To enable the forms of identity propositions to be expressed, a dyadic predicate constant is added to LPC, for which the most usual notation is = (written between, rather than before, its arguments).
We note here that quite similar expressions to the first identity of Proposition 3.7 are given by Kamano [[3], Proposition 2.4], Rim et al. [[8], Theorem 2.7] and Tsumura [[10], (1)].
Now, we claim that lim n → ∞ ∥ x n − J r x n ∥ = 0 for a fixed number r such that ε > r > 0. In fact, using the resolvent identity in Proposition 2.2, we have ∥ J r n x n − J r x n ∥ = ∥ J r ( r r n x n + ( 1 − r r n ) J r n x n ) − J r x n ∥ ≤ ( 1 − r r n ) ∥ x n − J r n x n ∥ ≤ ∥ x n − J r n x n ∥. (4.27).
Furthermore, we claim that lim n → ∞ ∥ x n − J r x n ∥ = 0 for a fixed number r such that ε > r > 0. In fact, taking into account the resolvent identity in Proposition 1.1, we have ∥ J r n x n − J r x n ∥ = ∥ J r ( r r n x n + ( 1 − r r n ) J r n x n ) − J r x n ∥ ≤ ( 1 − r r n ) ∥ x n − J r n x n ∥ ≤ ∥ x n − J r n x n ∥. (3.29).
But given that sets of possible worlds and sets of possible objects figure in important philosophical discussions concerning the identity of propositions and properties, these sets seem important.
If we set (g=I), the identity map, in Proposition 3.3, then Proposition 2.3 follows.
This results in Bolzano's identity criterion for propositions: Two propositions [A1 has b1] and [A2 has b2] are identical iff (i.e., if and only if) [A1] = [A2] and [b1] = [b2].
Second, proposition identity might be reduced to property identity in the manner of Myhill (1963) and Zalta (1983).
Certain wffs of LPC-with-identity express propositions about the number of things that possess a given property.
