Differential Propositional Calculus • Overview

The most fundamental concept in cybernetics is that of “difference”, either that two things are recognisably different or that one thing has changed with time.

W. Ross Ashby • An Introduction to Cybernetics

Here’s the outline of a sketch I wrote on differential propositional calculi, which extend propositional calculi by adding terms for describing aspects of change and difference, for example, processes taking place in a universe of discourse or transformations mapping a source universe to a target universe.  I wrote this as an intuitive introduction to differential logic, which is my best effort so far at dealing with the ancient and persistent problems of treating diversity and mutability in logical terms.  I’ll be looking at ways to improve this draft as I serialize it to my blog.

Part 1

Casual Introduction

Cactus Calculus

Part 2

Formal_Development

Elementary Notions

Special Classes of Propositions

Differential Extensions

Appendices

Appendices

Appendix 1. Propositional Forms and Differential Expansions

Table A1. Propositional Forms on Two Variables

Table A2. Propositional Forms on Two Variables

Table A3. Ef Expanded Over Differential Features

Table A4. Df Expanded Over Differential Features

Table A5. Ef Expanded Over Ordinary Features

Table A6. Df Expanded Over Ordinary Features

Appendix 2. Differential Forms

Table A7. Differential Forms Expanded on a Logical Basis

Table A8. Differential Forms Expanded on an Algebraic Basis

Table A9. Tangent Proposition as Pointwise Linear Approximation

Table A10. Taylor Series Expansion Df = df + d²f

Table A11. Partial Differentials and Relative Differentials

Table A12. Detail of Calculation for the Difference Map

Appendix 3. Computational Details

Operator Maps for the Logical Conjunction f8(u, v)

Computation of εf8
Computation of Ef8
Computation of Df8
Computation of df8
Computation of rf8
Computation Summary for Conjunction

Operator Maps for the Logical Equality f9(u, v)

Computation of εf9
Computation of Ef9
Computation of Df9
Computation of df9
Computation of rf9
Computation Summary for Equality

Operator Maps for the Logical Implication f11(u, v)

Computation of εf11
Computation of Ef11
Computation of Df11
Computation of df11
Computation of rf11
Computation Summary for Implication

Operator Maps for the Logical Disjunction f14(u, v)

Computation of εf14
Computation of Ef14
Computation of Df14
Computation of df14
Computation of rf14
Computation Summary for Disjunction

Appendix 4. Source Materials

Appendix 5. Various Definitions of the Tangent Vector

References

References

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Posted in Amphecks, Boolean Functions, C.S. Peirce, Cactus Graphs, Category Theory, Change, Computational Complexity, Cybernetics, Differential Analytic Turing Automata, Differential Calculus, Differential Logic, Discrete Dynamics, Dynamical Systems, Equational Inference, Functional Logic, Gradient Descent, Graph Theory, Group Theory, Hologrammautomaton, Indicator Functions, Logic, Logical Graphs, Mathematical Models, Mathematics, Minimal Negation Operators, Painted Cacti, Peirce, Propositional Calculus, Propositional Equation Reasoning Systems, Time, Visualization | Tagged , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , | Leave a comment

Survey of Differential Logic • 2

This is a Survey of previous blog and wiki posts on Differential Logic, material I plan to develop toward a more compact and systematic account.

Elements

Architectonics

Applications

Blog Dialogs

Explorations

Posted in Amphecks, Animata, Boolean Algebra, Boolean Functions, C.S. Peirce, Cactus Graphs, Category Theory, Change, Cybernetics, Differential Analytic Turing Automata, Differential Calculus, Differential Logic, Discrete Dynamics, Equational Inference, Frankl Conjecture, Functional Logic, Gradient Descent, Graph Theory, Hologrammautomaton, Indicator Functions, Inquiry Driven Systems, Leibniz, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Painted Cacti, Peirce, Propositional Calculus, Surveys, Time, Topology, Visualization, Zeroth Order Logic | Tagged , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , | Leave a comment

Pragmatic Semiotic Information • Discussion 19

Re: Differential Logic and Dynamic Systems
Re: FB | Systems SciencesKenneth Lloyd

An exchange on Facebook took me back to recent discussions of pragmatic truth and long-running discussions of pragmatic semiotic information.  Just by way of a note to myself and anyone who’s interested, I’ll copy my comment here and add a few links to keep the relevant gray cells warm.

Concepts of belief, fact, knowledge, opinion, etc. look rather different from a Peircean pragmatic perspective, in other words, when analyzed in terms of the pragmatic maxim.  In time the traditional conceptions begin to strike us as increasingly clumsy tools, better supplanted by Peirce’s concept of information.

Resources

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Posted in Abduction, Aristotle, C.S. Peirce, Comprehension, Deduction, Definition, Determination, Extension, Hypothesis, Induction, Inference, Information, Information = Comprehension × Extension, Inquiry, Intension, Intention, Logic, Logic of Science, Mathematics, Measurement, Observation, Peirce, Perception, Phenomenology, Physics, Pragmatic Semiotic Information, Pragmatism, Probability, Quantum Mechanics, Scientific Method, Semiotics, Sign Relations | Tagged , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , | Leave a comment

Cybernetics • Requisite Variety • Selection 13

Our venture into cybernetics, the study of systems whose time evolution signifies an object, brings us to the point of seeing how pragmatic, semiotic, and systems thinking all have triadic relations at their core.

Recall the game between R and D determined by the following data:

Ashby Cybernetics Table 11.3.1

We continue with Ashby’s analysis of how the game plays out.

Requisite Variety

11/3.[cont.]   Examination of the table soon shows that with this particular table R can win always.  Whatever value D selects first, R can always select a Greek letter that will give the desired outcome.  Thus if D selects 1, R selects \beta;  if D selects 2, R selects \alpha;  and so on.  In fact, if R acts according to the transformation

Ashby Cybernetics Figure 11.3.2

then he can always force the outcome to be a.

R\text{'s} position, with this particular table, is peculiarly favourable, for not only can R always force a as the outcome, but he can as readily force, if desired, b or c as the outcome.  R has, in fact, complete control of the outcome.

Reference

  • Ashby, W.R. (1956), An Introduction to Cybernetics, Chapman and Hall, London, UK.  Republished by Methuen and Company, London, UK, 1964.  Online.

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Posted in Adaptive Systems, Ashby, C.S. Peirce, Communication, Control, Cybernetics, Evolution, Information, Inquiry Driven Systems, Learning, Logic, Mathematics, Peirce, Purpose, Regulation, Survival, Truth Theory, W. Ross Ashby | Tagged , , , , , , , , , , , , , , , , , | Leave a comment

Cybernetics • Requisite Variety • Selection 12

Ashby now invites us to consider a series of games, beginning as follows.

Requisite Variety

11/3.   Play and outcome.  Let us therefore forget all about regulation and simply suppose that we are watching two players, R and D, who are engaged in a game.  We shall follow the fortunes of R, who is attempting to score an a.  The rules are as follows.  They have before them Table 11/3/1, which can be seen by both:

Ashby Cybernetics Table 11.3.1

D must play first, by selecting a number, and thus a particular row.  R, knowing this number, then selects a Greek letter, and thus a particular column.  The italic letter specified by the intersection of the row and column is the outcome.  If it is an a, R wins;  if not, R loses.

I’ll pause the play here and give readers a chance to contemplate strategies.

Reference

  • Ashby, W.R. (1956), An Introduction to Cybernetics, Chapman and Hall, London, UK.  Republished by Methuen and Company, London, UK, 1964.  Online.

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Posted in Adaptive Systems, Ashby, C.S. Peirce, Communication, Control, Cybernetics, Evolution, Information, Inquiry Driven Systems, Learning, Logic, Mathematics, Peirce, Purpose, Regulation, Survival, Truth Theory, W. Ross Ashby | Tagged , , , , , , , , , , , , , , , , , | Leave a comment

Pragmatic Theory Of Truth • 21

Re: Cybernetic CommunicationsKlaus Krippendorff

I appreciate the way Klaus Krippendorff immediately extracted one of the overarching themes of Peirce’s whole paper, indeed of his whole work.  That allows us to tread lightly past a lot of verbal nit-picking about the differences among traditional concepts like belief, fact, knowledge, opinion, etc. and get right down to systems-theoretic ideas about states of information and inquiry as a process that revises those states.

Here’s a bit I wrote a few years back rubricizing Peirce’s four ways of moving from doubt to belief — from a state of information so unsettled it puzzles the will to one secure enough on which to act, should the need for action arise.

My favorite polymathematician, Charles Sanders Peirce, gave a fourfold classification of what he called “methods of fixing belief”, or “settling opinion”, most notably and seminally in his paper, “The Fixation of Belief” (1877).  Adjusting his nomenclature very slightly, if only for the sake of preserving a mnemonic rhyme scheme, we may refer to his four types as Tenacity, Authority, Plausibility (à priori pleasing praiseworthiness), and full-fledged Scientific Inquiry.

Reference

Resource

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Posted in Aristotle, C.S. Peirce, Coherence, Concordance, Congruence, Consensus, Convergence, Correspondence, Dewey, Fixation of Belief, Information, Inquiry, John Dewey, Kant, Logic, Logic of Science, Method, Peirce, Philosophy, Pragmatic Maxim, Pragmatism, Semiotics, Sign Relations, Triadic Relations, Truth, Truth Theory, William James | Tagged , , , , , , , , , , , , , , , , , , , , , , , , , , | 1 Comment

Pragmatic Theory Of Truth • 20

Re: Renaissance MathematicusThony Christie
Re: Cybernetic CommunicationsLouis Kauffman
Re: FB | Charles S. Peirce SocietyJohn Corcoran

Various conceptions of belief in relation to pragmatic theories of inquiry, signs, and truth have come up recently in several discussion groups.  Some of the variations are too far off my present track but if I stay the pragmatic course I’d naturally recommend the novel fork taken by Peirce’s 1877 paper, “The Fixation of Belief”.

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Posted in Aristotle, C.S. Peirce, Coherence, Concordance, Congruence, Consensus, Convergence, Correspondence, Dewey, Fixation of Belief, Information, Inquiry, John Dewey, Kant, Logic, Logic of Science, Method, Peirce, Philosophy, Pragmatic Maxim, Pragmatism, Semiotics, Sign Relations, Triadic Relations, Truth, Truth Theory, William James | Tagged , , , , , , , , , , , , , , , , , , , , , , , , , , | Leave a comment