An Operational Logic of Proofs with Positive and Negative Information

The logic of proofs was introduced by Artemov in order to analize the formalization of the concept of proof rather than the concept of provability. In this context, some operations on proofs play a very important role. In this paper, we investigate some very natural operations, paying attention not only to positive information, but also to negative information (i.e. information saying that something cannot be a proof). We give a formalization for a fragment of such a logic of proofs, and we prove that our fragment is complete and decidable.     

Predicate Logics on Display

The paper provides a uniform Gentzen-style proof-theoretic framework for various subsystems of classical predicate logic. In particular, predicate logics obtained by adopting van Behthem's modal perspective on first-order logic are considered. The Gentzen systems for these logics augment Belnap's display logic by introduction rules for the existential and the universal quantifier. These rules for x and x are analogous to the display introduction rules for the modal operators and and do not themselves allow the Barcan formula or its converse to be derived. En route from the minimal modal predicate logic to full first-order logic, axiomatic extensions are captured by purely structural sequent rules.     

Logic and Philosophy of Logic in Wittgenstein

This essay discusses Wittgenstein's conception of logic, early and late, and some of the types of logical system that he constructed. The essay shows that the common view according to which Wittgenstein had stopped engaging in logic as a philosophical discipline by the time of writing Philosophical Investigations is mistaken. It is argued that, on the contrary, logic continued to figure at the very heart of later Wittgenstein's philosophy; and that Wittgenstein's mature philosophy of logic contains many interesting thoughts that have gone widely unnoticed.     

论八卦是八个逻辑范式

基于太极代数,本文证明八卦是八个逻辑范式,八卦中包含四对矛盾关系,其中"六子"构成辩证逻辑组。八卦是生命生产和思想生产都必须共同遵循的变化法则。学界似有这样的倾向,以为《周易》中只有类推逻辑而没有演绎逻辑,本文证明这种观点是不能成立的。八卦本质上就是演绎逻辑的,卦象的本质是逻辑法则。因此,基于卦象的联想或推理不能脱离八卦的逻辑内涵;否则,想象的灵活性必将导致卦象上的混淆,甚至使八卦沦为象数游戏的工具。     

Philosophy,Logic, Science,History

Analytic philosophy is sometimes said to have particularly close connections to logic and to science, and no particularly interesting or close relation to its own history. It is argued here that although the connections to logic and science have been important in the development of analytic philosophy, these connections do not come close to characterizing the nature of analytic philosophy, either as a body of doctrines or as a philosophical method. We will do better to understand analytic philosophy—and its relationship to continental philosophy—if we see it as a historically constructed collection of texts, which define its key problems and concerns. It is true, however, that analytic philosophy has paid little attention to the history of the subject. This is both its strength—since it allows for a distinctive kind of creativity—and its weakness—since ignoring history can encourage a philosophical variety of “normal science.”     

Cognitive technology to capture deep computational concepts with combinators

Computational activity is now recognized as a natural science, and computational and information processes have been discovered in the deep structures of many areas. Computations in the natural world were present long before the invention of computers, but a remarkable shift in understanding its fundamental nature occurs, in fact, before our eyes. The present moment, in fact, is a transition from the concept of computer science as an artificial science to the understanding that information processes are abundant in nature. Computing is recognized as a natural science that studies natural and artificial information processes.In everyday computing, operations are performed on the individual generators, with little attention paid to their internal structure. However, many common operations consist of more primitive constructions connected by a combination mode. The interaction of information processes and corresponding structures is carried out in an environment of “applicative interaction”, their applications to each other, and the study of the properties of this environment allows us to understand the nature of the computations.In the present work, the main attention is paid to elucidating the technological features of computations with individual generators, or objects. Their interaction is considered in an applicative environment, which allows us to elucidate the internal structure of ordinary operations, the knowledge of which allows us to understand their properties. The choice of initial constant generators, considered as generic ones and expressed by combinators, is discussed. These initial generators are used as the main “building blocks” that occur within the larger blocks of the applicative environment in interaction with each other. As a result of the interaction, constructions arise that give representative sets of ordinary operators and embedded computing systems.     

Sometimes it does hurt to ask: The constructive role of articulating impressions

Decisions can sometimes have a constructive role, so that the act of, for example, choosing one option over another creates a preference for that option (e.g., , ,  and ). In this work we explore the constructive role of just articulating an impression, for a presented visual stimulus, as opposed to making a choice (specifically, the judgments we employ are affective evaluations). Using quantum probability theory, we outline a cognitive model formalizing such a constructive process. We predict a simple interaction, in relation to how a second image is evaluated, following the presentation of a first image, depending on whether there is a rating for the first image or not. The interaction predicted by the quantum model was confirmed across three experiments and a variety of control manipulations. The advantages of using quantum probability theory to model the present results, compared with existing models of sequence order effects in judgment (e.g., Hogarth & Einhorn, 1992) or other theories of constructive processes when a choice is made (e.g.,  and ) are discussed.     

From Copenhagen to the consulting room: Pauli and Jung in Copenhagen

In this paper the city of Copenhagen is used as a starting point to highlight some critical historical events, both concerning the exchange of ideas between the physicist Wolfgang Pauli and C.G. Jung and the history of psychotherapy in Scandinavia. Pauli's years in Copenhagen under the influence of Niels Bohr and his philosophy prepared him mentally to receiveC.G. Jung's ideas. The paper also recounts the one occasion that Jung was in Scandinavia, attending the 9th conference of the General Medical Society for Psychotherapy in Copenhagen, just before going to New York in 1937 to give his final seminars on Pauli's dreams. The reason for focusing on these particular events is that they also constitute part of the history of the reception of psychodynamic psychology in Sweden and Scandinavia.     

Neuroscience,quantum indeterminism and the Cartesian soul

Quantum indeterminism is frequently invoked as a solution to the problem of how a disembodied soul might interact with the brain (as Descartes proposed), and is sometimes invoked in theories of libertarian free will even when they do not involve dualistic assumptions. Taking as example the Eccles–Beck model of interaction between self (or soul) and brain at the level of synaptic exocytosis, I here evaluate the plausibility of these approaches. I conclude that Heisenbergian uncertainty is too small to affect synaptic function, and that amplification by chaos or by other means does not provide a solution to this problem. Furthermore, even if Heisenbergian effects did modify brain functioning, the changes would be swamped by those due to thermal noise. Cells and neural circuits have powerful noise-resistance mechanisms, that are adequate protection against thermal noise and must therefore be more than sufficient to buffer against Heisenbergian effects. Other forms of quantum indeterminism must be considered, because these can be much greater than Heisenbergian uncertainty, but these have not so far been shown to play a role in the brain.     

Empiricism,probability, and knowledge of arithmetic: A preliminary defense

The topic of this paper is our knowledge of the natural numbers, and in particular, our knowledge of the basic axioms for the natural numbers, namely the Peano axioms. The thesis defended in this paper is that knowledge of these axioms may be gained by recourse to judgments of probability. While considerations of probability have come to the forefront in recent epistemology, it seems safe to say that the thesis defended here is heterodox from the vantage point of traditional philosophy of mathematics. So this paper focuses on providing a preliminary defense of this thesis, in that it focuses on responding to several objections. Some of these objections are from the classical literature, such as Frege's concern about indiscernibility and circularity (Section 2.1), while other are more recent, such as Baker's concern about the unreliability of small samplings in the setting of arithmetic (Section 2.2). Another family of objections suggests that we simply do not have access to probability assignments in the setting of arithmetic, either due to issues related to the ω-rule (Section 3.1) or to the non-computability and non-continuity of probability assignments (Section 3.2). Articulating these objections and the responses to them involves developing some non-trivial results on probability assignments (Appendix A–Appendix C), such as a forcing argument to establish the existence of continuous probability assignments that may be computably approximated (Theorem 4 Appendix B). In the concluding section, two problems for future work are discussed: developing the source of arithmetical confirmation and responding to the probabilistic liar.