Icon Index Symbol • 7

Questions Concerning Certain Faculties Claimed For Signs

Re: Peirce List Discussion • Helmut Raulien

Looking back over many previous discussions, I think one of the main things keeping people from being on the same page, or even being able to understand what others write on their individual pages, is the question of what makes a relation.

There’s a big difference between a single ordered tuple, say, (a_1, a_2, \ldots, a_k), and a whole set of ordered tuples that it takes to make up a k-place relation.  The language we use to get a handle on the structure of relations goes like this:

Say the variable x_1 ranges over the set X_1,
and the variable x_2 ranges over the set X_2,
\cdots
and the variable x_k ranges over the set X_k.

Then the set of all possible k-tuples (x_1, x_2, \ldots, x_k) ranges over a set that is notated as X_1 \times X_2 \times \ldots \times X_k and called the “cartesian product” of the “domains” X_1 to X_k.

There are two different ways in common use of defining a k-place relation.

  1. Some define a relation L on the domains X_1 to X_k as a subset of the cartesian product X_1 \times \ldots \times X_k, in symbols, L \subseteq X_1 \times \ldots \times X_k.
  2. Others like to make the domains of the relation an explicit part of the definition, saying that a relation L is a list of domains plus a subset of their cartesian product.

Sounds like a mess but it’s usually pretty easy to translate between the two conventions, so long as one watches out for the difference.

By way of a geometric image, the cartesian product X_1 \times \ldots \times X_k may be viewed as a space in which many different relations reside, each one cutting a different figure in that space.

To be continued …

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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.

One Response to Icon Index Symbol • 7

  1. Pingback: Icon Index Symbol • 8 | Inquiry Into Inquiry

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