Reduction and Elimination in Philosophy and the Sciences

tence is valid if and only if it is true in any naming structure.
A sentence is complete -valid if it is true in any complete
naming structure. ∆
It turns out that this semantics is in a sense
equivalent to set-theoretic semantics.
Theorem. For any QNL sentence A, A is S-valid if and
only if A is complete-valid. ∆
Next, let's take a look at the modal factor involved in
this semantics by comparing it to a certain two-sorted firstorder
modal logic of naming (MLN). The language of MLN
contains two sorts of variables: individual variables
and name variables
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x, y, z, x1, x2, ..., y1, y2, ..., z1, z2, ...
n, m, o, n1, n2, ..., m1, m2, ..., o1, o2, ...
Besides, it contains quantifiers ranging over objects of
those two sorts, the classical propositional connectives,
two modal operators (say, ◊ for possibility and � for necessity),
no predicate variables, the identity symbol and one
two-place predicate constant D. Formation rules are standard
(the only new thing is that D takes name variables as
first arguments and individual variables as second arguments).
Models of MLN are just naming structures.
Definition. An MLN-interpretation is a tuple ,
where M = is a naming structure, = w is in M
and i is (a) undefined if N is empty, and (b) maps all
individual variables into I and all name variables into N
otherwise. Satisfaction of MLN-formulas in interpretations
is defined as follows:
models D(n, x) iff i is defined and
is in d.
models t1 = t2 iff i(t1) = i(t2), where each ti is
one of the variables (arguments of = don’t have to
be of the same sort).
The clauses for negation and conjunction are standard.
models (∃t)A iff there is an interpretation i'
(mapping individual variables into I and name variables
into N) that differs from i only in what it assigns
to t and = w' in W such that w' is
at most of level k+1 (that is, (∀i > k) N'i = ∅), both Rww'
and:
⎛ ⎞
( ∃x∈N′ k+ 1)(
∀y) ⎜y ∈U Nn →( x, y ∈d ↔ x∈A) ⎟
⎝ n≥0, n≤k ⎠
M is said to be cumulatively complete iff for any w in W,
M is w-cumulatively complete. ∆
If w is of level k (Nk is the highest non-empty element
of w), then the domain of names of w (denoted by DN(w)) is
the union of of all Ni for 1 ≤ I ≤ k, and the domain of
objects of w (denoted by DO(w)) is the union of DN(w) and I.
Now, I will define a language that resembles the
language of set theory, and the satisfaction relation for this
language. The language of cumulative naming logic
(CNL) contains the standard (first-order) logical symbols
(including identity), variables xi that (under an
interpretation) will take pure individuals as values,