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Education is what is left after one has forgotten everything he learned in school

Albert Einstein

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this is a test to post from workflow

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joking

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text

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test

test

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Zen aims at a perfection of personhood. To this end, sitting meditation called “za-zen” is employed as a foundational method of prāxis across the different schools of this Buddha-Way, through which the Zen practitioner attempts to embody non-discriminatory wisdom vis-à-vis the meditational experience known as “satori” (enlightenment). A process of discovering wisdom culminates in the experiential dimension in which the equality of thing-events is apprehended in discerning them. The most distinguishing feature of this school of the Buddha-Way is seen in its contention that wisdom, accompanied by compassion, is expressed in the everyday “life-world” when associating with one’s self, people, and nature. The everyday “life-world” for most people is an evanescent transforming stage in which living is consumed, philosophically speaking, by an either-or, ego-logical, dualistic paradigm of thinking with its attendant psychological states such as stress and anxiety. Zen demands an overcoming of this paradigm by practically achieving an holistic perspective in cognition, so that the Zen practitioner can celebrate, with a stillness of mind, a life of tending toward the concrete thing-events of everyday life and nature. For this reason, the Zen practitioner is required to embody freedom expressive of the original human nature. Generally speaking, Zen cherishes simplicity and straightforwardness in grasping reality and acting on it “here and now,” for it believes that a thing-event that is immediately presencing before one’s eyes or under one’s foot is no other than an expression of suchness, i.e., it is such that it is showing its primordial mode of being. It also understands a specificity of thing-event to be a recapitulation of the whole; parts and the whole are to be lived in an inseparable relationship through an exercise of nondiscriminatory wisdom, without prioritizing the visible over the invisible, the explicit over the implicit, and vice versa. As such, Zen maintains a stance of “not one” and “not two,” i.e., “positionless position,” where “not two” signals a negation of the stance that divides the whole into two parts, i.e., dualism, while “not one” designates a negation of this stance when the Zen practitioner dwells in the whole as one, while suspending judgment in meditation, i.e., non-dualism. Free, bilateral movement between “not one” and “not two” characterizes Zen’s achievement of a personhood with a third perspective that cannot, however, be confined to either dualism or non-dualism (i.e., neither “not one” nor “not two”).

Japanese Zen Buddhist Philosophy (Stanford Encyclopedia of Philosophy)

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Bibliography A comprehensive bibliography on relevant logic was put together by Robert Wolff and can be found in Anderson, Belnap and Dunn 1992. The bibliography in Restall 2000 (see Other Internet Resources) is not as comprehensive as Wolff’s, but it does include material up to the present day. Books on Substructural Logic and Introductions to the Field Anderson, A.R., and Belnap, N.D., 1975, Entailment: The Logic of Relevance and Necessity, Princeton, Princeton University Press, Volume I. Anderson, A.R., Belnap, N.D. Jr., and Dunn, J.M., 1992, Entailment, Volume II, Princeton, Princeton University Press [This book and the previous one summarise the work in relevant logic in the Anderson–Belnap tradition. Some chapters in these books have other authors, such as Robert K. Meyer and Alasdair Urquhart.] Dunn, J. M. and Restall, G., 2000, “Relevance Logic” in F. Guenthner and D. Gabbay (eds.), Handbook of Philosophical Logic second edition; Volume 6, Kluwer, pp 1–136. [A summary of work in relevant logic in the Anderson–Belnap tradition.] Mares, Edwin D., 2004, Relevant Logic: a philosophical interpretation Cambridge University Press. [An introduction to relevant logic, proposing an information theoretic understanding of the ternary relational semantics.] Moortgat, Michael, 1988, Categorial Investigations: Logical Aspects of the Lambek Calculus Foris, Dordrecht. [Another introduction to the Lambek calculus.] Morrill, Glyn, 1994, Type Logical Grammar: Categorial Logic of Signs Kluwer, Dordrecht [An introduction to the Lambek calculus.] Paoli, Francesco, 2002, Substructural Logics: A Primer Kluwer, Dordrecht [A general introduction to substructural logics.] Read, S., 1988, Relevant Logic, Oxford: Blackwell. [An introduction to relevant logic motivated by considerations in the theory of meaning. Develops a Lemmon-style proof theory for the relevant logic R R .] Restall, Greg, 2000, An Introduction to Substructural Logics, Routledge. (online précis) [A general introduction to the field of substructural logics.] Routley, R., Meyer, R.K., Plumwood, V., and Brady, R., 1983, Relevant Logics and their Rivals, Volume I, Atascardero, CA: Ridgeview. [Another distinctive account of relevant logic, this time from an Australian philosophical perspective.] Schroeder-Heister, Peter, and Došen, Kosta, (eds), 1993, Substructural Logics, Oxford University Press. [An edited collection of essays on different topics in substructural logics, from different traditions in the field.] Troestra, Anne, 1992, Lectures on Linear Logic, CSLI Publications [A quick, easy-to-read introduction to Girard’s linear logic.] Other Works Cited Ackermann, Wilhelm, 1956, “Begründung Einer Strengen Implikation,” Journal of Symbolic Logic, 21: 113–128. Gianluigi Bellin, Martin Hyland, Edmund Robinson, and Christian Urban, 2006, “Categorical Proof Theory of Classical Propositional Calculus,” Theoretical Computer Science, 364: 146–165. Church, Alonzo, 1951, “The Weak Theory of Implication,” in Kontroliertes Denken: Untersuchungen zum Logikkalkül und zur Logik der Einzelwissenschaften, A. Menne, A. Wilhelmy and H. Angsil (eds.), Kommissions-Verlag Karl Alber, 22–37. Curry, Haskell B., 1977, Foundations of Mathematical Logic, New York: Dover (originally published in 1963). Dunn, J.M., 1991, “Gaggle Theory: An Abstraction of Galois Connections and Residuation with Applications to Negation and Various Logical Operations,” in Logics in AI, Proceedings European Workshop JELIA 1990 (Lecture notes in Computer Science, Volume 476), Berlin: Springer-Verlag. Dunn, J.M., 1993, “Star and Perp,” Philosophical Perspectives, 7: 331–357. Geach, P. T., 1955, “On Insolubilia,” Analysis, 15: 71–72. Gentzen, Gerhard, 1935, “Untersuchungen über das logische Schließen,” Mathematische Zeitschrift, 39: 176–210 and 405–431. [An English translation is found in Gentzen 1969.] Gentzen, Gerhard, 1969, The Collected Papers of Gerhard Gentzen, M. E. Szabo (ed.), Amsterdam: North Holland, 1969. Girard, Jean-Yves, 1987, “Linear Logic,” Theoretical Computer Science, 50: 1–101. Lambek, Joachim, 1958, “The Mathematics of Sentence Structure,” American Mathematical Monthly, 65: 154–170. Lambek, Joachim, 1961, “On the Calculus of Syntactic Types, ” in Structure of Language and its Mathematical Aspects (Proceedings of Symposia in Applied Mathematics, XII), R. Jakobson (ed.), Providence, RI: American Mathematical Society. Moh Shaw-Kwei, 1950, “The Deduction Theorems and Two New Logical Systems,” Methodos, 2: 56–75. Moortgat, Michael, 1995, “Multimodal Linguistic Inference,” Logic Journal of the IGPL, 3: 371–401.

Substructural Logics (Stanford Encyclopedia of Philosophy)

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Substructural logics are non-classical logics weaker than classical logic, notable for the absence of structural rules present in classical logic. These logics are motivated by considerations from philosophy (relevant logics), linguistics (the Lambek calculus) and computing (linear logic). In addition, techniques from substructural logics are useful in the study of traditional logics such as classical and intuitionistic logic. This article provides a brief overview of the field of substructural logic. For a more detailed introduction, complete with theorems, proofs and examples, the reader can consult the books and articles in the Bibliography.

Substructural Logics (Stanford Encyclopedia of Philosophy)

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logical systems are, in a sense, miscellaneous class or set.

for it seems to me that a system describes an ordinary inference (Minimum logic), other system captures a behaviour of computres (intuitionistic logic) and there are a system which is a moral of our speech acts (classical logic). in addition, there are much more systems which is called 'logical', such as substructural logic, modal logic, andc... How and Why do we consider them as a certain class of system ? i know only a few reasons they share some mathematical properties, for example cut-elimination, and they are studied in a context which has begun with Frege, Gentzen, Goedel, and more.

what i wish is to classify them and to give a united view of them.

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論理体系というものは、ある意味で雑多である。

つまり、その意味が、推論のモデルであったり計算機の記述であったり、言語的規範であったり、そこに統一性がない。 しかしながら (数学的な) 性質を共有し、cut除去とか、その上、FregeやGentzenやGoedelとかから始まる研究脈絡にあるから、 何かしら統一的なものとして、「論理的」と呼ばれる記号処理の体系��あるクラスが存在しているようにぼくには見える。

そうした雑多であるものらを、分類し統一したい。

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Finally, I have found what to be written in my graduate thesis

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Scholarly Sundholm B.G. (2014), Constructive Recursive Functions, Church's Thesis, and Brouwer's Theory of the Creating Subject: Afterthoughts on a Paris Joint Session. In: Dubucs Jaques, Bordeau Michel (Eds.) Constructivity and Computability in Historical and Philosophical Perspective Logic, Epistemology, and the Unity of Science no. 34. Dordrecht: Springer. 1-35. book chapter Atten Mark van & Sundholm Göran (2014), Intuitionistische Logica en het Scheppend Subject, Nieuw Archief voor Wiskunde 15(2): 124-130. article in journal: refereed Atten Mark van, Sundholm Göran , Bordeau Michel & Atten Vanessa van (2014), "Que les principes de la logique ne sont pas fiables": Novelle traduction francaise et commentaire de l'article de 1908 de L. E. J. Brouwer, Studium : tijdschrift voor wetenschaps- en universiteitsgeschiedenis / Revue d'histoire des sciences et des universités 67(2): 257-281. article in journal: refereed Sundholm B.G. (2013), Demonstrations versus Proofs, Being an Afterword to Constructions, Proofs and the Meaning of the Logical Constants. In: M. van der Schaar (Ed.) Judgement and the Epistemic Foundation of Logic Logic, Epistemology, and the Unity of Science no. 31. Dordrecht: Springer Netherlands. 15-22. book chapter Sundholm & B. G. (2013), Containment and Variation; Two Strands in the Development of Analyticity from Aristotle to Martin-Löf. In: Schaar M. van der (Ed.) Judgement and the Epistemic Foundation of Logic Logic, Epistemology, and the Unity of Science no. 31. Dordrecht: Springer Netherlands. 23-35. book chapter Sundholm B.G. (2012), Error, Topoi : An International Review of Philosophy 31(1): 87-92. article in journal: refereed Sundholm B.G. (2012), “Inference versus consequence” revisited: inference, consequence, conditional, implication, Synthese 187(3): 943-956. article in journal: refereed Dybjer P., Lindström S., Palmgren E. & Sundholm G. (eds.) (2012), Epistemology versus Ontology Essays on the Philosophy and Foundations of Mathematics in Honour of Per Martin-Löf Logic, Epistemology, and the Unity of Science no. 27. Dordrecht: Springer Netherlands. book Sundholm B.G. (2012), On the Philosophical Work of Per Martin-Löf. In: Dybjer P., Lindström S., Palmgren E., Sundholm G. (Eds.) Epistemology versus Ontology Essays on the Philosophy and Foundations of Mathematics in Honour of Per Martin-Löf Logic, Epistemology, and the Unity of Science no. 27. Dordrecht: Springer Netherlands. xvii-xxiv. book chapter Sundholm B.G. (2011), A Garden of Grounding Trees. In: Cellucci C., Grosholz E., Ippoliti E. (Eds.) Logic and Knowledge. Cambridge: Cambridge Scholars. 53-64. book chapter Sundholm B.G. (2009), A Century of Judgment and Inference: 1837-1936. In: Haaparanta L. (Ed.) The Devlopment of Modern Logic. Oxford: Oxford University Press. 263-317. book chapter Sundholm B.G. & Van Atten Mark (2008), The proper explanation of intuitionistic logic: on Brouwer's demonstartion of the Bar Theorem. In: M. van Atten P., Boldini P., Bourdeau M. (Eds.) One Hundred Years of Intuitionism (1907-2007). Basel: Birkhäuser. 60-77. conference paper: refereed Sundholm B.G. (2008), A novel(?) paradox. In: Dégremont Cédric, Keiff Laurent, Rückert Helge (Eds.) Dialogues, Logics and Other Strange Things. Essays in Honour of Shahid Rahman. London: College Publications (Tributes 7). 375-377. conference paper: refereed Sundholm B.G. (2007), Semantic values of natural deduction derivations, Synthese 148(3): 623-638. article in journal: refereed Sundholm B.G. (2007), A Century of Judgement and Inference: 1836-1937, The History of Modern Logic. article in journal Sundholm B.G. (2004), Antirealism and the Roles of Truth. In: Niniluoto I, Sintonen M (Eds.) Handbook of Epistemology. Dordrecht: Kluwer. 437-466. book chapter Sundholm B.G. (2004), The proof-explanation is logically neutral, Revue internationale de philosophie 58(4): 401-410. article in journal Sundholm B.G. (2004), Heinrich Scholz between Frege and Hilbert. In: Schmidt am Busch H.C., Wehmeier K.F. (Eds.) Heinrich Scholz. Logiker, Philosoph, Theologe, Mentis Verlag: Paderborn. 103-117. book chapter Sundholm B.G. (2003), Tarski and Lesniewski on Languages with Meaning versus Languages without Use: A 60th Birthday Provocation for Jan Wolenski. In: Hintikka J., Czarnecki T., Kijania-Placek K., Placek T., Rojszczak A. (Eds.) Philosophy and Logic. In Search of the polish Tradition. Dordecht: Kluwer. 109-128. conference paper Sundholm B.G. (2003), "Mind your P'ds and Q's". On the proper interpretation of modal logic. In: Logica Yearbook 2002. Czech Academy of Science, Prague: Filosofia Publishers. 101-111. conference paper Bos E.P. & Sundholm B.G. (2002), History of Logic: Medieval. In: Jacquette D. (Ed.) A Companion to Philosophical Logic. Malden (Mass.) - Oxford (UK). 24-34. conference paper: refereed Sundholm B.G. (2002), Gottlob Frege, August Bebel, and the Return of Alsace-Lorraine: on the dating of the distinction between 'Sinn' and 'Bedeutung', History and Philosophy of Logic 22(1). article in journal: refereed Sundholm B.G. (2002), What is an expression?. In: Logica Yearbook 2001. Prague: Filosofia Publishers, Czech Academy of Science. 181-194. conference paper: refereed Sundholm B.G. (2002), A Century of Inference: 1837-1936. In: Gärdenfors P., Wolenski J., Kijania-Placek K. (Eds.) In the Scope of Logic, Methodology and Philosophy of Science. Dordrecht: Kluwer. 565-580. conference paper: refereed Sundholm B.G. (2002), Varieties of Consequence. In: Jacquette D. (Ed.) A Companion to Philosophical Logic. 241-255. conference paper: refereed Sundholm B.G. (2001), A Plea for Logical Atavism. In: The Logica Yearbook 2000. Prague: Filosofia Publishers, Czech Academy of Science. 151-162. conference paper: refereed Sundholm B.G. (2001), Systems of Deduction (Chapter 1:2). In: Gabbay D., Guenthner F. (Eds.) Handbook of Philosophical Logic, Vol. I: Elements of Classical Logic Synthese library : Studies in Epistemology, Logic, Methodology, and Philosophy of Science no. 164. Dordrecht: D. Reidel Publishing Company. 133-188. book chapter Sundholm B.G. (2001), The Proof Theory of Stig Kanger: a personal recollection. In: Holmström-Hintikka G., Lindström S., Sliwinski R. (Eds.) Collected Papers of Stig Kanger with Essays on his Life and Work. Dordrecht: Kluwer. 31-42. conference paper Sundholm B.G. (2000), Proofs as Acts versus Proofs as Objects: Some Questions for Dag Prawitz, Theoria 64: 187-216. article in journal: refereed Sundholm B.G. (2000), When, and Why, did Frege read Bolzano?. In: The Logica Yearbook 1999. Prague: Filosofia Publishers. 164-174. conference paper: refereed Sundholm B.G. (2000), Virtues and Vices of Interpreteted 'Classical' Formalisms: Some Impertinent Questions for Pavel Materna on the occasion of his 70th Birthday. In: Childers T., Palomäki J. (Eds.) Between Worlds and Words. Prague: Filosofia Publishers. 3-12. conference paper Sundholm B.G. (1999), Identity: Propositional, Criterial, Absolute. In: The Logica Yearbook 1998. Prague: Filosofia Publishers, Czech Academy of Science. 20-26. conference paper: refereed Sundholm B.G. (1999), Maccoll on Judgement and Inference, Nordic Journal of Philosophy 3: 119-132. article in journal: refereed Sundholm B.G. (1998), Inference, Consequence, Implication: A Constructivist's Perspective, Philosophia Mathematica, series III 6: 178-194. article in journal: refereed Sundholm B.G. (1998), Inference versus Consequence, The Logica Yearbook: 26-36. article in journal: refereed Sundholm B.G. (1998), Intuitionism and Logical Tolerance, Vienna Circle Institute Yearbook 6: 135-145. article in journal: refereed Sundholm B.G. (1997), Implicit epistemic aspects of constructive logic, Journal of Logic, Language, and Information 6: 191-212. article in journal: refereed Sundholm B.G. (1994), Vestiges of Realism. In: McGuinness B., Oliveri G. (Eds.) The Philosophy of Michael Dummett: Kluwer. 137-165. book chapter Sundholm B.G. (1994), Ontologic versus Epistemologic: Some Strands in the Development of Logic, 1837-1957. In: Prawitz D., Westerstahl D. (Eds.) Logic and Philosophy of Science in Uppsala: KLuwer. 373-384. book chapter Sundholm B.G. (1994), Proof-Theoretical Semantics and Pregean Identity Criteria for Propositions, The Monist 77(3): 294-314. article in journal: refereed Sundholm B.G. (1994), Existence, Proof and Truth-Making: A Perspective on the Intuitionistic Conception of Truth, Topoi : An International Review of Philosophy 13: 117-126. article in journal: refereed

Göran Sundholm - Leiden University

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FLcwで交換が成立することの代数的対応物

主張

単位元が最大元であるような順序モノイドAについて以下が成立するなら、Aは可換である：

(a)x ≤ xx。

証明

条件 (a) より、任意のx, y ∈ Aについて、

xy ≤ xyxy。

単位元1が最大元なので、 単調性より

x(yx)≤1(yx)

つまり、

x(yx)≤yx。

さらに、yについても同様に

x(yx)y ≤ xy1

つまり、

xyxy ≤ xy。

bかつcより、 推移律を用いて、

xy ≤ yx

を任意のx, y ∈ Aについて得る。

任意性より、yx ≤ xyなので、 反対称性より、

xy = yx。

よって、Aは、可換モノイド。

証明終わり。

参考文献

古森・小野『現代数理論理学序説』 (日本評論社、2010)

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test4 via IFTTT

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test2 from Dropbox http://ift.tt/2iqdO4F via IFTTT