Relations & Their Relatives • Discussion 5

Posted on March 22, 2015 by Jon Awbrey

Re: Peirce List • Howard Pattee

At this point we can distinguish two forms of decomposability or reducibility — along with their corresponding negations, indecomposability or irreducibility – that commonly arise.

  • Reducibility under relational composition P \circ Q.

All triadic relations are irreducible in this sense.  This is because relational compositions of monadic and dyadic relations can produce only more monadic and dyadic relations.

  • Reducibility under projections.  For this we need a few definitions:

Every triadic relation, say L contained as a subset of the cartesian product X \times Y \times Z, determines three dyadic relations, namely, the projections of L on the three “planes” X \times Y, X \times Z, and Y \times Z.

In particular:

Every sign relation, say M contained as a subset of the cartesian product O \times S \times I, those being the sets of objects, signs, and interpretant signs respectively under discussion, determines three dyadic relations, which we may notate as follows:

  • \mathrm{proj}_{OS}{M}, the projection of M on the O \times S plane;
  • \mathrm{proj}_{OI}{M}, the projection of M on the O \times I plane;
  • \mathrm{proj}_{SI}{M}, the projection of M on the S \times I plane.

The following Figure illustrates the situation.

Aspects of a Sign Relation

Here is the critical point.  The triadic relation always determines the three dyadic projections but the three dyadic projections may or may not determine the triadic relation.  Thus we have two cases:

  • If the dyadic projections determine the triadic relation, that is, there is only one triadic relation that has those three projections, then the triadic relation is said to be projectively reducible to those three dyadic relations.
  • If the dyadic projections do not determine the triadic relation, that is, there is more than one triadic relation that has those same three projections, then the triadic relation is said to be projectively irreducible.

See the following article for concrete examples of both possibilities, a pair of generic triadic relations that are projectively irreducible and a pair of triadic sign relations that are projectively reducible to their dyadic projections.

Resources

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Relations & Their Relatives • Discussion 4

Posted on March 19, 2015 by Jon Awbrey

Re: Peirce List Discussion • Howard Pattee

We use this or that species of diagrams to represent a fraction of the properties, hardly ever all the properties, of the objects in an object domain.  The diagrams that Peirce developed to represent propositions about relations are quite handy so long as one grasps the conventions of representation, manipulation, and interpretation.  They are not all that different in kind from Feynman interaction diagrams or Penrose twistor diagrams.  Iconicity is nice when you can get it but one has to keep in mind that the map is not the territory, as the saying goes.

What do I see in a picture like this?

         s  
        /   
  o---<L    
        \   
         i

The ``L" brings to mind a triadic relation L, which collateral knowledge tells me is a set of triples.  What sort of triples?  The picture sets a place for them by means of the place-names ``o", ``s", ``i", in no particular order.  Without loss of generality I can take them up in the ordered triple (o, s, i).  All of this is just mnemonic machination meant to say that a typical element is (o, s, i) in L.  It’s up to me to remember that L is a subset of O \times S \times I, with o \in O, s \in S, and i \in I.  The diagram is just a mnemonic catalyst.  You have to know the codebook to decode it.

Pictures can victimize people, as Wittgenstein remarked and often exemplified.  One way people fall victim to pictures like the one depicted above is when they confuse a relation with a single one of its tuples.  That would represent a misunderstanding of what the picture is intended to represent.

Resources

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Animated Logical Graphs • 6

Posted on March 13, 2015 by Jon Awbrey

Re: Peirce List • Jim Willgoose

At root we are dealing with a genre of very abstract formal systems.  They have grammars that determine their well-formed expressions and rules that determine the permissible transformations among expressions, but they lack all logical meaning until we supply an interpretation.

The formal system Peirce developed for propositional logic and Spencer Brown resurrected for his Laws of Form admits a formal duality which allows it to be fleshed out with logical meanings in two distinct ways.  The two interpretations are employed in Peirce’s entitative graphs and existential graphs, respectively.  It is clear from everything they write that both authors are well aware of both interpretations, but Peirce would come to found his later developments on the existential sense while Spencer Brown favored the entitative sense in his expositions.

See the following readings for further discussion.

References

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Peirce’s 1880 “Algebra Of Logic” Chapter 3 • Selection 8

Posted on March 12, 2015 by Jon Awbrey

Chapter 3. The Logic of Relatives (cont.)

§4. Classification of Relatives (cont.)

227.   These different classes have the following relations.  Every negative of a concurrent and every alio-relative is both an opponent and the negative of a self-relative.  Every concurrent and every negative of an alio-relative is both a self-relative and the negative of an opponent.

There is only one relative which is both a concurrent and the negative of an alio-relative;  this is ‘identical with ──’.

There is only one relative which is at once an alio-relative and the negative of a concurrent;  this is the negative of the last, namely, ‘other than ──’.

The following pairs of classes are mutually exclusive, and divide all relatives between them:

Alio-relatives and self-relatives,
Concurrents and opponents,
Negatives of alio-relatives and negatives of self-relatives,
Negatives of concurrents and negatives of opponents.

No relative can be at once either an alio-relative or the negative of a concurrent, and at the same time either a concurrent or the negative of an alio-relative.

228.   We may append to the symbol of any relative a semicolon to convert it into an alio-relative of a higher order.  Thus (l;\!) will denote a ‘lover of ── that is not ──’.

References

  • Peirce, C.S. (1880), “On the Algebra of Logic”, American Journal of Mathematics 3, 15–57.  Collected Papers (CP 3.154–251), Chronological Edition (CE 4, 163–209).
  • Peirce, C.S., Collected Papers of Charles Sanders Peirce, vols. 1–6, Charles Hartshorne and Paul Weiss (eds.), vols. 7–8, Arthur W. Burks (ed.), Harvard University Press, Cambridge, MA, 1931–1935, 1958.  Volume 3 : Exact Logic, 1933.
  • Peirce, C.S., Writings of Charles S. Peirce : A Chronological Edition, Peirce Edition Project (eds.), Indiana University Press, Bloomington and Indianapolis, IN, 1981–.  Volume 4 (1879–1884), 1986.

Resources

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Scientific Attitude • 1

Posted on March 10, 2015 by Jon Awbrey

There is an outlook on the world I call the Scientific Attitude (SA).  There are times when the letter “A” is better served by apperception, approach, or attunement, but attitude will do for a start.

The scientific attitude accepts appearances, as appearances, but it does not stop there — it inquires after the possible realities that would both save and solve the appearances.

Reality is what persists and the scientific attitude accepts its limitation to what persists.  Thisness and thatness may come and go, but scientific knowledge rests on results that are reproducible.  It is knowledge of particulars in general terms.

Posted in Appearance, Epistemology, Generality, Haecceity, Inquiry, Inquiry Into Inquiry, Knowledge, Logic, Logic of Science, Philosophy of Science, Pragmatic Maxim, Reality, Reproducibility, Science, Scientific Attitude, Scientific Method | Tagged , , , , , , , , , , , , , , , | Leave a comment

Relations & Their Relatives • Discussion 3

Posted on March 5, 2015 by Jon Awbrey

Re: Peirce List • Edwina Taborsky • Howard Pattee

In the best mathematical terms, a triadic relation is a cartesian product of three sets together with a specified subset of that cartesian product.

Alternatively, one may think of a triadic relation as a set of 3-tuples contained in a specified cartesian product.

It is important to recognize that sets have formal properties that their elements do not.  The greatest number of category mistakes that bedevil errant discussions of relations and especially triadic sign relations arise from a failure to grasp this fact.

For example, the irreducibility (or indecomposability) of triadic relations is a property of sets-of-triples, not of individual triples.

See the articles under the following heading for concrete examples and further discussion.

Additional Resources

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Relations & Their Relatives • Discussion 2

Posted on March 4, 2015 by Jon Awbrey

Re: Peirce List • Helmut Raulien

In systems theory and engineering there is a well-recognized duality or complementarity between the dimensions of Control and Information, frequently cast in terms of action and perception, actuators and detectors, effectors and sensors, and a variety of other aliases.  There we find the dual devices of reachability matrices, representing the operations it takes to put a system in a given state, and observability matrices, representing the operations it takes to identify a system as being in a given state.

The appearance of matrices at this point, understood in the sense of 2-dimensional arrays of coefficients, may give us a clue to the dyadic character of the analysis and design models issuing from them.  And yet there is every opportunity in systems theory and engineering to open up the additional “elbow room” that triadic relations provide.

Resources

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

Posted in C.S. Peirce, Category Theory, Control, Cybernetics, Dyadic Relations, Information, Inquiry, Logic, Logic of Relatives, Mathematics, Peirce, Relation Theory, Semiosis, Semiotics, Sign Relations, Systems Theory, Triadic Relations | Tagged , , , , , , , , , , , , , , , , | 14 Comments

Peirce’s 1880 “Algebra Of Logic” Chapter 3 • Selection 7

Posted on February 28, 2015 by Jon Awbrey

Chapter 3. The Logic of Relatives (cont.)

§4. Classification of Relatives

225.   Individual relatives are of one or other of the two forms

\begin{array}{lll} \mathrm{A : A} & \qquad & \mathrm{A : B}, \end{array}

and simple relatives are negatives of one or other of these two forms.

226.   The forms of general relatives are of infinite variety, but the following may be particularly noticed.

Relatives may be divided into those all whose individual aggregants are of the form \mathrm{A : A} and those which contain individuals of the form \mathrm{A : B}.  The former may be called concurrents, the latter opponents.

Concurrents express a mere agreement among objects.  Such, for instance, is the relative ‘man that is ──’, and a similar relative may be formed from any term of singular reference.  We may denote such a relative by the symbol for the term of singular reference with a comma after it;  thus (m,\!) will denote ‘man that is ──’ if (m) denotes ‘man’.  In the same way a comma affixed to an n-fold relative will convert it into an (n + 1)-fold relative.  Thus,  (l) being ‘lover of ──’,  (l,\!) will be ‘lover that is ── of ──’.

The negative of a concurrent relative will be one each of whose simple components is of the form \mathrm{\overline{A : A}}, and the negative of an opponent relative will be one which has components of the form \mathrm{\overline{A : B}}.

We may also divide relatives into those which contain individual aggregants of the form \mathrm{A : A} and those which contain only aggregants of the form \mathrm{A : B}.  The former may be called self-relatives, the latter alio-relatives.  We also have negatives of self-relatives and negatives of alio-relatives.

References

  • Peirce, C.S. (1880), “On the Algebra of Logic”, American Journal of Mathematics 3, 15–57.  Collected Papers (CP 3.154–251), Chronological Edition (CE 4, 163–209).
  • Peirce, C.S., Collected Papers of Charles Sanders Peirce, vols. 1–6, Charles Hartshorne and Paul Weiss (eds.), vols. 7–8, Arthur W. Burks (ed.), Harvard University Press, Cambridge, MA, 1931–1935, 1958.  Volume 3 : Exact Logic, 1933.
  • Peirce, C.S., Writings of Charles S. Peirce : A Chronological Edition, Peirce Edition Project (eds.), Indiana University Press, Bloomington and Indianapolis, IN, 1981–.  Volume 4 (1879–1884), 1986.

Resources

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Relations & Their Relatives • Discussion 1

Posted on February 27, 2015 by Jon Awbrey

Re: Peirce List • Helmut Raulien

The divisor of relation signified by x|y is a dyadic relation on the set of positive integers \mathbb{M} and thus may be understood as a subset of the cartesian product \mathbb{M} \times \mathbb{M}.  It is an example of a partial order, while the less than or equal to relation signified by x \le y is an example of a total order relation.

The mathematics of relations can be applied most felicitously to semiotics but there we must bump the adicity or arity up to three.  We take any sign relation L to be subset of a cartesian product O \times S \times I, where O is the set of objects under consideration in a given discussion, S is the set of signs, and I is the set of interpretant signs involved in the same discussion.

One thing we need to understand is the sign relation L \subseteq O \times S \times I relevant to a given level of discussion may be rather more abstract than what we would call a sign process proper, that is, a structure extended through a dimension of time.  Indeed, many of the most powerful sign relations generate sign processes through iteration or recursion or similar operations.  In that event, the most penetrating analysis of the sign process or semiosis in view is achieved through grasping the generative sign relation at its core.

Resources

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

Posted in C.S. Peirce, Category Theory, Dyadic Relations, Logic, Logic of Relatives, Logical Graphs, Mathematics, Nominalism, Peirce, Pragmatism, Realism, Relation Theory, Semiotics, Sign Relations, Triadic Relations, Visualization | Tagged , , , , , , , , , , , , , , , | 15 Comments

Mathematical Demonstration and the Doctrine of Individuals • 2

Posted on February 23, 2015 by Jon Awbrey

Selection from C.S. Peirce’s “Logic Of Relatives” (1870)

In reference to the doctrine of individuals, two distinctions should be borne in mind.  The logical atom, or term not capable of logical division, must be one of which every predicate may be universally affirmed or denied.  For, let \mathrm{A} be such a term.  Then, if it is neither true that all \mathrm{A} is \mathrm{X} nor that no \mathrm{A} is \mathrm{X}, it must be true that some \mathrm{A} is \mathrm{X} and some \mathrm{A} is not \mathrm{X};  and therefore \mathrm{A} may be divided into \mathrm{A} that is \mathrm{X} and \mathrm{A} that is not \mathrm{X}, which is contrary to its nature as a logical atom.

Such a term can be realized neither in thought nor in sense.

Not in sense, because our organs of sense are special — the eye, for example, not immediately informing us of taste, so that an image on the retina is indeterminate in respect to sweetness and non-sweetness.  When I see a thing, I do not see that it is not sweet, nor do I see that it is sweet;  and therefore what I see is capable of logical division into the sweet and the not sweet.  It is customary to assume that visual images are absolutely determinate in respect to color, but even this may be doubted.  I know no facts which prove that there is never the least vagueness in the immediate sensation.

In thought, an absolutely determinate term cannot be realized, because, not being given by sense, such a concept would have to be formed by synthesis, and there would be no end to the synthesis because there is no limit to the number of possible predicates.

A logical atom, then, like a point in space, would involve for its precise determination an endless process.  We can only say, in a general way, that a term, however determinate, may be made more determinate still, but not that it can be made absolutely determinate.  Such a term as “the second Philip of Macedon” is still capable of logical division — into Philip drunk and Philip sober, for example;  but we call it individual because that which is denoted by it is in only one place at one time.  It is a term not absolutely indivisible, but indivisible as long as we neglect differences of time and the differences which accompany them.  Such differences we habitually disregard in the logical division of substances.  In the division of relations, etc., we do not, of course, disregard these differences, but we disregard some others.  There is nothing to prevent almost any sort of difference from being conventionally neglected in some discourse, and if I be a term which in consequence of such neglect becomes indivisible in that discourse, we have in that discourse,

[I] = 1.

This distinction between the absolutely indivisible and that which is one in number from a particular point of view is shadowed forth in the two words individual (τὸ ἄτομον) and singular (τὸ καθ᾿ ἕκαστον);  but as those who have used the word individual have not been aware that absolute individuality is merely ideal, it has come to be used in a more general sense.  (CP 3.93)

Note

Peirce explains his use of the square bracket notation at CP 3.65.

I propose to denote the number of a logical term by enclosing the term in square brackets, thus, [t].

The number of an absolute term, as in the case of I, is defined as the number of individuals it denotes.

References

  • Peirce, C.S. (1870), “Description of a Notation for the Logic of Relatives, Resulting from an Amplification of the Conceptions of Boole’s Calculus of Logic”, Memoirs of the American Academy of Arts and Sciences 9, 317–378, 26 January 1870.  Reprinted, Collected Papers 3.45–149, Chronological Edition 2, 359–429.  Online (1) (2) (3).
  • Peirce, C.S., Collected Papers of Charles Sanders Peirce, vols. 1–6, Charles Hartshorne and Paul Weiss (eds.), vols. 7–8, Arthur W. Burks (ed.), Harvard University Press, Cambridge, MA, 1931–1935, 1958.
  • Peirce, C.S., Writings of Charles S. Peirce : A Chronological Edition, Peirce Edition Project (eds.), Indiana University Press, Bloomington and Indianapolis, IN, 1981–.

Resources

Posted in C.S. Peirce, Deduction, Doctrine of Individuals, Foundations of Mathematics, Identity, Information = Comprehension × Extension, Logic, Logic of Relatives, Mathematical Demonstration, Mathematics, Relation Theory | Tagged , , , , , , , , , , | 6 Comments