All Process, No Paradox • 2

Posted on January 15, 2014 by Jon Awbrey

These are the forms of time, which imitates eternity and revolves according to a law of number.

Plato • Timaeus

Re: Lou Kauffman • Iterants, Imaginaries, Matrices

As serendipity would have it, Lou Kauffman, who knows a lot about the lines of inquiry Charles Sanders Peirce and George Spencer Brown pursued into graphical syntaxes for logic, just last month opened a blog and his very first post touched on perennial questions of logic and time — Logos and Chronos — puzzling the wits of everyone who has thought about them for as long as anyone can remember.  Just locally and recently these questions have arisen in the following contexts:

Kauffman’s treatment of logic, paradox, time, and imaginary truth values led me to make the following comments I think are very close to what I’d been struggling to say before.

Let me get some notational matters out of the way before continuing.

I use \mathbb{B} for a generic 2-point set, usually \{ 0, 1 \} and typically but not always interpreted for logic so that 0 = \mathrm{false} and 1 = \mathrm{true}.  I use “teletype” parentheses \texttt{(} \ldots \texttt{)} for negation, so that \texttt{(} x \texttt{)} = \lnot x for x ~\text{in}~ \mathbb{B}.  Later on I’ll be using teletype format lists \texttt{(} x_1 \texttt{,} \ldots \texttt{,} x_k \texttt{)} for minimal negation operators.

As long as we’re reading x as a boolean variable (x \in \mathbb{B}) the equation x = \texttt{(} x \texttt{)} is not paradoxical but simply false.  As an algebraic structure \mathbb{B} can be extended in many ways but it remains a separate question what sort of application, if any, such extensions might have to the normative science of logic.

On the other hand, the assignment statement x := \texttt{(} x \texttt{)} makes perfect sense in computational contexts.  The effect of the assignment operation on the value of the variable x is commonly expressed in time series notation as x' = \texttt{(} x \texttt{)} and the same change is expressed even more succinctly by defining \mathrm{d}x = x' - x and writing \mathrm{d}x = 1.

Now suppose we are observing the time evolution of a system X with a boolean state variable x : X \to \mathbb{B} and what we observe is the following time series.

\begin{array}{c|c} t & x \\ \hline 0 & 0 \\ 1 & 1 \\ 2 & 0 \\ 3 & 1 \\ 4 & 0 \\ 5 & 1 \\ 6 & 0 \\ 7 & 1 \\ 8 & 0 \\ 9 & 1 \\ \ldots & \ldots \end{array}

Computing the first differences we get:

\begin{array}{c|cc} t & x & \mathrm{d}x \\ \hline 0 & 0 & 1 \\ 1 & 1 & 1 \\ 2 & 0 & 1 \\ 3 & 1 & 1 \\ 4 & 0 & 1 \\ 5 & 1 & 1 \\ 6 & 0 & 1 \\ 7 & 1 & 1 \\ 8 & 0 & 1 \\ 9 & 1 & 1 \\ \ldots & \ldots & \ldots \end{array}

Computing the second differences we get:

\begin{array}{c|cccc} t & x & \mathrm{d}x & \mathrm{d}^2 x & \ldots \\ \hline 0 & 0 & 1 & 0 & \ldots \\ 1 & 1 & 1 & 0 & \ldots \\ 2 & 0 & 1 & 0 & \ldots \\ 3 & 1 & 1 & 0 & \ldots \\ 4 & 0 & 1 & 0 & \ldots \\ 5 & 1 & 1 & 0 & \ldots \\ 6 & 0 & 1 & 0 & \ldots \\ 7 & 1 & 1 & 0 & \ldots \\ 8 & 0 & 1 & 0 & \ldots \\ 9 & 1 & 1 & 0 & \ldots \\ \ldots & \ldots & \ldots & \ldots & \ldots \end{array}

This leads to thinking of the system X as having an extended state (x, \mathrm{d}x, \mathrm{d}^2 x, \ldots, \mathrm{d}^k x), and this additional language gives us the facility of describing state transitions in terms of the various orders of differences.  For example, the rule x' = \texttt{(} x \texttt{)} can now be expressed by the rule \mathrm{d}x = 1.

The following article has a few more examples along these lines.

Resources

cc: Cybernetics • Laws of Form • Ontolog Forum • Peirce List
cc: FB | Cybernetics • Structural Modeling • Systems Science

Posted in Animata, Boolean Functions, C.S. Peirce, Cybernetics, Differential Logic, Discrete Dynamics, Laws of Form, Logic, Logical Graphs, Lou Kauffman, Mathematics, Paradox, Peirce, Plato, Process, Spencer Brown, Timaeus, Time | Tagged , , , , , , , , , , , , , , , , , | 11 Comments

All Process, No Paradox • 1

Posted on January 13, 2014 by Jon Awbrey


This thing all things devours:
Birds, beasts, trees, flowers;
Gnaws iron, bites steel;
Grinds hard stones to meal;
Slays king, ruins town,
And beats high mountain down.

Tolkien • The Hobbit

Talking about time is a waste of time.  Time is merely an abstraction from process and what is needed are better languages and better pictures for describing process in all its variety.  In the sciences the big breakthrough in describing process came with the differential and integral calculus, that made it possible to shuttle between quantitative measures of state and quantitative measures of change.  But every inquiry into a new phenomenon begins with the slimmest grasp of its qualitative features and labors long and hard to reach as far as a tentative logical description.  What can avail us in the mean time, still tuning up before the first measure, to reason about change in qualitative terms?

Et sic deinceps … (So it begins …)

cc: FB | Cybernetics • Structural Modeling • Systems Science
cc: Cybernetics • Laws of Form • Ontolog Forum • Peirce List

Posted in Animata, C.S. Peirce, Change, Cybernetics, Differential Logic, Graph Theory, Laws of Form, Logic, Logical Graphs, Mathematics, Paradox, Peirce, Process, Process Thinking, Spencer Brown, Systems Theory, Time, Tolkien | Tagged , , , , , , , , , , , , , , , , , | 11 Comments

Precursors Of Category Theory • 3

Posted on January 3, 2014 by Jon Awbrey

Act only according to that maxim by which you can at the same time will that it should become a universal law.

Immanuel Kant (1785)

Precursors Of Category Theory

Peirce

Cued by Kant’s idea on the function of concepts in general, Peirce locates his categories on the highest level of abstraction affording a meaningful measure of traction in practice, reserving judgment on the absolute unity of perfect ambiguity and the numerous dualisms which taken together may well converge on the same conception as Peirce’s trinity.

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.  (CP 1.545).

§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.  (CP 1.546).

C.S. Peirce, “On a New List of Categories” (1867)

Selection 2

I will now say a few words about what you have called Categories, but for which I prefer the designation Predicaments, and which you have explained as predicates of predicates.

That wonderful operation of hypostatic abstraction by which we seem to create entia rationis that are, nevertheless, sometimes real, furnishes us the means of turning predicates from being signs that we think or think through, into being subjects thought of.  We thus think of the thought-sign itself, making it the object of another thought-sign.

Thereupon, we can repeat the operation of hypostatic abstraction, and from these second intentions derive third intentions.  Does this series proceed endlessly?  I think not.  What then are the characters of its different members?

My thoughts on this subject are not yet harvested.  I will only say that the subject concerns Logic, but that the divisions so obtained must not be confounded with the different Modes of Being:  Actuality, Possibility, Destiny (or Freedom from Destiny).

On the contrary, the succession of Predicates of Predicates is different in the different Modes of Being.  Meantime, it will be proper that in our system of diagrammatization we should provide for the division, whenever needed, of each of our three Universes of modes of reality into Realms for the different Predicaments.  (CP 4.549).

C.S. Peirce, “Prolegomena to an Apology for Pragmaticism”, The Monist 16, 492–546 (1906), CP 4.530–572.

The first thing to extract from this passage is that Peirce’s Categories, or “Predicaments”, are predicates of predicates.  Considerations like these tend to generate hierarchies of subject matters, extending through what is traditionally called the logic of second intentions, or what is handled very roughly by second order logic in contemporary parlance, and continuing onward through higher intentions, or higher order logic and type theory.

Peirce arrived at his own system of three categories after a thoroughgoing study of his predecessors, with special reference to the categories of Aristotle, Kant, and Hegel.  The names he used for his own categories varied with context and occasion, but ranged from moderately intuitive terms like quality, reaction, and symbolization to maximally abstract terms like firstness, secondness, and thirdness.  Taken in full generality, k-ness may be understood as referring to those properties all k-adic relations have in common.  Peirce’s distinctive claim is that a type hierarchy of three levels is generative of all we need in logic.

Part of the justification for Peirce’s claim that three categories are necessary and sufficient appears to arise from mathematical facts about the reducibility of k-adic relations.  With regard to necessity, triadic relations cannot be completely analyzed in terms or monadic and dyadic predicates.  With regard to sufficiency, all higher arity k-adic relations can be analyzed in terms of triadic and lower arity relations.

Posted in Abstraction, Ackermann, Aristotle, C.S. Peirce, Carnap, Category Theory, Hilbert, Kant, Logic, Logic of Relatives, Mathematics, Peirce, Peirce's Categories, Relation Theory, Saunders Mac Lane, Sign Relations, Type Theory | Tagged , , , , , , , , , , , , , , , , | 8 Comments

Precursors Of Category Theory • 2

Posted on December 30, 2013 by Jon Awbrey

Thanks to art, instead of seeing one world only, our own, we see that world multiply itself and we have at our disposal as many worlds as there are original artists …

☙ Marcel Proust

Precursors Of Category Theory

When it comes to looking for the continuities of the category concept across different systems and systematizers, we don’t expect to find their kinship in the names or numbers of categories, since those are legion and their divisions deployed on widely different planes of abstraction, but in their common function.

Aristotle

Things are equivocally named, when they have the name only in common, the definition (or statement of essence) corresponding with the name being different.  For instance, while a man and a portrait can properly both be called animals (ζωον), these are equivocally named.  For they have the name only in common, the definitions (or statements of essence) corresponding with the name being different.  For if you are asked to define what the being an animal means in the case of the man and the portrait, you give in either case a definition appropriate to that case alone.

Things are univocally named, when not only they bear the same name but the name means the same in each case — has the same definition corresponding.  Thus a man and an ox are called animals.  The name is the same in both cases;  so also the statement of essence.  For if you are asked what is meant by their both of them being called animals, you give that particular name in both cases the same definition.

— Aristotle, Categories, 1.1a1–12.

Translator’s Note.  “Ζωον in Greek had two meanings, that is to say, living creature, and, secondly, a figure or image in painting, embroidery, sculpture.  We have no ambiguous noun.  However, we use the word ‘living’ of portraits to mean ‘true to life’.”

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 the fact 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.

References

  • Aristotle, “The Categories”, Harold P. Cooke (trans.), pp. 1–109 in Aristotle, Volume 1, Loeb Classical Library, William Heinemann, London, UK, 1938.
  • Karpeles, Eric (2008), Paintings in Proust, Thames and Hudson, London, UK.
Posted in Abstraction, Ackermann, Aristotle, C.S. Peirce, Carnap, Category Theory, Hilbert, Kant, Logic, Logic of Relatives, Mathematics, Peirce, Peirce's Categories, Relation Theory, Saunders Mac Lane, Sign Relations, Type Theory | Tagged , , , , , , , , , , , , , , , , | 6 Comments

Precursors Of Category Theory • 1

Posted on December 20, 2013 by Jon Awbrey

A few years back I began a sketch on the Precursors of Category Theory, aiming to trace the continuities of the category concept from Aristotle, thorough Kant and Peirce, Hilbert and Ackermann, to contemporary mathematical use.  Perhaps a few will find these notes of use in the current context.

Precursors Of Category Theory

Preamble

Now the discovery of ideas as general as these is chiefly the willingness to make a brash or speculative abstraction, in this case supported by the pleasure of purloining words from the philosophers:  “Category” from Aristotle and Kant, “Functor” from Carnap (Logische Syntax der Sprache), and “natural transformation” from then current informal parlance.

— Saunders Mac Lane, Categories for the Working Mathematician, 29–30.

Posted in Abstraction, Ackermann, Aristotle, C.S. Peirce, Carnap, Category Theory, Hilbert, Kant, Logic, Logic of Relatives, Mathematics, Peirce, Peirce's Categories, Relation Theory, Saunders Mac Lane, Sign Relations, Type Theory | Tagged , , , , , , , , , , , , , , , , | 10 Comments

Alpha Now, Omega Later • 7

Posted on December 19, 2013 by Jon Awbrey

Re: R.J. Lipton and K.W. Regan • Theorems From Physics?

In a way, the relation between “physics space” and “information space” is one of the topics I address in my work on inquiry driven systems.  Here is a pertinent place in medias res, from which point one may happily sample both backwards and forwards:

Prospects for Inquiry Driven Systems • Unabridgements

Part of my task in the projected work is to make a bridge, in theory and practice, from simple physical systems to the more complex systems, also physical but in which new orders of features have become salient, that begin to exhibit what is recognized as intelligence.  At the moment it seems that a good way to do this is to anchor the knowledge component of intelligent systems in the tangent and co-tangent spaces that are founded on the base space of a dynamic manifold.  This means finding a place for knowledge representations in the residual part of the initial factorization.  This leads to a consideration of the questions:  What makes the difference between these supposedly different factors of the total manifold?  What properties mark the distinction as commonly intended?

From a naturalistic perspective everything falls equally under the prospective heading of physis, signifying nothing more than the first inklings of natural process, though not everything is necessarily best explained in detail by those fragments of natural law which are currently known to us.  So it falls to any science that pretends to draw a distinction between the more and the less basic physics to describe it within nature and without trying to get around nature.  In this context the question may now be rephrased:  What natural terms distinguish every system’s basic processes from the kinds of coping processes that support and crown the intelligent system’s personal copy of the world?  What protocols attach to the sorting and binding of these two different books of nature?  What colophon can impress the reader with a need to read them?  What instinct can motivate a basis for needing to know?

Previous Discussion

Posted in C.S. Peirce, Differential Logic, Dynamical Systems, Equational Inference, Laws of Form, Logic, Logical Graphs, Mathematics, Propositional Calculus, Semiotics, Sign Relations, Spencer Brown | Tagged , , , , , , , , , , , | 4 Comments

Icon, Likeness, Likely Story, Likelihood, Probability • 3

Posted on December 12, 2013 by Jon Awbrey

Re: Peirce List • Phyllis Chiasson

A more complete excerpt and the translator’s notes are very helpful here.

A probability (εικος) is not the same as a sign (σηµειον).  The former is a generally accepted premiss ;  for that which people know to happen or not to happen, or to be or not to be, usually in a particular way, is a probability :  e.g., that the envious are malevolent or that those who are loved are affectionate.  A sign, however, means a demonstrative premiss which is necessary or generally accepted.1  That which coexists with something else, or before or after whose happening something else has happened, is a sign of that something’s having happened or being.

An enthymeme is a syllogism from probabilities or signs ;  and a sign can be taken in three ways — in just as many ways as there are of taking the middle term in the several figures :  either as in the first figure or as in the second or as in the third.

  • E.g., the proof that a woman is pregnant because she has milk is by the first figure ;  for the middle term is ‘having milk’.  A stands for ‘pregnant’, B for ‘having milk’, and C for ‘woman’.
  • The proof that the wise are good because Pittacus was good is by the third figure.  A stands for ‘good’, B for ‘the wise’, and C for Pittacus.  Then it is true to predicate both A and B of C ;  only we do not state the latter, because we know it, whereas we formally assume the former.
  • The proof that a woman is pregnant because she is sallow is intended to be by the middle figure ;  for since sallowness is a characteristic of woman in pregnancy, and is associated with this particular woman, they suppose that she is proved to be pregnant.  A stands for ‘sallowness’, B for ‘being pregnant’, C for ‘woman’.

If only one premiss is stated, we get only a sign ;  but if the other premiss is assumed as well, we get a syllogism,2 e.g., that Pittacus is high-minded, because those who love honour are high-minded, and Pittacus loves honour ;  or again that the wise are good, because Pittacus is good and also wise.

In this way syllogisms can be effected ;  but whereas a syllogism in the first figure cannot be refuted if it is true, since it is universal, a syllogism in the last figure can be refuted even if the conclusion is true, because the syllogism is neither universal nor relevant to our purpose.3  For if Pittacus is good, it is not necessary for this reason that all other wise men are good.  A syllogism in the middle figure is always and in every way refutable, since we never get a syllogism with the terms in this relation4 ;  for it does not necessarily follow, if a pregnant woman is sallow, and this woman is sallow, that she is pregnant.  Thus truth can be found in all signs, but they differ in the ways which have been described.

We must either classify signs in this way, and regard their middle term as an index (τεκµηριον)5 (for the name ‘index’ is given to that which causes us to know, and the middle term is especially of this nature), or describe the arguments drawn from the extremes6 as ‘signs’, and that which is drawn from the middle as an ‘index’.  For the conclusion which is reached through the first figure is most generally accepted and most true.  (Aristotle, Prior Analytics 2.27, 70a3–70b6).

Translator’s Notes

  1. If referable to one phenomenon only, a sign has objective necessity ;  if to more than one, its value is a matter of opinion.
  2. Strictly an enthymeme.
  3. If the signs of an enthymeme in the first figure are true, the conclusion is inevitable.  Aristotle does not mean that the conclusion is universal, but that the universality of the major premiss implies the validity of the minor and conclusion.  The example (<all> those who have honour, etc.) quoted for the third figure contains no universal premiss or sign, and fails to establish a universal conclusion.
  4. i.e. when both premisses are affirmative.
  5. Signs may be classified as irrefutable (1st figure) and refutable (2nd and 3rd figures), and the name ‘index’ may be attached to their middle terms, either in all figures or (more probably) only in the first, where the middle is distinctively middle.
  6. Alternatively the name ‘sign’ may be restricted to the 2nd and 3rd figures, and may be replaced by ‘index’ in the first.

Reference

  • Aristotle, “Prior Analytics”, Hugh Tredennick (trans.), pp. 181–531 in Aristotle, Volume 1, Loeb Classical Library, William Heinemann, London, UK, 1938.

Resource

Posted in Analogy, Aristotle, C.S. Peirce, Icon Index Symbol, Induction, Inquiry, Likelihood, Likely Story, Likeness, Logic, Mathematics, Probability, Probable Reasoning, Semiotics, Sign Relations | Tagged , , , , , , , , , , , , , , | 1 Comment

Icon, Likeness, Likely Story, Likelihood, Probability • 2

Posted on December 11, 2013 by Jon Awbrey

Re: Peirce List • Phyllis Chiasson

I’m still a bit fuzzy on how Aristotle’s account relates to Peirce’s usage, though I’m pretty sure Peirce must have taken Aristotle’s usage into account, but it does seem that Aristotle drew some sort of distinction here, using a term “tekmerion” which gets translated as “index” to make the following remark later on in that chapter.

We must either classify signs in this way, and regard their middle term as an index [τεκµηριον] (for the name ‘index’ is given to that which causes us to know, and the middle term is especially of this nature), or describe the arguments drawn from the extremes as ‘signs’, and that which is drawn from the middle as an ‘index’.  For the conclusion which is reached through the first figure is most generally accepted and most true.  (Aristotle, Prior Analytics, 2.27.70b1–6).

Reference

  • Aristotle, “Prior Analytics”, Hugh Tredennick (trans.), pp. 181–531 in Aristotle, Volume 1, Loeb Classical Library, William Heinemann, London, UK, 1938.

Resource

Posted in Analogy, Aristotle, C.S. Peirce, Icon Index Symbol, Induction, Inquiry, Likelihood, Likely Story, Likeness, Logic, Mathematics, Probability, Probable Reasoning, Semiotics, Sign Relations | Tagged , , , , , , , , , , , , , , | 1 Comment

Icon, Likeness, Likely Story, Likelihood, Probability • 1

Posted on December 11, 2013 by Jon Awbrey

Re: Peirce List • Benjamin Udell • Michael Shapiro

Here’s a likely locus classicus for “icon” in its logical sense —

A probability (εικος) is not the same as a sign (σηµειον).  The former is a generally accepted premiss;  for that which people know to happen or not to happen, or to be or not to be, usually in a particular way, is a probability:  e.g., that the envious are malevolent or that those who are loved are affectionate.  A sign, however, means a demonstrative premiss which is necessary or generally accepted.  That which coexists with something else, or before or after whose happening something else has happened, is a sign of that something’s having happened or being.  (Aristotle, Prior Analytics, 2.27.70a3–10).

Reference

  • Aristotle, “Prior Analytics”, Hugh Tredennick (trans.), pp. 181–531 in Aristotle, Volume 1, Loeb Classical Library, William Heinemann, London, UK, 1938.

Resource

Posted in Analogy, Aristotle, C.S. Peirce, Icon Index Symbol, Induction, Inquiry, Likelihood, Likely Story, Likeness, Logic, Mathematics, Probability, Probable Reasoning, Semiotics, Sign Relations | Tagged , , , , , , , , , , , , , , | 2 Comments

Alpha Now, Omega Later • 6

Posted on December 6, 2013 by Jon Awbrey

Re: Alpha Now, Omega Later • Theorems From Physics? • Isomorphism Is Where It’s At

In the late 1970s a number of problems in combinatorics and graph theory that I really wanted to know the answers to had driven me to the desperate measures of trying to write a theorem-proving program to help with the work.  Being familiar with the conceptual efficiencies of Peirce’s logical graphs and inspired by George Spencer Brown’s more recent resurrection of Peirce’s ideas, I naturally turned to those resources for the initial implements of my computational prospecting. The succession of computers and programming languages that I quested with over the years taught me a lot about the things that work and the things that do not. Dissolving for now to the present scene, I will use the next few posts to outline, as succinctly as I can, the basic constructs that developed through that line of inquiry.

Minimal Negation Operators and Painted Cacti

Let \mathbb{B} = \{ 0, 1 \}.

The objects of penultimate interest are the boolean functions f : \mathbb{B}^n \to \mathbb{B} for n \in \mathbb{N}.

A minimal negation operator \nu_k for k \in \mathbb{N} is a boolean function \nu_k : \mathbb{B}^k \to \mathbb{B} defined as follows:

  • \nu_0 = 0.
  • \nu_k (x_1, \ldots, x_k) = 1 if and only if exactly one of the arguments x_j equals 0.

The first few of these operators are already enough to generate all boolean functions f : \mathbb{B}^n \to \mathbb{B} via functional composition but the rest of the family is worth keeping around for many practical purposes.

In most contexts \nu (x_1, \ldots, x_k) may be written for \nu_k (x_1, \ldots, x_k) since the number of arguments determines the rank of the operator.  In some contexts even the letter \nu may be omitted, writing just the argument list (x_1, \ldots, x_k), in which case it helps to use a distinctive typeface for the list delimiters, as \texttt{(} x_1 \texttt{,} \ldots \texttt{,} x_k \texttt{)}.

A logical conjunction of k arguments can be expressed in terms of minimal negation operators as \nu_{k+1} (x_1, x_2, \ldots, x_{k-1}, x_k, 0) and this is conveniently abbreviated as a concatenation of arguments x_1 x_2 \ldots x_{k-1} x_k.

See the following article for more information.

To be continued …

Posted in C.S. Peirce, Differential Logic, Dynamical Systems, Equational Inference, Laws of Form, Logic, Logical Graphs, Mathematics, Propositional Calculus, Semiotics, Sign Relations, Spencer Brown | Tagged , , , , , , , , , , , | 5 Comments