RE: SUO: Irreducible Triadicity
>Dear Lee,
>
>I thought we had gotten beyond this. Indeed I thought we had got to the
>point where it was demonstrated (by Pat, but by me previously) that in
>representation terms anything can be reduced to dyadic relations.
>
>This caused a change in tack to saying that there are axioms that require
>more than 2 arguments (not a particular surprise to me). The question
>remaining is then whether there are any axioms that cannot be reduced to
>some set of independent axioms that involve only 3 elements.
That has to depend on what is meant by 'independent'. If it means not
sharing any vocabulary at all, then it is impossible, in general, to
reduce an assertion involving an n-ary relation to a set of
completely disconnected axioms. If the same name can occur in more
than one axiom, then it is always possible to express any n-ary
relation assertion as a set of axioms using EITHER some trinary
relations OR at most binary relations, but mentioning some name at
least three times. These are the two ways that the graph-theoretic
result can be applied to logic. If you think of the graph nodes as
relations (Peircian relational graphs), you get the trinary
relations; if you think of the arcs as relations (semantic networks)
you get the binary/3names rule. So there is always a threeness in
there somewhere, indeed. Whether you think this reflects anything
important about the nature of things (or of signs) must I guess be a
personal matter. I don't, myself, but then I like sushi more than
chocolate.
Pat
PS I solemnly SWEAR I will not send anything else to SUO about triadicityness.
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