A Minimal Identity Relation

4.2 Identity in P, cont.

Minimal
Identity

For particulars in P, as for individuals and haecceities in HI and complexes in C, identity is weakly reflexive (T34), symmetric (T35) and transitive (T36); identicals are indiscernible (T37#); and self-identical indiscernibles identical (T38#).

T34 x(y(x = y)x = x)

T35 xy(x = yy = x)

T36 xyz(x = y & y = zx = z)

T37# xyz(x = y(x emb Zy emb Z))

T38# xy((x = x & y = y)(z(x emb Zy emb Z)x = y))

Identity &
Unity

As in HI and C, moreover, in P it cannot be established whether identity is reflexive or indiscernibles are identical. For the self-identity of a particular is bound up in P with the unity of the individual and haecceity which constitute it.

T39 x(x = xx ex X)

This unity depends in turn upon a haecceity's being exemplified,

T40 x(x ex Xy(y ex X))

so that every particular is self-identical, and identity totally reflexive if, and only if, every haecceity is exemplified.

T41 x(x = x)xy(y ex X)

However, whether every, or any, haecceity is exemplified is not specified in P, for P specifies just those relations of individuals and haecceities which ground properties that are criterial for identity. Consequently, whether identity is totally or partially reflexive, or indiscernibles are identical, cannot be established in P. For these properties differentiate species within a genus without being relevant to the genus itself.

Classical and Non-Classical Objects