Post : Peirce's 1870 “Logic Of Relatives” • Comment 12.1
http://inquiryintoinquiry.com/2014/06/10/peirces-1870-logic-of-relatives-%e2%80%a2-comment-12-1/
Posted : June 10, 2014 at 12:00 am
Author : Jon Awbrey
Peircers,
To get a better sense of why the above formulas mean what they do, and to prepare the ground for
understanding more complex relational expressions, it will help to assemble the following materials
and definitions:
• X is a set singled out in a particular discussion
as the ''universe of discourse''.
• W ⊆ X is the monadic relation, or set, whose elements fall under
the absolute term w = woman. The elements of W are referred to
as the ''denotation'' or the ''extension'' of the term w.
• L ⊆ X × X is the dyadic relation associated with
the relative term l = lover of____.
• S ⊆ X × X is the dyadic relation associated with
the relative term s = servant of____.
• *W* = (*W*_x) = Mat(W) = Mat(w) is the 1-dimensional matrix
representation of the set W and the term w.
• *L* = (*L*_xy) = Mat(*L*) = Mat(l) is the 2-dimensional matrix
representation of the relation L and the relative term l.
• *S* = (*S*_xy) = Mat(S) = Mat(s) is the 2-dimensional matrix
representation of the relation S and the relative term s.
Recalling a few definitions, the ''local flags'' of the relation L are given as
follows:
• u ⋆ L
= L_{u@1}
= {(u, x) ∈ L}
= the ordered pairs in L that have u in the 1st place.
• L ⋆ v
= L_{v@2}
= {(x, v) ∈ L}
= the ordered pairs in L that have v in the 2nd place.
The ''applications'' of the relation L are defined as follows:
• u ∙ L
= proj_2 (u ⋆ L)
= {x ∈ X : (u, x) ∈ L}
= loved by u.
• L ∙ v
= proj_1 (L ⋆ v)
= {x ∈ X : (x, v) ∈ L}
= lover of v.
Regards,
Jon
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