There are theoretical limitations to what can be implemented by a computer program. In this paper we are concerned with a limitation on the strength of computer implemented deduction. We use a version of the Curry paradox to arrive at this limitation.

A. N. Turing’s 1936 concept of computability, computing machines, and computable binary digital sequences, is subject to Turing’s Cardinality Paradox. The paradox conjoins two opposed but comparably powerful lines of argument, supporting the propositions that the cardinality of dedicated Turing machines outputting all and only the computable binary digital sequences can only be denumerable, and yet must also be nondenumerable. Turing’s objections to a similar kind of diagonalization are answered, and the implications of the paradox for the concept of a (...) Turing machine, computability, computable sequences, and Turing’s effort to prove the unsolvability of the Entscheidungsproblem, are explained in light of the paradox. A solution to Turing’s Cardinality Paradox is proposed, positing a higher geometrical dimensionality of machine symbol-editing information processing and storage media than is available to canonical Turing machine tapes. The suggestion is to add volume to Turing’s discrete two-dimensional machine tape squares, considering them instead as similarly ideally connected massive three-dimensional machine information cells. Three-dimensional computing machine symbol-editing information processing cells, as opposed to Turing’s two-dimensional machine tape squares, can take advantage of a denumerably infinite potential for parallel digital sequence computing, by which to accommodate denumerably infinitely many computable diagonalizations. A three-dimensional model of machine information storage and processing cells is recommended on independent grounds as better representing the biological realities of digital information processing isomorphisms in the three-dimensional neural networks of living computers. (shrink)

Recent scholarship has helped to demythologise the life and work of Georg Philipp Friedrich von Hardenberg who, as the poet “Novalis”, had come to instantiate the nineteenth-century’s stereotype of the romantic poet. Among Hardenberg’s interests that seem to sit uneasily with this literary persona were his interests in science and mathematics, and especially in the idea, traceable back to Leibniz, of a mathematically based computational approach to language. Hardenberg’s approach to language, and his attempts to bring mathematics to bear on (...) poetry, is examined in relation to debates that developed late in the eighteenth century over the relation of language to thought—debates which share many features with contemporary ones in this area. (shrink)

The paper presents a paradoxical feature of computational systems that suggests that computationalism cannot explain symbol grounding. If the mind is a digital computer, as computationalism claims, then it can be computing either over meaningful symbols or over meaningless symbols. If it is computing over meaningful symbols its functioning presupposes the existence of meaningful symbols in the system, i.e. it implies semantic nativism. If the mind is computing over meaningless symbols, no intentional cognitive processes are available prior to symbol (...) grounding. In this case, no symbol grounding could take place since any grounding presupposes intentional cognitive processes. So, whether computing in the mind is over meaningless or over meaningful symbols, computationalism implies semantic nativism. (shrink)

We argued [Since this argument appeared in other journals, I am reprising it here, almost verbatim.] (Fulda in J Law Info Sci 2:230–232, 1991/AI & Soc 8(4):357–359, 1994) that the paradox of the preface suggests a reason why machines cannot, will not, and should not be allowed to judge criminal cases. The argument merely shows that they cannot now and will not soon or easily be so allowed. The author, in fact, now believes that when—and only when—they are ready they (...) actually should be so allowed, in the interests of justice. Both the original argument applied and this detailed reconsideration applies exclusively to trial courts, and both specifically exclude(d) sentencing. The argument highlights some key relevant differences between minds and machines and attempts, also, to explain why automation is of far greater import for the first-level justice system (trial courts) than for higher courts. A final section discusses why sentencing was, is, and should be excluded. (shrink)

This volume began as a remembrance of Alonzo Church while he was still with us and is now finally complete. It contains papers by many well-known scholars, most of whom have been directly influenced by Church's own work. Often the emphasis is on foundational issues in logic, mathematics, computation, and philosophy - as was the case with Church's contributions, now universally recognized as having been of profound fundamental significance in those areas. The volume will be of interest to logicians, (...) computer scientists, philosophers, and linguists. The contributions concern classical first-order logic, higher-order logic, non-classical theories of implication, set theories with universal sets, the logical and semantical paradoxes, the lambda-calculus, especially as it is used in computation, philosophical issues about meaning and ontology in the abstract sciences and in natural language, and much else. The material will be accessible to specialists in these areas and to advanced graduate students in the respective fields. (shrink)

In the future, policies surrounding end-of-life decisions will be faced with the question of whether competent people in a completely locked-in state should be enabled to make end-of-life decisions via brain-computer interfaces (BCI). This article raises ethical issues with acting through BCIs in the context of these decisions, specifically self-administration requirements within assisted suicide policies. We argue that enabling patients to end their life even once they have entered completely locked-in state might, paradoxically, prolong and uphold their quality of life.

This paper examines the paradoxical transparency involved in training machine-learning models. Existing literature typically critiques the opacity of machine-learning models such as neural networks or collaborative filtering, a type of critique that parallels the black-box critique in technology studies. Accordingly, people in power may leverage the models’ opacity to justify a biased result without subjecting the technical operations to public scrutiny, in what Dan McQuillan metaphorically depicts as an “algorithmic state of exception”. This paper attempts to differentiate the black-box (...) abstraction that wraps around complex computational systems from the opacity of machine-learning models. It contends that the degree of asymmetry in knowledge is greater in the former than the latter. In the case of software systems, software codes are difficult to understand as only software experts with sufficient domain knowledge are equipped to formulate a sound critique. In contrast, the meanings of trained parameters in a machine-learning model are obscure even to the data scientists who configure and train the model. Hence, the asymmetry of knowledge lies only in how data examples are collected, the choice and configuration of machine-learning models, and the specification of features in model design. Under the trend of algorithmic decision-making proliferating with machine-learning heuristics, the paper contends that the more symmetric distribution of knowledge in machine learning could lead to a more transparent production process if proper policies are in place. (shrink)

Although computation and the science of physical systems would appear to be unrelated, there are a number of ways in which computational and physical concepts can be brought together in ways that illuminate both. This volume examines fundamental questions which connect scholars from both disciplines: is the universe a computer? Can a universal computing machine simulate every physical process? What is the source of the computational power of quantum computers? Are computational approaches to solving physical problems and paradoxes always (...) fruitful? Contributors from multiple perspectives reflecting the diversity of thought regarding these interconnections address many of the most important developments and debates within this exciting area of research. Both a reference to the state of the art and a valuable and accessible entry to interdisciplinary work, the volume will interest researchers and students working in physics, computer science, and philosophy of science and mathematics. (shrink)

Any physical explanation of consciousness seems to leave unresolved the ‘explanatory gap': Isn't it conceivable that all the elements in that explanation could occur, with the same information processing outcomes as in a conscious process, but in the absence of consciousness? E.g. any digital computational process could occur in the absence of consciousness. To resolve this dilemma, we propose a biological-process-oriented physiological- phenomenological characterization of consciousness that addresses three ‘paradoxical’ qualities seemingly incompatible with the empirical realm: The dual location (...) of phenomenal properties ‘out there’ yet ‘in here’ in consciousness; the mysterious ‘thickness’ of the specious present; the feeling of ‘free agency', that we can voluntarily direct our actions, including the act of conscious attention, while at the same time attention and the emotions that direct it seem responsive to physiological substrates with physical causes. These paradoxes are then resolved by relating three elements of consciousness: organismically interested anticipation; sensory and proprioceptive imagery generated by the interested anticipation rather than by sensory input; resonating of these activities with activity stimulated by sensory data, where the interested anticipation precedes the processing of the input. Each of these elements is bridged to physiological processes such that, if they occur in a certain relation to each other, we can understand why they would inevitably be accompanied by the corresponding elements of conscious experience. (shrink)

Scriven’s paradox of predictability arises from the combination of two ideas: first, that everything in a deterministic universe is, in principle, predictable; second, that it is possible to create a system that falsifies any prediction that is made of it. Recently, the paradox has been used by Rummens and Cuypers to argue that there is a fundamental difference between embedded and external predictors; and by Ismael to argue against a governing conception of laws. The present paper defends a new diagnosis (...) of the roots of the paradox. First, it is argued that the unpredictability has to be understood in the light of Turing’s famous results about computability, in particular his proof that there is no solution to the ‘halting problem.’ This allows us to see that previous analyses of the paradox were either mistaken or not fully adequate. Second, the sense of paradox that nevertheless remains is traced to the idea that rational behaviour is not dependent on contingent environmental circumstances: that it is always up to us to engage in activities such as rational prediction or rational belief. The paradox of predictability teaches us that this idea, natural though it may be, is mistaken. (shrink)

This book provides a general survey of the main concepts, questions and results that have been developed in the recent interactions between quantum information, quantum computation and logic. Divided into 10 chapters, the books starts with an introduction of the main concepts of the quantum-theoretic formalism used in quantum information. It then gives a synthetic presentation of the main “mathematical characters” of the quantum computational game: qubits, quregisters, mixtures of quregisters, quantum logical gates. Next, the book investigates the puzzling (...) entanglement-phenomena and logically analyses the Einstein–Podolsky–Rosen paradox and introduces the reader to quantum computational logics, and new forms of quantum logic. The middle chapters investigate the possibility of a quantum computational semantics for a language that can express sentences like “Alice knows that everybody knows that she is pretty”, explore the mathematical concept of quantum Turing machine, and illustrate some characteristic examples that arise in the framework of musical languages. The book concludes with an analysis of recent discussions, and contains a Mathematical Appendix which is a survey of the definitions of all main mathematical concepts used in the book. (shrink)

The authors investigated the ontological argument computationally. The premises and conclusion of the argument are represented in the syntax understood by the automated reasoning engine PROVER9. Using the logic of definite descriptions, the authors developed a valid representation of the argument that required three non-logical premises. PROVER9, however, discovered a simpler valid argument for God's existence from a single non-logical premise. Reducing the argument to one non-logical premise brings the investigation of the soundness of the argument into better focus. Also, (...) the simpler representation of the argument brings out clearly how the ontological argument constitutes an early example of a ?diagonal argument? and, moreover, one used to establish a positive conclusion rather than a paradox. (shrink)

This book will examine paradoxes in diverse areas of thought: philosophy, mathematics, physics, economics, political science, psychology, computer science, logic, statistics, linguistics, law, etc. Though the treatment of each paradox is rigorous, the book will be written accessibly with a lighthearted and humorous tone so as to keep the reader engaged. Each chapter will focus on a single paradox, structured roughly like so: 1. A question is asked in the context of a story. As an answer, the paradox is presented (...) (which often results in an aha moment). The historical background of the paradox is recounted. 2. The dénouement explains how the paradox is resolved or why there is no resolution. 3. The chapter ends with further remarks, usually contemporary real-world examples or applications of said paradox. Some examples of the paradoxes covered are the Axiom of Choice (Mathematics), Monty Hall Problem (Statistics), Morgenbesser's Paradox (Linguistics), Tea Leaves Paradox (Physics), The Ultimatum Game (Economics), and The Chicken or Egg Question (Evolution). (shrink)

A proof-theoretic test for paradoxicality was famously proposed by Tennant: a paradox must yield a closed derivation of absurdity with no normal form. Drawing on the remark that all derivations of a given proposition can be transformed into derivations in normal form of a logically equivalent proposition, we investigate the possibility of paradoxes in normal form. We compare paradoxes à la Tennant and paradoxes in normal form from the viewpoint of the computational interpretation of proofs and from the viewpoint of (...) proof-theoretic semantics. (shrink)

Philosophy and Computing explores each of the following areas of technology: the digital revolution; the computer; the Internet and the Web; CD-ROMs and Mulitmedia; databases, textbases, and hypertexts; Artificial Intelligence; the future of computing. Luciano Floridi shows us how the relationship between philosophy and computing provokes a wide range of philosophical questions: is there a philosophy of information? What can be achieved by a classic computer? How can we define complexity? What are the limits of quantam computers? Is the Internet (...) an intellectual space or a polluted environment? What is the paradox in the Strong Artificial Intelligence program? Philosophy and Computing is essential reading for anyone wishing to fully understand both the development and history of information and communication technology as well as the philosophical issues it ultimately raises. (shrink)

The authors investigated the ontological argument computationally. The premises and conclusion of the argument are represented in the syntax understood by the automated reasoning engine PROVER9. Using the logic of definite descriptions, the authors developed a valid representation of the argument that required three non-logical premises. PROVER9, however, discovered a simpler valid argument for God's existence from a single non-logical premise. Reducing the argument to one non-logical premise brings the investigation of the soundness of the argument into better focus. Also, (...) the simpler representation of the argument brings out clearly how the ontological argument constitutes an early example of a ‘diagonal argument’ and, moreover, one used to establish a positive conclusion rather than a paradox. (shrink)

The purpose of this study is to investigate the relationship between a firm’s strategy and consumers’ decisions in the presence of the paradox of choice and sharing personal information. The paradox of choice implies that having too many choices does not necessarily ensure happiness and sometimes having less is more. A new model is constructed introducing a factor of information sharing into the model of a previous study that embedded the paradox of choice only (Kinjo and Ebina in AI Soc (...) 30(2):291–297, 2015). A key feature of the model is its disutility function. It is demonstrated that if the sign of the cross derivative of the function is positive (negative) at the optimum, there is a positive (negative) correlation between the degree of sharing personal information chosen by the consumers and the number of products offered by the firm in its recommendation systems. It is also numerically indicated that the profit function of the firm becomes convex or concave depending on the shape of the disutility function. These results suggest that firms should carefully investigate the shape of the disutility function, under the paradox of choice and sharing personal information. (shrink)

Reasonning in naive set theory (with unlimited comprehension), we derive a paradox (a formal contradiction) which can be seen as a variant of the Burali-Forti paradox.

In a world plagued by disagreement and conflict one might expect that the exact sciences of logic and mathematics would provide a safe harbor. In fact these disciplines are rife with internal divisions between different, often incompatible, systems. Do these disagreements admit of resolution? Can such resolution be achieved without disturbing assumptions that the theorems of logic and mathematics state objective truths about the real world? In this original and historically rich book John Woods explores apparently intractable disagreements in logic (...) and the foundations of mathematics and sets out conflict resolution strategies that evade or disarm these stalemates. An important sub-theme of the book is the extent to which pluralism in logic and the philosophy of mathematics undermines realist assumptions. This book makes an important contribution to such areas of philosophy as logic, philosophy of language and argumentation theory. It will also be of interest to mathematicians and computer scientists. (shrink)

Accelerating Turing machines have attracted much attention in the last decade or so. They have been described as “the work-horse of hypercomputation” (Potgieter and Rosinger 2010: 853). But do they really compute beyond the “Turing limit”—e.g., compute the halting function? We argue that the answer depends on what you mean by an accelerating Turing machine, on what you mean by computation, and even on what you mean by a Turing machine. We show first that in the current literature the (...) term “accelerating Turing machine” is used to refer to two very different species of accelerating machine, which we call end-stage-in and end-stage-out machines, respectively. We argue that end-stage-in accelerating machines are not Turing machines at all. We then present two differing conceptions of computation, the internal and the external, and introduce the notion of an epistemic embedding of a computation. We argue that no accelerating Turing machine computes the halting function in the internal sense. Finally, we distinguish between two very different conceptions of the Turing machine, the purist conception and the realist conception; and we argue that Turing himself was no subscriber to the purist conception. We conclude that under the realist conception, but not under the purist conception, an accelerating Turing machine is able to compute the halting function in the external sense. We adopt a relatively informal approach throughout, since we take the key issues to be philosophical rather than mathematical. (shrink)

Two critical questions spun the web of the Turing test debate. First, can an appropriately programmed machine pass the Turing test? Second, is passing the test by such a machine, ipso facto, considered proof that it is intelligent and hence “minded”? While the first question is technological, the second is purely philosophical. Focusing on the second question, this article interrogates the implication of John Searle’s Chinese room denial of machine intelligence. The thrust of Searle’s argument is that a machine lacks (...) intentionality, so it can only simulate intelligence, not duplicate it. In his thinking, whatever a machine inputs to generate an intelligent output has no bearing on humanlike intelligence. Incidentally, Searle did not classify such a machine’s output as simulated and non-intelligent, nor did he explain how this output is actualised with mere simulation. The connection between “unintelligent machine’s input” and “intelligent machine’s output” is at this point shrouded in mystery. Consequently, the more Searle attempts a denial of machine intelligence in the Chinese room, the more he mystifies it. Using the method of critical analysis, this article advances three fundamental arguments to prove a machines’ obscurity in the Chinese room thought experiment. On the ground of Searle’s conviction, the first argument queries the absurdity in bypassing intentionality to produce intelligence; the second points out the obfuscation in generating intelligence with mere computation, and the third draws attention to the dilemma of classifying a machine’s output either as real-life intelligent behaviour or simulated intelligent behaviour. (shrink)

This chapter presents a new semantics for inductive empirical knowledge. The epistemic agent is represented concretely as a learner who processes new inputs through time and who forms new beliefs from those inputs by means of a concrete, computable learning program. The agent’s belief state is represented hyper-intensionally as a set of time-indexed sentences. Knowledge is interpreted as avoidance of error in the limit and as having converged to true belief from the present time onward. Familiar topics are re-examined within (...) the semantics, such as inductive skepticism, the logic of discovery, Duhem’s problem, the articulation of theories by auxiliary hypotheses, the role of serendipity in scientific knowledge, Fitch’s paradox, deductive closure of knowability, whether one can know inductively that one knows inductively, whether one can know inductively that one does not know inductively, and whether expert instruction can spread common inductive knowledge—as opposed to mere, true belief—through a community of gullible pupils. (shrink)

The theory of consciousness proposed by O'Brien & Opie is open to the Sorites paradox, for it defines a consciousness system internally in terms of computationally relevant units which add up to consciousness only if sufficient in number. The Sorites effect applies on the assumed level of features.

The paper shows how ideas that explain the sense of an expression as a method or algorithm for finding its reference, preshadowed in Frege’s dictum that sense is the way in which a referent is given, can be formalized on the basis of the ideas in Thomason (1980). To this end, the function that sends propositions to truth values or sets of possible worlds in Thomason (1980) must be replaced by a relation and the meaning postulates governing the behaviour of (...) this relation must be given in the form of a logic program. The resulting system does not only throw light on the properties of sense and their relation to computation, but also shows circular behaviour if some ingredients of the Liar Paradox are added. The connection is natural, as algorithms can be inherently circular and the Liar is explained as expressing one of those. Many ideas in the present paper are closely related to those in Moschovakis (1994), but receive a considerably lighter formalization. (shrink)

There is no consensus as to whether a Liar sentence is meaningful or not. Still, a widespread conviction with respect to Liar sentences (and other ungrounded sentences) is that, whether or not they are meaningful, they are useless . The philosophical contribution of this paper is to put this conviction into question. Using the framework of assertoric semantics , which is a semantic valuation method for languages of self-referential truth that has been developed by the author, we show that certain (...) computational problems, called query structures , can be solved more efficiently by an agent who has self-referential resources (amongst which are Liar sentences) than by an agent who has only classical resources; we establish the computational power of self-referential truth . The paper concludes with some thoughts on the implications of the established result for deflationary accounts of truth. (shrink)

Bennett and Schumacher’s postselected quantum teleportation is a model of closed timelike curves (CTCs) that leads to results physically different from Deutsch’s model. We show that even a single qubit passing through a postselected CTC (P-CTC) is sufficient to do any postselected quantum measurement with certainty, and we discuss an important difference between “Deutschian” CTCs (D-CTCs) and P-CTCs in which the future existence of a P-CTC might affect the present outcome of an experiment. Then, based on a suggestion of Bennett (...) and Smith, we explicitly show how a party assisted by P-CTCs can distinguish a set of linearly independent quantum states, and we prove that it is not possible for such a party to distinguish a set of linearly dependent states. The power of P-CTCs is thus weaker than that of D-CTCs because the Holevo bound still applies to circuits using them, regardless of their ability to conspire in violating the uncertainty principle. We then discuss how different notions of a quantum mixture that are indistinguishable in linear quantum mechanics lead to dramatically differing conclusions in a nonlinear quantum mechanics involving P-CTCs. Finally, we give explicit circuit constructions that can efficiently factor integers, efficiently solve any decision problem in the intersection of NP and coNP, and probabilistically solve any decision problem in NP. These circuits accomplish these tasks with just one qubit traveling back in time, and they exploit the ability of postselected closed timelike curves to create grandfather paradoxes for invalid answers. (shrink)

Computational philosophy (CP) aims at investigating many important concepts and problems of the philosophical and epistemological tradition in a new way by taking advantage of information-theoretic, cognitive, and artificial intelligence methodologies. I maintain that the results of computational philosophy meet the classical requirements of some Peircian pragmatic ambitions. Indeed, more than a 100 years ago, the American philosopher C.S. Peirce, when working on logical and philosophical problems, suggested the concept of pragmatism(pragmaticism, in his own words) as a logical criterion to (...) analyze what words and concepts express through their practical meaning. Many words have been spent on creative processes and reasoning, especially in the case of scientific practices. In fact, many philosophers have usually offered a number of ways of construing hypotheses generation, but they aim at demonstrating that the activity of generating hypotheses is paradoxical, obscure, and thus not analyzable. Those descriptions are often so far from Peircian pragmatic prescription and so abstract to result completely unknowable and obscure. To dismiss this tendency and gain interesting insight about the so-called logic of scientific discovery we need to build constructive procedures, which could play a role in moving the problem-solving process forward by implementing them in some actual models. The computational turn gives us a new way to understand creative processes in a strictly pragmatic sense. In fact, by exploiting artificial intelligence and cognitive science tools, computational philosophy allows us to test concepts and ideas previously conceived only in abstract terms. It is in the perspective of these actual computational models that I find the central role of abduction in the explanation of creative reasoning in science. I maintain that the computational philosophy analysis of model-based and manipulative abduction and of external and epistemic mediators is important not only to delineate the actual practice of abduction, but also to further enhance the development of programs computationally adequate in rediscovering, or discovering for the first time, for example, scientific hypotheses or mathematical theorems. The last part of the paper is devoted to illustrating the problem of the extra-theoretical dimension of reasoning and discovery from the perspective of some mathematical cases derived from calculus and geometry. (shrink)

There is significant interest in type-free systems that allow flexible self-application. Such systems are of interest in property theory, natural language semantics, the theory of truth, theoretical computer science, the theory of classes, and category theory. While there are a variety of proposed type-free systems, there is a particularly natural type-free system that we believe is prototypical: the logic of recursive algorithms. Algorithmic logic is the study of basic statements concerning algorithms and the algorithmic rules of inference between such statements. (...) As shown in [1], the threat of paradoxes, such as the Curry paradox, requires care in implementing rules of inference in this context. As in any type-free logic, some traditional rules will fail. The first part of the paper develops a rich collection of inference rules that do not lead to paradox. The second part identifies traditional rules of logic that are paradoxical in algorithmic logic, and so should be viewed with suspicion in type-free logic generally. (shrink)

Philosophical modeling has a long and distinguished history, but the computer offers new and powerful prospects for the creation and manipulation of models. It seems inevitable that the computer will become a major tool in future philosophical research. Here I offer an overview of explorations in philosophical computer modeling that we in the Group for Logic and Formal Semantics at SUNY Stony Brook have undertaken: explorations regarding (1) the potential emergence of cooperation in a society of egoists, (2) self‐reference and (...) paradox in fuzzy logic, (3) a fractal approach to formal systems, and (4) on‐going explorations with models for the emergence of communication. The attempt is not to give a detailed report on any of these but to give a bird s‐eye view of them all, emphasizing the general form that the models take, the philosophical motivations behind them, and the kinds of philosophical results that they may have to offer. (shrink)

Although Fuzzy logic and Fuzzy Mathematics is a widespread subject and there is a vast literature about it, yet the use of Fuzzy issues like Fuzzy sets and Fuzzy numbers was relatively rare in time concept. This could be seen in the Fuzzy time series. In addition, some attempts are done in fuzzing Turing Machines but seemingly there is no need to fuzzy time. Throughout this article, we try to change this picture and show why it is helpful to consider (...) the instants of time as Fuzzy numbers. In physics, though there are revolutionary ideas on the time concept like B theories in contrast to A theory also about central concepts like space, momentum… it is a long time that these concepts are changed, but time is considered classically in all well-known and established physics theories. Seemingly, we stick to the classical time concept in all fields of science and we have a vast inertia to change it. Our goal in this article is to provide some bases why it is rational and reasonable to change and modify this picture. Here, the central point is the modified version of “Unexpected Hanging” paradox as it is described in "Is classical Mathematics appropriate for theory of Computation".This modified version leads us to a contradiction and based on that it is presented there why some problems in Theory of Computation are not solved yet. To resolve the difficulties arising there, we have two choices. Either “choosing” a new type of Logic like “Para-consistent Logic” to tolerate contradiction or changing and improving the time concept and consequently to modify the “Turing Computational Model”. Throughout this paper, we select the second way for benefiting from saving some aspects of Classical Logic. In chapter 2, by applying quantum Mechanics and Schrodinger equation we compute the associated fuzzy number to time. These, provides a new interpretation of Quantum Mechanics.More exactly what we see here is "Particle-Fuzzy time" interpretation of quantum Mechanics, in contrast to some other interpretations of Quantum Mechanics like " Wave-Particle" interpretation. At the end, we propound a question about the possible solution of a paradox in Physics, the contradiction between General Relativity and Quantum Mechanics. (shrink)

In this paper we show how Dummett-Prawitz-style proof-theoretic semantics has to be modified in order to cope with paradoxical phenomena. It will turn out that one of its basic tenets has to be given up, namely the definition of the correctness of an inference as validity preservation. As a result, the notions of an argument being valid and of an argument being constituted by correct inference rules will no more coincide. The gap between the two notions is accounted for (...) by introducing the distinction between sense and denotation in the proof-theoretic-semantic setting. (shrink)

This book is an introduction, entirely by example, to the possibilities of using computer models as tools in phosophical research in general and in philosophical logic in particular. Topics include chaos, fractals, and the semantics of paradox; epistemic dynamics; fractal images of formal systems; the evolution of generosity; real-valued game theory; and computation and undecidability in the spatialized Prisoner's Dilemma.

Since it was presented in 1963, Chisholm’s paradox has attracted constant attention in the deontic logic literature, but without the emergence of any definitive solution. We claim this is due to its having no single solution. The paradox actually presents many challenges to the formalization of deontic statements, including (1) context sensitivity of unconditional oughts, (2) formalizing conditional oughts, and (3) distinguishing generic from nongeneric oughts. Using the practical interpretation of ‘ought’ as a guideline, we propose a linguistically motivated logical (...) solution to each of these problems, and explain the relation of the solution to the problem of contrary-to-duty obligations. (shrink)

Most discussions frame the Liar Paradox as a formal logical-linguistic puzzle. Attempts to resolve the paradox have focused very little so far on aspects of cognitive psychology and processing, because semantic and cognitive-psychological issues are generally assumed to be disjunct. I provide a motivation and carry out a cognitive-computational treatment of the liar paradox based on a cognitive-computational model of language and conceptual knowledge within the Predictive Processing framework. I suggest that the paradox arises as a failure of synchronization between (...) two ways of generating the liar situation in two different PP sub-models, one corresponding to language processing and the other to the processing of meaning and world-knowledge. In this way, I put forward the claim that the liar sentence is meaningless but has an air of meaningfulness. I address the possible objection that the proposal violates the Principle of Unrestricted Compositionality, which purportedly regulates the conceptual competence of thinkers. (shrink)

The paper attempts to give a solution to the Fitch's paradox though the strategy of the reformulation of the paradox in temporal logic, and a notion of knowledge which is a kind of ceteris paribus modality. An analogous solution has been offered in a different context to solve the problem of metaphysical determinism.

Most discussions frame the Liar Paradox as a formal logical-linguistic puzzle. Attempts to resolve the paradox have focused very little so far on aspects of cognitive psychology and processing, because semantic and cognitive-psychological issues are generally assumed to be disjunct. I provide a motivation and carry out a cognitive-computational treatment of the liar paradox based on a model of language and conceptual knowledge within thePredictive Processing(PP) framework. I suggest that the paradox arises as a failure of synchronization between two ways (...) of generating the liar situation in two different (idealized) PP sub-models, one corresponding to language processing and the other to the processing of meaning and world-knowledge. In this way, I put forward the claim that the liar sentence is meaningless but has an air of meaningfulness. I address the possible objection that the proposal violates thePrinciple of Unrestricted Compositionality, which purportedly regulates the conceptual competence of thinkers. (shrink)

Most discussions frame the Liar Paradox as a formal logical-linguistic puzzle. Attempts to resolve the paradox have focused very little so far on aspects of cognitive psychology and processing, because semantic and cognitive-psychological issues are generally assumed to be disjunct. I provide a motivation and carry out a cognitive-computational treatment of the liar paradox based on a model of language and conceptual knowledge within the Predictive Processing framework. I suggest that the paradox arises as a failure of synchronization between two (...) ways of generating the liar situation in two different PP sub-models, one corresponding to language processing and the other to the processing of meaning and world-knowledge. In this way, I put forward the claim that the liar sentence is meaningless but has an air of meaningfulness. I address the possible objection that the proposal violates the Principle of Unrestricted Compositionality, which purportedly regulates the conceptual competence of thinkers. (shrink)

Fitch’s paradox shows, from fairly innocent-looking assumptions, that if there are any unknown truths, then there are unknowable truths. This is generally thought to deliver a blow to antirealist positions that imply that all truths are knowable. The present paper argues that a probabilistic version of antirealism escapes Fitch’s result while still offering all that antirealists should care for.

Hardy’s non-locality paradox is a proof without inequalities showing that certain non-local correlations violate local realism. It is ‘possibilistic’ in the sense that one only distinguishes between possible outcomes (positive probability) and impossible outcomes (zero probability). Here we show that Hardy’s paradox is quite universal: in any (2,2,l) or (2,k,2) Bell scenario, the occurrence of Hardy’s paradox is a necessary and sufficient condition for possibilistic non-locality. In particular, it subsumes all ladder paradoxes. This universality of Hardy’s paradox is not true (...) more generally: we find a new ‘proof without inequalities’ in the (2,3,3) scenario that can witness non-locality even for correlations that do not display the Hardy paradox. We discuss the ramifications of our results for the computational complexity of recognising possibilistic non-locality. (shrink)

Three systems of double extension set theory have been proposed by Andrzej Kisielewicz in two papers. In this paper, it is shown that the two stronger systems are inconsistent, and that the third, weakest system does not admit extensionality for general sets or the use of general sets as parameters in its comprehension scheme. The parameter-free version of the comprehension principle of double extension set theory is also shown to be inconsistent with extensionality. The definitions of the systems and a (...) self-contained exposition of their properties is given, sufficient to develop the inconsistency proofs. (shrink)

If not a paradox, consciousness is at least an enigma. Many believe consciousness is hard to have, whereas others are panpsychists. Many hold that consciousness is hard to understand, perhaps impossibly so, whereas others believe we already have available an adequate general understanding of consciousness. Rocco Gennaro belongs to the second camp, and in this important work he explains why.In The Paradox of Consciousness, Gennaro develops and defends a higher-order thought (HOT) theory of consciousness. A HOT theory is an alternative (...) to theories that consciousness depends on a “global workplace” that integrates and processes data from multiple sense modalities, or that what makes a conscious first-order mental state is what the state represents (“representationalism”). Any of these causal/computational theories has the great advantage over, say, quantum approaches, of providing a large-scale blueprint for actually designing a system that, if the theory is correct, would be conscious. T .. (shrink)

Although Fuzzy logic and Fuzzy Mathematics is a widespread subject and there is a vast literature about it, yet the use of Fuzzy issues like Fuzzy sets and Fuzzy numbers was relatively rare in time concept. This could be seen in the Fuzzy time series. In addition, some attempts are done in fuzzing Turing Machines but seemingly there is no need to fuzzy time. Throughout this article, we try to change this picture and show why it is helpful to consider (...) the instants of time as Fuzzy numbers. In physics, though there are revolutionary ideas on the time concept like B theories in contrast to A theory also about central concepts like space, momentum… it is a long time that these concepts are changed, but time is considered classically in all well-known and established physics theories. Seemingly, we stick to the classical time concept in all fields of science and we have a vast inertia to change it. Our goal in this article is to provide some bases why it is rational and reasonable to change and modify this picture. Here, the central point is the modified version of “Unexpected Hanging” paradox as it is described in "Is classical Mathematics appropriate for theory of Computation".This modified version leads us to a contradiction and based on that it is presented there why some problems in Theory of Computation are not solved yet. To resolve the difficulties arising there, we have two choices. Either “choosing” a new type of Logic like “Para-consistent Logic” to tolerate contradiction or changing and improving the time concept and consequently to modify the “Turing Computational Model”. Throughout this paper, we select the second way for benefiting from saving some aspects of Classical Logic. In chapter 2, by applying quantum Mechanics and Schrodinger equation we compute the associated fuzzy number to time. These, provides a new interpretation of Quantum Mechanics.More exactly what we see here is "Particle-Fuzzy time" interpretation of quantum Mechanics, in contrast to some other interpretations of Quantum Mechanics like " Wave-Particle" interpretation. At the end, we propound a question about the possible solution of a paradox in Physics, the contradiction between General Relativity and Quantum Mechanics. (shrink)

Ensemble perception—the encoding of objects by their group properties—is known to be resistant to outlier noise. However, this resistance is somewhat paradoxical: how can the visual system determine which stimuli are outliers without already having derived statistical properties of the ensemble? A simple solution would be that ensemble perception is not a simple, one-step process; instead, outliers are detected through iterative computations that identify items with high deviance from the mean and reduce their weight in the representation over time. (...) Here we tested this hypothesis. In Experiment 1, we found evidence that outliers are discounted from mean orientation judgments, extending previous results from ensemble face perception. In Experiment 2, we tested the timing of outlier rejection by having participants perform speeded judgments of sets with or without outliers. We observed significant increases in reaction time (RT) when outliers were present, but a decrease compared to no-outlier sets of matched range suggesting that range alone did not drive RTs. In Experiment 3 we tested the timing by which outlier noise reduces over time. We presented sets for variable exposure durations and found that noise decreases linearly over time. Altogether these results suggest that ensemble representations are optimized through iterative computations aimed at reducing noise. The finding that ensemble perception is an iterative process provides a useful framework for understanding contextual effects on ensemble perception. (shrink)

This paper brings the intellectual tools of cognitive science to bear on resolving the “paradox of the active user” [Interfacing Thought: Cognitive Aspects of Human–Computer Interaction, Cambridge, MIT Press, MA, USA]—the persistent use of inefficient procedures in interactive tasks by experienced or even expert users when demonstrably more efficient procedures exist. The goal of this paper is to understand the roots of this paradox by finding regularities in these inefficient procedures. We examine three very different data sets. For each data (...) set, we first satisfy ourselves that the preferred procedures used by some subjects are indeed less efficient than the recommended procedures. We then amass evidence, for each set, and conclude that when a preferred procedure is used instead of a more efficient, recommended procedure, the preferred procedure tends to have two major characteristics: (1) the preferred procedure is a well‐practiced, generic procedure that is applicable either within the same task environment in different contexts or across different task environments, and (2) the preferred procedure is composed of interactive components that bring fast, incremental feedback on the external problem states. The support amassed for these characteristics leads to a new understanding of the paradox. In interactive tasks, people are biased towards the use of general procedures that start with interactive actions. These actions require much less cognitive effort as each action results in an immediate change to the external display that, in turn, cues the next action. Unfortunately for the users, the bias to use interactive unit tasks leads to a path that requires more effort in the long run. Our data suggest that interactive behavior is composed of a series of distributed choices; that is, people seldom make a once‐and‐for‐all decision on procedures. This series of biased selection of interactive unit tasks often leads to a stable suboptimal level of performance. (shrink)

Adam Brandenburger and H. Jerome Keisler have recently discovered a two person Russell-style paradox. They show that the following configurations of beliefs is impossible: Ann believes that Bob assumes that Ann believes that Bob’s assumption is wrong. In [7] a modal logic interpretation of this paradox is proposed. The idea is to introduce two modal operators intended to represent the agents’ beliefs and assumptions. The goal of this paper is to take this analysis further and study this paradox from the (...) point of view of a modal logician. In particular, we show that the paradox can be seen as a theorem of an appropriate hybrid logic. (shrink)

Kripke (1982, Wittgenstein on rules and private language. Cambridge, MA: MIT Press) presents a rule-following paradox in terms of what we meant by our past use of “plus”, but the same paradox can be applied to any other term in natural language. Many responses to the paradox concentrate on fixing determinate meaning for “plus”, or for a small class of other natural language terms. This raises a problem: how can these particular responses be generalised to the whole of natural language? (...) In this paper, I propose a solution. I argue that if natural language is computable in a sense defined below, and the Church–Turing thesis is accepted, then this auxiliary problem can be solved. (shrink)