Re: “On the Paradigm of Experience Appro priate for Semiotic”

* Comments on the Peirce List slow reading of Joseph Ransdell,
   "On the Paradigm of Experience Appropriate for Semiotic",
   http://www.cspeirce.com/menu/library/aboutcsp/ransdell/paradigm.htm

Re: Comments by Claudio Guerri (cont.)

I realize that many of us have been through these sorts of discussions
many times before, so let me just highlight what I consider to be some
of the most important points.

1. We must not confuse the roles in a sign relation or the components
    of a sign relational 3-tuple, that is, Object, Sign, Interpretant,
    with the Peircean categories of Firstness, Secondness, Thirdness.
    These two sets of concepts reside at very different logical levels,
    as one can tell from the fact that Peirce described his Categories
    as "Predicaments", that is, predicates of predicates.

To make the shortest possible shrift of the matter, Category k is the
category of k-adic predicates or k-adic relations. Viewing categories
as Aristotle initially described them, as disambiguating references or
devices for resolving the equivocation of terms by indicating the type
of object intended for their interpretation, Peirce's claim that three
categories are necessary and sufficient for the purposes of logic says
that a properly designed system of logic can resolve all equivocation
in just three levels or steps.

For a more detailed discussion, here is an excerpt from the section I wrote
on Peircean categories for the Wikipedia article on Peirce several years ago.
This material can now be found at the MyWikiBiz article on Peirce.

Cf. http://mywikibiz.com/Charles_Sanders_Peirce#Theory_of_categories

In the logic of Aristotle categories are adjuncts to reasoning that are designed
to resolve equivocations and thus to prepare ambiguous signs, that are otherwise
recalcitrant to being ruled by logic, for the application of logical laws. An
equivocation is a variation in meaning, or a manifold of sign senses, and so
Peirce's claim that three categories are sufficient amounts to an assertion
that all manifolds of meaning can be unified in just three steps.

The following passage is critical to the understanding of Peirce's Categories:

CSP: 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.

CSP: 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.

CSP: 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?

CSP: 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).

CSP: 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.

CSP: Peirce, CP 4.549, "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 the fact that Peirce's Categories,
or "Predicaments", are predicates of predicates. Meaningful predicates have both
extension and intension, so predicates of predicates get their meanings from at
least two sources of information, namely, the classes of relations and the
qualities of qualities to which they refer. Considerations like these tend
to generate hierarchies of subject matters, extending through what is
traditionally called the logic of second intensions, or what is handled
very roughly by second order logic in contemporary parlance, and continuing
onward through higher intensions, 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 that 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,
respectively. Taken in full generality, k-ness may be understood as referring to
those properties that all k-adic relations have in common. Peirce's distinctive
claim is that a type hierarchy of three levels is generative of all that we need
in logic.

Part of the justification for Peirce's claim that three categories are both
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.

Jon

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