The Difference That Makes A Difference That Peirce Makes : 18

Re: Peter SmithWhich Is The Quantifier?

From a functional point of view it was a step backward when we passed from Peirce’s \sum and \prod to the current convention of \exists and \forall for logical quantifiers.  There’s a rough indication of what I mean at the following location:

Functional Logic : Higher Order Propositions

Just a reminder to get back to this later …

Posted in C.S. Peirce, Category Theory, Complementarity, Duality, Formal Languages, Higher Order Propositions, Indicator Functions, Inquiry, Logic, Logic of Relatives, Logical Graphs, Mathematics, Peirce, Pragmatism, Predicate Calculus, Propositional Calculus, Propositions, Quantifiers, Relation Theory, Semiotics, Type Theory, Zeroth Order Logic | Tagged , , , , , , , , , , , , , , , , , , , , , | Leave a comment

Charles Sanders Peirce, George Spencer Brown, and Me • 2

Re: Laws Of Form DiscussionJAJBAM

I’m making an effort to present this material in a more gradual and logical order than I’ve ever managed to do before.  There are issues about the relationship between episodic and semantic memory that are giving me trouble as I try to remember how I came to look at things the way I do … but never mind that now.  I’ll eventually get around to explaining the forces that drove me to generalize the forms of logical graphs from trees to cacti, as graph theorists call them, and how that made the transition to differential logic so much easier than it would have been otherwise, but I think it would be better now to begin at the beginning with the common core of forms introduced by CSP and GSB.

Here’s a couple of articles I wrote up for that purpose:

There are versions of those articles at several other places on the web that may be better formatted or more convenient for discussion:

One big issue that comes up at the beginning is the question of “duality”.  Both C.S. Peirce and Spencer Brown understood they were dealing with a very abstract calculus, one that could be interpreted for the purposes of ordinary propositional logic in two different ways.  Peirce called the two different ways of interpreting the abstract graphs his entitative and existential graphs.  He started out with a system of graphs he chose to read in the entitative manner but switched over to the existential choice as he developed his logical graphs beyond the purely propositional level.  Spencer Brown elected to emphasize the entitative reading in his main exposition but he was very clear in the terminology he used that the forms and transformations themselves are independent of their interpretations.

Table 1 at either of the locations linked below has columns for the graph-theoretic forms and the parenthesis-string forms of several basic expressions, reading them under the existential interpretation.

  • Table 1. Syntax and Semantics of a Calculus for Propositional Logic • (a)(b)

The Tables linked below serve to compare the existential and entitative interpretations of logical graphs by providing translations into familiar notations and English paraphrases for a few of the most basic and commonly occurring forms.

Posted in Abstraction, Amphecks, Analogy, Animata, Automated Research Tools, Boolean Algebra, Boolean Functions, C.S. Peirce, Cactus Graphs, Cybernetics, Deduction, Diagrammatic Reasoning, Differential Logic, Duality, Form, Graph Theory, Iconicity, Information Theory, Inquiry, Laws of Form, Logic, Logical Graphs, Mathematics, Model Theory, Painted Cacti, Peirce, Pragmatic Maxim, Proof Theory, Propositional Calculus, Semiotics, Sign Relational Manifolds, Spencer Brown, Symbolism, Systems Theory, Theorem Proving, Topology, Visualization, Zeroth Order Logic | Tagged , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , | 2 Comments

Charles Sanders Peirce, George Spencer Brown, and Me • 1

Re: Laws Of Form DiscussionJon Awbrey

It’s almost 50 years now since I first encountered the volumes of Peirce’s Collected Papers in the math library at Michigan State, and shortly afterwards a friend called my attention to the entry for Spencer Brown’s Laws of Form in the Whole Earth Catalog and I sent off for it right away.  I would spend the next decade just beginning to figure out what either one of them was talking about in the matter of logical graphs and I would spend another decade after that developing a program, first in Lisp and then in Pascal, that turned graph-theoretic data structures formed on their ideas to good purpose in the mechanics of its propositional reasoning engine.  I thought it might contribute to a number of ongoing discussions if I could articulate what I think I learned from that experience.

Posted in Abstraction, Amphecks, Analogy, Animata, Automated Research Tools, Boolean Algebra, Boolean Functions, C.S. Peirce, Cactus Graphs, Cybernetics, Deduction, Diagrammatic Reasoning, Differential Logic, Duality, Form, Graph Theory, Iconicity, Information Theory, Inquiry, Laws of Form, Logic, Logical Graphs, Mathematics, Model Theory, Painted Cacti, Peirce, Pragmatic Maxim, Proof Theory, Propositional Calculus, Semiotics, Sign Relational Manifolds, Spencer Brown, Symbolism, Systems Theory, Theorem Proving, Topology, Visualization, Zeroth Order Logic | Tagged , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , | Leave a comment

Charles Sanders Peirce, George Spencer Brown, and Me

Re: Laws Of Form DiscussionJames Bowery

James Bowery left a comment on my blog and opened a thread in the Yahoo! group devoted to discussing the mathematics of George Spencer Brown’s Laws of Form.  I’ve been meaning to join that discussion as soon as I could work up the time and concentration to think about it … at long last I think I can do that now.  I’ll use the above heading on this blog to post any bits from my side of the conversation that I think might serve a wider audience.

It’s been a long time since I joined a new discussion group so I thought I’d start by posting a bit of the old-fashioned self-intro.

Posted in Abstraction, Amphecks, Analogy, Animata, Automated Research Tools, Boolean Algebra, Boolean Functions, C.S. Peirce, Cactus Graphs, Cybernetics, Deduction, Diagrammatic Reasoning, Differential Logic, Duality, Form, Graph Theory, Iconicity, Information Theory, Inquiry, Laws of Form, Logic, Logical Graphs, Mathematics, Model Theory, Painted Cacti, Peirce, Pragmatic Maxim, Proof Theory, Propositional Calculus, Semiotics, Sign Relational Manifolds, Spencer Brown, Symbolism, Systems Theory, Theorem Proving, Topology, Visualization, Zeroth Order Logic | Tagged , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , | Leave a comment

¿Shifting Paradigms? • 5

Re: Peter CameronInfinity and Foundation

We always encounter a multitude of problems whenever we try to rationalize mathematics by reducing it to logic, where logic itself is reduced to a purely deductive style.  A number of thinkers have proposed it is time — well past time — to stop counting so heavily on that idea and to join a Declaration of Independence for Mathematics.

Posted in Algorithms, Boole, C.S. Peirce, Combinatorics, Computation, Foundations of Mathematics, Inquiry, Laws of Form, Leibniz, Logic, Mathematics, Model Theory, Paradigms, Peirce, Proof Theory, Spencer Brown | Tagged , , , , , , , , , , , , , , , | 2 Comments

{ Information = Comprehension × Extension } • Discussion 6

Re: Peirce List Discussion • JAJFSJA

What interests me so much about Peirce’s first legislation of the “laws of information” in his 1865–1866 “Logic of Science” is that the primal twins of Inquiry and Semiotics nestle so closely in their first nest that we can see their kinship far better and more easily than ever we will again.  (I am cautiously optimistic their further development won’t go the same way it did for Rome.)

More than that, whatever disclaimers Peirce may issue about his own originality, I don’t think anyone can fairly encounter his definition of a term’s information as “the measure of its superfluous comprehension” without being downright shocked at its novelty.

Posted in Abduction, C.S. Peirce, Comprehension, Deduction, Extension, Hypothesis, Icon Index Symbol, Induction, Inference, Information, Information = Comprehension × Extension, Information Theory, Inquiry, Intension, Logic, Logic of Science, Peirce, Peirce's Categories, Pragmatism, Science, Scientific Method, Semiotic Information, Semiotics, Sign Relations | Tagged , , , , , , , , , , , , , , , , , , , , , , , | Leave a comment

{ Information = Comprehension × Extension } • Discussion 5

Re: Peirce List Discussion • John Sowa

What you say goes to the heart of a problem I saw in Natural Propositions, whether it was Peirce’s account or Stjernfelt’s analysis I did not have time to decide as the schedule of the slow reading went too fast for me to take it up on the List.  I marked the critical passages and my copy of the book is around here someplace but I am trying to stay focused on the subject matter and the set of problems I introduced under the above subject line.

There are many issues about cross-disciplinary communication, the varieties of quasi-religious belief about the uses of words in the whole proposition/sentence/statement complex, the various uses Peirce and others use across contexts, disciplines, historical time, and even within the same discussion.  But I think it’s best to hold the forte on that for now.

Posted in Abduction, C.S. Peirce, Comprehension, Deduction, Extension, Hypothesis, Icon Index Symbol, Induction, Inference, Information, Information = Comprehension × Extension, Information Theory, Inquiry, Intension, Logic, Logic of Science, Peirce, Peirce's Categories, Pragmatism, Science, Scientific Method, Semiotic Information, Semiotics, Sign Relations | Tagged , , , , , , , , , , , , , , , , , , , , , , , | Leave a comment