuronal ensembles. (shrink)

In this research I will focus on a basis for a formal model based on an alternative kind of logic invented by Hans Götzsche: Occurrence Logic (Occ Log), which is not based on truth values and truth functionality. Also, I have taken into account tense logic developed and elaborated by A. N. Prior. In this article I will provide a conceptual and logical foundation for formal Occurrence Logic based on symbolic logic and will illustrate the most important relations between (...) symbolic logic and Occurrence Logic. There will be a special need to focus on arguments based on symbolic occurrence logic. The interrelationships and dependencies between symbolic logic and Occ Log could be conducive to an approach that can express and analyse ‘the occurrences of the world’s descriptions based on various logical principles at different moments’. Symbolic Occ Log checks the occurrence conditioning and the priority of occurrences by employing occurrence values. I shall conclude that the desired approach could gradually support me in providing a truth-functional independent first-order predicate calculus that could work on semantic analysis of different propositions within world descriptions. (shrink)

People can be taught to manipulate symbols according to formal mathematical and logical rules. Cognitive scientists have traditionally viewed this capacity—the capacity for symbolic reasoning—as grounded in the ability to internally represent numbers, logical relationships, and mathematical rules in an abstract, amodal fashion. We present an alternative view, portraying symbolic reasoning as a special kind of embodied reasoning in which arithmetic and logical formulae, externally represented as notations, serve as targets for powerful perceptual and sensorimotor systems. Although (...) class='Hi'>symbolic reasoning often conforms to abstract mathematical principles, it is typically implemented by perceptual and sensorimotor engagement with concrete environmental structures. (shrink)

We establish the thesis that in moral education, particularly in the first years of the child’s development, unreflexive acts or unreflexiveness in certain behaviours of adults is a condition for the development of the personality structure and virtues that enable autonomous ethical reflection and a relation to the Other. With the notion of unreflexiveness we refer to resolvedness in the response of adults when it is necessary to establish a limit, or cut, in the child’s demand for pleasure, as well (...) as to resolvedness as one of the structurally essential elements in behaviours with which the adult subordinates the child to the symbolic order or language. In philosophy, a symptomatic position in this context is represented by Kant’s theory of education. On the basis of the “traditional” concept of “negative education”, and through Freud’s and Lacan’s psychoanalytic concepts, we identify two principles that should, in contemporary times, be an essential part of the moral educational behaviours of adults: the principle of a delimited response to the child’s demand for the satisfaction of pleasure, and the principle of reasoned, reflected, but nevertheless certain, persevering, resolved unreflexiveness in the subordination of the child’s desire to the symbolic order . On the basis of the preceding analysis, we highlight certain consequences of this structure of the subject for the contexts of particular theoretical discussions and for school practice. (shrink)

No one need deny the importance of language to thought and cognition. At the same time, there is a tendency in studies of mind and mental functioning to assume that properties and principles of linguistic, or language-like, forms of representation must hold of forms of thought and representation in general. Consideration of a wider range of symbol systems shows that this is not so. In turn, various claims and arguments in cognitive theory that depend on assumptions applicable only to (...) linguistic systems, do not go through or become difficult to state in a manner that makes them both interesting and plausible. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Symbolic Classification and The Emergence of a Metaphysics of CausalityOwen Goldinwhat is distinctive about metaphysics as a mode of thought that emerged in the fifth century before the Common Era? How did it emerge out of early ways of conceptualizing the world as a whole, and why? Many answers have been proposed. One common view is that earlier modes of thought personify natural agencies; once this is abandoned, (...) the way is open for discerning more complex regularities among natural events.1 Another view is that the process of "rationalization" introduced by the Greeks involves an implicit appeal to the individual's natural intelligence, in contrast to a dependence on an "inherited stock of wisdom."2 A third view is that the first Greek philosophers introduced inference as a way of navigating conceptual relationships, independent of the sorts of personal interactions among divinities related in mythical narrative.3 According to a fourth account, the mode of rationality that allowed for the development of ancient Greek metaphysics was a social [End Page 3] practice with the aim of persuasion, brought on by more democratic involvement in societal deliberations concerning law and policy.4To these I add a fifth way of answering the question. Crucial to the emergence of Greek metaphysics is explicit distinction of ontological/causal priority and posteriority. Ancient Greek scientific explanations developed out of older, archaic modes of conceptualization, which I maintain have deep parallels with modes of conceptualization shared by many nonliterate cultures. Such modes of thought rest on an implicit worldview which can be called a metaphysics, in a loose sense, and are still worthy of philosophical consideration.I begin with a red flag that will be raised for many as soon as we start talking about the metaphysical traditions of nonliterate cultures. What counts as a metaphysical tradition? I make some distinctions. In the loosest sense, an account or tradition is metaphysical if it reflects a coherent worldview concerning the kinds of beings that there are and how they are related to one another. According to a stricter sense of the term, a tradition is metaphysical if it involves working through this worldview in an explicit and systematic matter. In an even stricter sense, a tradition is metaphysical if, in addition to being explicit and systematic, it merits being classified as "philosophical," itself not an unproblematic term. I here stipulate that an account is philosophical if it does not rest on premises that are specific to certain cultural traditions and employs inferential arguments that are in principle subject to refutation.Generally speaking, there are two ways in which a tradition that is metaphysical in the strictest sense can listen to one that is metaphysical in a looser sense. First, a metaphysician might examine the latter tradition in order to explore how, historically, it served as the root for the former. An example is Aristotle's interest in Hesiod as expressing metaphysical intuitions that later philosophers unpacked.5 By studying [End Page 4] the ontological presuppositions of the sort of account from which our own metaphysics is derived, one learns more about our own ontological presuppositions and what we gain from them.6 This sort of dialogue carries with it the danger that the later thinker presupposes that—in the spirit of "you have to start somewhere"—the earlier thinkers were simply doing a bad job working through their own theories; Aristotle has often been accused of precisely this.7Another sort of dialogue is possible as well. An alternative metaphysical tradition can reveal the possibility of a radically different way of conceptualizing the causal and ontological structure of the world, one with its own advantages. In such a case there is shared set of problems at issue, shared points of reference within the philosophical literature, and a shared understanding of what philosophical argument is. But philosophical insight might also emerge from the kind of dialogue in which a developed metaphysics that conforms to certain standards of inferential argument confronts a way of thinking that is metaphysical in a looser sense. A substance-based ontologist can be led to reflect on whether the world has to be cut up as she does by studying Aztec... (shrink)

It is well-documented that mathematics achievement is an important predictor of many positive life outcomes like college graduation, career opportunities, salary, and even citizenship. As such, it is important for researchers and educators to help students succeed in mathematics. Although there are undoubtedly many factors that contribute to students' success in mathematics, much of the research and intervention development has focused on variations in instructional techniques. Indeed, even a cursory glance at many educational journals and granting agencies reveals that there (...) is a large amount of time, energy, and resources being spent on determining the best way to convey information through direct, declarative instruction. The proposed project is motivated by recent calls to expand the focus of research in mathematics education beyond direct, declarative instruction. The overarching goal of the presented experiment is to evaluate the efficacy of a novel mathematics intervention designed using principles taken from the literature on non-declarative learning. The intervention combines errorless learning and structured cue fading to help second grade students improve their understanding of symbolic magnitude. Results indicate that students who learned about symbolic magnitude using the novel intervention did better than students who were provided with extensive declarative support. These findings offer preliminary evidence in favor of using learning combination of errorless learning and cue fading techniques in the mathematics classroom. (shrink)

On the one hand, the concept of truth is a major research subject in analytic philosophy. On the other hand, mathematical logicians have developed sophisticated logical theories of truth and the paradoxes. Recent developments in logical theories of the semantical paradoxes are highly relevant for philosophical research on the notion of truth. And conversely, philosophical guidance is necessary for the development of logical theories of truth and the paradoxes. From this perspective, this volume intends to reflect and promote deeper interaction (...) and collaboration between philosophers and logicians investigating the concept of truth than has existed so far.Aside from an extended introductory overview of recent work in the theory of truth, the volume consists of articles by leading philosophers and logicians on subjects and debates that are situated on the interface between logical and philosophical theories of truth. The volume is intended for graduate students in philosophy and in logic who want an introduction to contemporary research in this area, as well as for professional philosophers and logicians. (shrink)

People often solve spatially presented cognitive problems more easily than their nonspatial counterparts. We explain this phenomenon by characterizing space as an inter-modality that provides common structure to different specific perceptual modalities. The usefulness of spatial structure for knowledge processing on different levels of granularity and for interaction between internal and external processes is described. Map representations are discussed as examples in which the usefulness of spatially organized symbols is particularly evident. External representations and processes can enhance internal representations and (...) processes effectively when the same structures and principles can be implicitly assumed. (shrink)

We give a level-by-level analysis of the Weak Vopěnka Principle for definable classes of relational structures ( $\mathrm {WVP}$ ), in accordance with the complexity of their definition, and we determine the large-cardinal strength of each level. Thus, in particular, we show that $\mathrm {WVP}$ for $\Sigma _2$ -definable classes is equivalent to the existence of a strong cardinal. The main theorem (Theorem 5.11) shows, more generally, that $\mathrm {WVP}$ for $\Sigma _n$ -definable classes is equivalent to the existence of (...) a $\Sigma _n$ -strong cardinal (Definition 5.1). Hence, $\mathrm {WVP}$ is equivalent to the existence of a $\Sigma _n$ -strong cardinal for all $n<\omega $. (shrink)

In his Knowledge and Belief (1962), Hintikka establishes his system of epistemic logic with the KK (Knowing that One Knows, in symbols, Kp→KKp) principle (KK for short). However, his system of epistemic logic and the KK principle are grounded upon his strong notion of knowledge, which requires that knowledge is infallible, that is, it makes further inquiry pointless, and becomes ‘discussion-stopper’; knowledge implies truth, to wit, cognitive agents will not be mistaken in their knowledge; cognitive agents will be ‘perfect logicians’, (...) i.e. they have infinitive capability of logical inference. Hintikka calls the argument for KK from the strong notion of knowledge as the ‘transcendental argument’ for KK. Obviously, the strong notion of knowledge is far away from our ordinary conception of knowledge; based on such a strong sense of knowledge, epistemic logic has met the problem of logical omnipotence, and is difficult to apply in our ordinary cognitive practice. Moreover, there is no close connection of the KK principle with internalism or externalism. (shrink)

Leibnizian-Newtonian calculus was a theory that dealt with geometrical objects; the figure continued to play one of the fundamental roles it had played in Greek geometry: it susbstituted a part of reasoning. During the eighteenth century a process of de-geometrization of calculus took place, which consisted in the rejection of the use of diagrams and in considering calculus as an 'intellectual' system where deduction was merely linguistic and mediated. This was achieved by interpreting variables as universal quantities and introducing the (...) notion of function, which replaced the study of curves. However, the emancipation of calculus from its basis in geometry was not comprehensive. In fact, the geometrical properties of curves were attributed de facto to functions and thus eighteenth-century calculus continued implicitly to use principles borrowed from geometry. There was therefore no transition to a purely syntactical theory based on axiomatically introduced terms, a shift which only took place subsequently in modern times. (shrink)

In 1981, Takeuti introduced quantum set theory as the quantum counterpart of Boolean valued models of set theory by constructing a model of set theory based on quantum logic represented by the lattice of closed subspaces in a Hilbert space and showed that appropriate quantum counterparts of ZFC axioms hold in the model. Here, Takeuti's formulation is extended to construct a model of set theory based on the logic represented by the lattice of projections in an arbitrary von Neumann algebra. (...) A transfer principle is established that enables us to transfer theorems of ZFC to their quantum counterparts holding in the model. The set of real numbers in the model is shown to be in one-to-one correspondence with the set of self-adjoint operators affiliated with the von Neumann algebra generated by the logic. Despite the difficulty pointed out by Takeuti that equality axioms do not generally hold in quantum set theory, it is shown that equality axioms hold for any real numbers in the model. It is also shown that any observational proposition in quantum mechanics can be represented by a corresponding statement for real numbers in the model with the truth value consistent with the standard formulation of quantum mechanics, and that the equality relation between two real numbers in the model is equivalent with the notion of perfect correlation between corresponding observables (self-adjoint operators) in quantum mechanics. The paper is concluded with some remarks on the relevance to quantum set theory of the choice of the implication connective in quantum logic. (shrink)

The purpose of this dissertation is to provide a defence of Aristotle's principle of contradiction against the critique made on it by Jan Lukasiewicz in an article he wrote in 1910 which was translated and published in the March 1971 number of The Review of Metaphysics. Lukasiewicz maintains in general that the law of contradiction has no logical worth. Specifically, he charges Aristotle with having several laws of contradiction instead of one as Aristotle claims; with attempting to prove the law (...) despite his claim that this is impossible and finally with failing in the very attempt to prove the law, or at least one of its formulations. In 20 chapters, each dealing with a respective section of Lukasiewicz's article, I attempt to show that all of Lukasiewicz's allegations are unfounded. My methodology is simply to follow Lukasiewicz through his 20 sections and compare what he says with what Aristotle actually wrote. If successful, this dissertation will show, in basic agreement with Professor Joseph Owens, that the best way to read Aristotle is on his own terms and not, as in the case of Lukasiewicz, via the latest developments in symbolic logic. I reiterate, with Aristotle, that the law of contradiction is the basic principle of being qua being and has logical worth precisely because of that fact. (shrink)

In this book, George Schedler offers moral and legal perspectives on two legacies of the Civil War: the adoption of the Confederate flag by Southern states and the question of African American reparations. Schedler's analysis of reparations focuses on the principle that whatever the enslaved would have earned and enjoyed had they not been enslaved should determine compensation.