Inquiry Into Inquiry

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Re: R.J. Lipton and K.W. Regan IBM Conference on the Informational LensA little bit of history recoded It may be worth noting the Information Revolution in our understanding of science began in the mid 1860s when C.S.Peirce laid down what he called the Laws of Information inhis lectures on the Logic of Science at Harvard University and the Lowell Institute. Peirce took up the puzzle of the validity of scientific inference and claimed it was entirely removed by a consideration of the laws of information .Here s a collection of excerpts and commentary I assembled on the subject.Information = Comprehension × ExtensionResourceSurvey of Pragmatic Semiotic Informationcc: Cybernetics Ontolog Forum Peirce List Structural Modeling Systems Science Re: Richard J. Lipton Logical Complexity Of ProofsRe: Animated Logical Graphs (35) (36) (37) (38) (39) (40)Last time we looked at a formula of propositional logic Leibniz called a Praeclarum Theorema (PT). We don t concur it s a theorem, of course, until there s a proof it s identically true and Leibniz gave an argument to demonstrate that. Written out in oneofour more current formalisms, PT takes the following form.Somewhat in the spirit of Reduced Instruction Set Computing, we reformulated PT in a propositional calculus using just two primitive operations, writing the logical negation of aproposition as and the logical conjunction of two propositions as That gave us a text string in teletype parentheses and proposition letters, formatted two ways below.Our next transformation of the theorem s expression exploits a standard correspondence in combinatorics and computer science between parenthesized symbol strings and trees with symbols attached to the nodes.We can see the correspondence between text and tree in the case of PT by starting at the root of the tree and reading off the characters of the text string as we traverse the edges and nodes of the tree in the following manner. The initial tells us to ascend the first edge, the next tells us to ascend the next edge on the left, where we find the letter from the string checks with the letter attached to the node of the tree where we are. Another takes us up another edge, where we find the letter from the string checks with the letter on the current tree node. Reading the first on the string entitles us todescend an edge and reading another gives us licence to descend another. The way of things is most likely clear by this point at any rate, I leave the exercise to the reader.On the scene of the general correspondence between formulas and graphs the action may be summed up as follows. The tree, called a parse tree or parse graph, is constructed in the process of checking whether the text string is syntactically well-formed, in other words, whether it satisfies the prescriptions of the associated formal grammar and is therefore a member in good standing of the prescribed formal language. If the text string checks out, grammatically speaking, we call it a traversal string of the corresponding parse graph, because it can be reconstructed from the graph by a process like that illustrated above called traversing the graph.To be continued ReferenceLeibniz, Gottfried W. (1679 1686?), Addenda to the Specimen of the Universal Calculus , pp. 40 46 in G.H.R. Parkinson (ed., trans., 1966), Leibniz : Logical Papers, Oxford University Press, London, UK.Praeclarum TheoremaThis BlogOEIS Wiki (1) (2) (3)PlanetMath Praeclarum TheoremaMetamath Proof Explorer Praeclarum TheoremaFrithjof Dau Computer Animated Proof of Leibniz s Praeclarum TheoremaResourcesLogic SyllabusLogical GraphsCactus LanguageFutures Of Logical GraphsMinimal Negation OperatorsSurvey of Theme One ProgramSurvey of Animated Logical GraphsPropositional Equation Reasoning SystemsHow To Succeed In Proof Business Without Really TryingApplicationsApplications of a Propositional Calculator Constraint Satisfaction ProblemsExploratory Qualitative Analysis of Sequential Observation Datacc: Cybernetics Ontolog Peirce (1) (2) (3) (4) (5) (6) Structural Modeling Systems Re: Richard J. Lipton Logical Complexity Of ProofsRe: Animated Logical Graphs (35) (36) (37) (38) (39)One way to see the difference between insight proofs and routine proofs is to pick a single example of a theorem in propositional calculus and prove it two ways, one more insightful and one more routine.The praeclarum theorema, or splendid theorem, is a theorem of propositional calculus noted and named by G.W.Leibniz, who stated and proved it in the following manner.If a is b and d is c, then ad will be bc.This is a fine theorem, which is proved in this way:a is b, therefore ad is bd (by what precedes),d is c, therefore bd is bc (again by what precedes),ad is bd, and bd is bc, therefore ad is bc. Q.E.D. Leibniz Logical Papers, p. 41.Expressed in contemporary logical notation, the theorem may be written as follows.Using teletype parentheses for the logical negation of a proposition and simple concatenation for the logical conjunction of propositions and enables writing the theorem in the following in-line and lispish ways.InlineLispishReferenceLeibniz, Gottfried W. (1679 1686?), Addenda to the Specimen of the Universal Calculus , pp. 40 46 in G.H.R. Parkinson (ed., trans., 1966), Leibniz : Logical Papers, Oxford University Press, London, UK.Praeclarum TheoremaThis BlogOEIS Wiki (1) (2) (3)PlanetMath Praeclarum TheoremaMetamath Proof Explorer Praeclarum TheoremaFrithjof Dau Computer Animated Proof of Leibniz s Praeclarum TheoremaResourcesLogic SyllabusLogical GraphsCactus LanguageFutures Of Logical GraphsMinimal Negation OperatorsSurvey of Theme One ProgramSurvey of Animated Logical GraphsPropositional Equation Reasoning SystemsHow To Succeed In Proof Business Without Really TryingApplicationsApplications of a Propositional Calculator Constraint Satisfaction ProblemsExploratory Qualitative Analysis of Sequential Observation Datacc: Cybernetics Ontolog Peirce (1) (2) (3) (4) (5) (6) Structural Modeling SystemsTake your place on The Great MandalaAs it moves through your brief moment of time.Win or lose now you must choose nowAnd if you lose you re only losing your life.Peter YarrowRe: Ontolog Forum Alex ShkotinI like Kant s criticism of Aristotle versions of categories. Finding these basic concepts such a proposal was worthy of such an astute thinker like Aristotle. But since he did not have any principle, he picked them up as they came across to him, and first typed ten concepts, which he called categories (predicates). Then it seemed to him that he found five more such concepts, which he added to the previous ones under the name of post-predicate. However, his table was still insufficient. (Kant, Critique of Pure Reason, 10. About Pure Rational Concepts, or Categories).Dear Alex,My sketch on Precursors Of Category Theory shows a big ellipsis under the heading for Kant. I meant to get back to him, as I used to do every half-decade or so, but it s been a long time since I kept to that schedule. Kant is a lodestar in the Peircean constellation Peirce s New List of Categories invokes his guidance on the function of concepts just as he tries his own hand at the wheel. I quoted that passage in my selections from Peirce.Selection 1 1. This paper is based upon the theory already established, that the function of conceptions is to reduce the manifold of sensuous impressions to unity, and that the validity of a conception consists in the impossibility of reducing the content of consciousness to unity without the introduction of it. (CP1.545). 2. This theory gives rise to a conception of gradation among those conceptions which are universal. For one such conception may unite the manifold of sense and yet another may be required to unite the conception and the manifold to which it is applied; and so on. (CP1.546).C.S. Peirce, On a New List of Categories (1867)To be continued ReferencesAristotle, The Categories , Harold P. Cooke (trans.), pp. 1 109 in Aristotle, Volume1, Loeb Classical Library, William Heinemann, London, UK, 1938.Peirce, C.S., On a New List of Categories , Proceedings of the American Academy of Arts and Sciences 7 (1868), 287 298. Presented May 14, 1867. Online.Collection Of Source Materials Determination Peirce (CP5.447) (CP5.448)Foundations Of Mathematics List C.S. Peirce on General and Vague (1) (2) (3)ResourcesPrecursors Of Category TheorySurvey of Precursors Of Category TheoryLane, R. (2001), Principles of Excluded Middle and Contradiction , in M. Bergman and J. Queiroz (eds.), The Commens Encyclopedia : The Digital Encyclopedia of Peirce Studies New Edition, Pub. 140731-0107a. Online.cc: Cybernetics Ontolog Forum Peirce List Structural Modeling Systems Science A few years ago I began a sketch on the Precursors of Category Theory , aiming to trace the continuities of the category concept from Aristotle, to Kant and Peirce, through Hilbert and Ackermann, to contemporary mathematical practice. A Survey of blog and wiki notes on the subject is given below, still very rough and incomplete, but perhaps a few will find it of use.Wiki NotesPrecursors Of Category TheoryPropositions As Types AnalogyBlog PostsNotes On Categories (1)Precursors Of Category Theory (1) (2) (3)Discussions (1) (2) (3) Re: FB | Medieval Logic EB JA JA EB JA EB JA JA EBJA: In the logic of Aristotle categories are adjuncts to reasoning designed to resolve ambiguities and thus to prepare equivocal signs, otherwise recalcitrant to being ruled by logic, for the application of logical laws. The example of ζωον illustrates that we don t need categories to make generalizations so much as we need them to control generalizations, to reign in abstractions and analogies that are stretched too far.EB: Aristotelian categories are adjuncts to reasoning ?I ve been exploring a particular type of commonality or continuity in the way categorical references function in various systems of categories from Aristotle, through Kant and Peirce, to contemporary mathematical category theory. The following posts give the background on that.Precursors Of Category Theory (1) (2) (3)Viewing the discussion of Excluded Middle and Non-Contradiction in the light of Peirce s observations about their relation to the General and the Vague, the upshot is that elements of natural language, indeed, all species of representation in the wild, do not as a rule obey the usual laws of logic, but violate them in various ways. It is only after signs and symbols have been categorized, their equivocations driven down or reduced by reference to the appropriate category, that they become subject to logical laws.ReferencesAristotle, The Categories , Harold P. Cooke (trans.), pp. 1 109 in Aristotle, Volume1, Loeb Classical Library, William Heinemann, London, UK, 1938.Collection Of Source Materials Determination Peirce (CP5.447) (CP5.448)Foundations Of Mathematics List C.S. Peirce on General and Vague (1) (2) (3)ResourcesPrecursors Of Category TheorySurvey of Precursors Of Category TheoryLane, R. (2001), Principles of Excluded Middle and Contradiction , in M. Bergman and J. Queiroz (eds.), The Commens Encyclopedia : The Digital Encyclopedia of Peirce Studies New Edition, Pub. 140731-0107a. Online.cc: Cybernetics Ontolog Forum Peirce List Structural Modeling Systems Science Re: Richard J. Lipton Logical Complexity Of ProofsHappy Peirce s Birthday, Everyone We ve been discussing aspects of proof style arising in connection with the complexity of proofs. In previous posts we took up (1) the aspect of formal duality, reflecting in passing on the prospect of higher symmetries, and (2) the spectrum ranging from information-reducing to information-preserving inference rules. Here s a quick recap Animated Logical Graphs (35) (36) (37) (38)A third aspect of proof style arising in this connection is the degree of insight demanded and demonstrated in the performance of a proof. Generally speaking, the same endpoint can be reached in many different ways from given starting points, by paths ranging from those exhibiting appreciable insight to those exercising little more than persistence in sticking to a set routine.A modicum of insight suffices to suggest the quality of insight resists pinning down in asuccinct definition but we do tend to recognize it when we see it, so let me inch forward by highlighting its salient features in a graded series of examples.To be continued ResourcesLogic SyllabusLogical GraphsCactus LanguageFutures Of Logical GraphsMinimal Negation OperatorsSurvey of Theme One ProgramSurvey of Animated Logical GraphsPropositional Equation Reasoning SystemsApplicationsApplications of a Propositional Calculator Constraint Satisfaction ProblemsExploratory Qualitative Analysis of Sequential Observation Datacc: Cybernetics Ontolog Peirce (1) (2) (3) (4) (5) (6) Structural Modeling Systems Enter your email address to follow this blog and receive notifications of new posts by email.

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