Icon Index Symbol • 5

Questions Concerning Certain Faculties Claimed For Signs

Re: Peirce List Discussion • Helmut Raulien

Given that sign relations are special cases of triadic relations, we can get significant insight into the structures of both cases by examining a few simple examples of triadic relations, without getting distracted by all the extra features that come into play with sign relations.

When I’m talking about a k-place relation L I’ll always be thinking about a set of k-tuples.  Each k-tuple has the form:

(x_1, x_2, \ldots, x_{k-1}, x_k),

or, as Peirce often wrote them:

x_1 : x_2 : \ldots : x_{k-1} : x_k.

Of course, L could be a set of one k-tuple but that would be counted a trivial case.

That sums up the extensional view of k-place relations, so far as we need it for now.

Using a single letter like ``L" to refer to a set of k-tuples is already the genesis of an intensional view, since we now think of the elements of L as having some property in common, even if it’s only their membership in L.  When we turn to devising some sort of formalism for working with relations in general, whether it’s an algebra, logical calculus, or graph-theoretic notation, it’s in the nature of the task to “unify the manifold”, to represent a many as a one, to express a set of many tuples by means of a single sign.  That can be a great convenience, producing formalisms of significant power, but failing to discern the many in the one can lead to no end of confusion.

To be continued …


This entry was posted in Abduction, Algorithms, Artificial Intelligence, Automated Research Tools, C.S. Peirce, Cognitive Science, Computer Science, Data Structures, Deduction, Functional Logic, Icon Index Symbol, Induction, Inquiry, Inquiry Driven Systems, Inquiry Into Inquiry, Intelligent Systems, Interpretive Frameworks, Knowledge Representation, Logic, Logic of Relatives, Logic of Science, Logical Graphs, Mathematics, Objective Frameworks, Peirce, Relation Theory, Relative Membership, Scientific Method, Semiotics, Set Theory, Sign Relations, Systems Theory, Triadic Relations and tagged , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , . Bookmark the permalink.

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