Indicator Functions • Discussion 1

Peter Smith, on his Logic Matters blog, asks the question, “What Is A Predicate?”, and considers a number of answers.

There are of course other possible answers, and one I learned quite early on, arising very naturally in applying mathematical logic to what were generally known as “AI problems”, like perception and pattern recognition, and the one I found increasingly useful as I took up the reflective stance on symbolic computation afforded by Peirce’s pragmatic semiotics, may be described as follows:

In many applications a predicate is a function from a universe of discourse X to a binary value in \mathbb{B} = \{0, 1\}, that is, a characteristic function or indicator function f : X \to \mathbb{B}, and f^{-1}(1), the fiber of 1 under f, is the set of elements denoted or indicated by the predicate.  That is the semantics, anyway.  As far as syntax goes, there are many formal languages whose syntactic expressions serve as names for those functions and nominally speaking one may call those names predicates.

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Posted in Boole, Boolean Functions, C.S. Peirce, Category Theory, Indication, Indicator Functions, Logic, Mathematics, Peirce, Propositional Calculus, Propositional Equation Reasoning Systems, Set Theory, Venn Diagrams | Tagged , , , , , , , , , , , , | Leave a comment

{ Information = Comprehension × Extension } • Comment 8

So what is all this fuss about the relation between inquiry and signs, as analyzed in Peirce’s theories of their structure and function and synthesized in his theory of information?

The best way I’ve found to see where the problem lies is to run through a series of concrete examples of the sort Peirce used to illustrate his notions of information, inquiry, and signs, examples just complex enough to show the interplay of main ideas.

There is an enlightening set of examples in Peirce’s early lectures on the Logic of Science.  Here is the blog post I wrote to set up their discussion:

Another angle from which to approach the incidence of signs and inquiry is by way of C.S. Peirce’s “laws of information” and the corresponding theory of information he developed from the time of his lectures on the “Logic of Science” at Harvard University (1865) and the Lowell Institute (1866).

When it comes to the supposed reciprocity between extensions and intensions, Peirce, of course, has another idea, and I would say a better idea, partly because it forms the occasion for him to bring in his new-fangled notion of “information” to mediate the otherwise static dualism between the other two.  The development of this novel idea brings Peirce to enunciate the formula:

\mathrm{Information} = \mathrm{Comprehension} \times \mathrm{Extension}

But comprehending what in the world that might mean is a much longer story, the end of which your present teller has yet to reach.  So, this time around, I will take up the story near the end of the beginning of Peirce’s own telling of it, for no better reason than that’s where I myself initially came in, or, at least, where it all started making any kind of sense to me.  And from this point we will find it easy enough to flash both backward and forward, to and fro, as the occasions arise for doing so.

Readings

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

Differential Logic, Dynamic Systems, Tangent Functors • Discussion 2

Re: Cybernetic CommunicationsSU

One thing that interests me here is the relation between narratives and navigation.  Navigation has to do with how we move through actual state spaces while narratives are the tales we tell about past adventures and what we may have learned from them by way of guiding future ventures.

Navigation has its local (individual, immediate) and global (general, ultimate) aspects but it tends to lose its point if it does not keep at least one eye to present business.  Purloining a paradigm from physics it keeps watch over the bearings of local and global purposes on each other with instruments analogous to differential and integral calculus.

Narratives, in contrast, inhabiting as they do the semiotic plane of signs and symbols, have a tendency to detach themselves from the matter at hand, to become autonomous, to create worlds of fantasy all their own, and even to spin altogether out of control.

So we have to watch out for that …

Resources

Posted in Amphecks, Boolean Functions, C.S. Peirce, Cactus Graphs, Category Theory, Cybernetics, Differential Analytic Turing Automata, Differential Calculus, Differential Logic, Discrete Dynamical Systems, Dynamical Systems, Graph Theory, Hill Climbing, Hologrammautomaton, Information Theory, Inquiry Driven Systems, Intelligent Systems, Knowledge Representation, Laws of Form, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Painted Cacti, Peirce, Propositional Calculus, Propositional Equation Reasoning Systems, Spencer Brown, Systems, Visualization | Tagged , , , , , , , , , , , , , , , , , , , , , , , , , , , , , | 1 Comment

Pragmatic Semiotic Information • Discussion 9

Re: Ontolog ForumJS

I’ll be experiencing intermittent interruptions from now until the moving vans come and probably a while after but I’m hoping things will settle down by Thanksgiving.

For the moment I’ll simply post a few links to matters I’ve been trying to get back to and hope to develop further as time goes on.

The topic named in the title is the same as what I used to call Semiotic Information but I added Pragmatic to emphasize the continuity with Aristotle’s pragmata and to point up the object dimension of sign relations, encompassing objects both actual and intentional.  Various excursions along those lines are linked on the following Survey page:

Reference

  • Awbrey and Awbrey (1995), “Interpretation as Action : The Risk of Inquiry” (1) (2)

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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

{ Information = Comprehension × Extension } • Comment 7

One of the most tantalizing puzzles in Peirce’s work is the relation between his theory of inquiry and his theory of signs.  From the outset I found it useful to return to his early ventures where the two theories work most closely in tandem, indeed as offshoots of a single conception, namely, information.

Peirce’s inquiry into “the laws of information”, going back to his lectures of 1865 and 1866, marks one of those occasions when he leapt far ahead of his time, anticipating ideas we’d not see again until much later in the Twentieth Century.

So I’ve long found it well worth the effort to tease out the hints of information theory Peirce sketched in those early days.  In that spirit I’m going to make another try at returning to a line of inquiry I started two years ago.  Here is where I left off —

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

Differential Logic, Dynamic Systems, Tangent Functors • Discussion 1

Re: Cybernetic CommunicationsAnalyzing Differences of Perceptions/Constructions

Susan and I just returned from our annual dramatic immersion in Stratford Ontario and the last play we saw was a playfully dramatic rendition of Milton’s Paradise Lost.  That and the spectacular immorality play currently embroiling Washington DC moved me to meditate on the differences that narratives bear in relation to their objects (objectives, intentions, goals, aims), especially the difference between narratives aimed at the truth that sets us free and narratives propagating the lies that enslave humanity.

Resources

Posted in Amphecks, Boolean Functions, C.S. Peirce, Cactus Graphs, Category Theory, Cybernetics, Differential Analytic Turing Automata, Differential Calculus, Differential Logic, Discrete Dynamical Systems, Dynamical Systems, Graph Theory, Hill Climbing, Hologrammautomaton, Information Theory, Inquiry Driven Systems, Intelligent Systems, Knowledge Representation, Laws of Form, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Painted Cacti, Peirce, Propositional Calculus, Propositional Equation Reasoning Systems, Spencer Brown, Systems, Visualization | Tagged , , , , , , , , , , , , , , , , , , , , , , , , , , , , , | Leave a comment

Pragmatic Semiotic Information • Discussion 8

Re: Ontolog ForumJS

The concept of a triadic sign relation, in typical form L \subseteq O \times S \times I, where O is the object domain (think universe of discourse) and S and I are domains of signs (think channels or languages) being used to talk and think about O, is most often applied in one of two ways.

  1. S and I are really the same channel, language, medium, set of signs, or state space of a system we are using to convey information about O.  In cases where S = I we are often concerned with transformations taking place within a single set of signals and we may write I = S^\prime to signify our focus on sign relational triples of the form (o, s, s^\prime) where s^\prime is a sign that follows s in a logical or temporal sequence, in short, where s^\prime is contemplated as a next state of s.
  2. S and I are two different channels, languages, media, sets of signs, or state spaces of systems being used to convey information about O.  In this case the issue is one of translation or interoperability.

Reference

  • Awbrey and Awbrey (1995), “Interpretation as Action : The Risk of Inquiry” (1) (2)

cc: Systems ScienceStructural Modeling

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