I 140
Identity/Bigelow/Pargetter: we understand this here as a 2-digit predicate, we do not need to expand the language.
I 141
Axioms:
A19. (x)(x = x)
A20. (a u ~a (σ/λ) > σ unequal λ)
Everyday language translation: if something is true of something and not true of something, then these two things cannot be identical.
I 141
Contingent Identity/Bigelow/Pargetter: these two axioms have a surprising consequence: namely that all identity is necessary.
Cf. >
Identity/Kripke.
There is then no contingent identity. Non-identity is then also necessary. So the following can be proved as theorems:
NI. (x = y) > N(x = y)
NNI. (x unequal y) > N(x unequal y)
Semantic rule: then causes an identity statement to be true in all possible worlds or true in none.
>
Possible worlds, >
Necessity, >
Truth.
Valuation rule/identity/Bigelow/Pargetter: V (=) (c, c) = W
W: is the set of all possible worlds.
Identity statements/Bigelow/Pargetter: are then either necessary or impossible.
This is surprising and shows another illustration of the interplay between semantics and ontology.
>
Semantics, >
Ontology.
Ontology/Bigelow/Pargetter: is what is suggested to us by a streamlined and plausible semantics.
Identity/Science/Bigelow/Pargetter: in the history of science there have often been discoveries that have shown us that things we thought were different are identical.
Cf. >
Natural kinds/Putnam, >
Progress, >
Science, >
Knowledge.
I 143
Now one should think that these are contingent identities.
>
Contingency.
Contingent Identity/Semantics/Bigelow/Pargetter: if they like contingent identity, they would have to change the semantics. And that is not hard:
Def Diversity/new: instead of saying that two things are different, if something is true of one but not true of the other, we could say that something non-modal is true of one, but not true of the other.
That brings out some new systems.
>Cf. >
Leibniz principle, >
Indistinguishability,
>
Distinctions.
It is interesting to note that some of these systems verify NNI while they continue to falsify NI.
For example, it is more difficult to allow New York and Miami to be one and the same city than to allow Miami to be two cities.
Identity/BigelowVsContingent Identity/Bigelow/Pargetter: we should let the semantics decide and say that there is simply no contingent identity.
Contingent Identity/Bigelow/Pargetter: instead of changing the semantics and then to allow it nevertheless, we should rather explain why they seem to exist: e.g.
Theory of descriptions/Russell/Bigelow/Pargetter: provides a means to reconcile contingently with necessary identities: assertions of the form
the F = the G
can be analyzed as contingent by saying that the properties F and G are co-instantiated by a single thing. This is still compatible with the necessary self-identity.
>
Theory of descriptions/Russell, >
Descriptions.
Bigelow/Pargetter: through descriptions most contingent identities are explained away.
I 144
Introverted Realism/Bigelow/Pargetter: (see above Chapter 1) introverted realism, as can be seen here, can reinforce the extroverted realism from which it originated.
>
Realism/Bigelow, >
Realism.
