Preproceedings 2006 - Austrian Ludwig Wittgenstein Society

2) ϕ( P ℑ ∪ Ρ ℑ )=ϕ( P ℑ ) ∪ ϕ( R ℑ )
ϕ is therefore a Boolean isomorphism, i.e. a bijection
between sets that preserve complement, union, and
product.
We further require that for each relation P ℑ there be:
2.3) ϕ( P ℑ )≠∅ if and only if P ℑ ≠∅
If two S-structures are Boolean isomorphic and
satisfy 2.3) there exists a subset e(F) of the set of Sformulas
such that, given two formulas F and G, there are
satisfied the conditions:
2.4) i) e(¬F) ⇔ ¬e(Fi)
ii) e(F∨ G)⇔ e(F ) ∨ e(G)
iii) e(∃x (Fx)) ⇔ ∃x e(Fx)
iv) F⇒G if and only if e(F)⇒e(G).
In particular let At ℑ 1...... At ℑ n, the formulas that
appear as disjoint in the normal form of H 1
ℑ and At ℜ 1......
A ℜ n, and those that appear in the normal form of H 1
ℜ,, and
let us give the definition:
2.5) Given a monadic S-signature for each S-formula F:
e(F) ⇔∨{e(Ati)| Ati ∈ AtF}
where:
e(Ati)⇔Ati if and only if Ati∉{ At ℑ 1,..…,At ℑ n ,At ℜ 1,..…, At ℜ n
};
e(At ℜ i)⇔At ℑ i ;
e(At ℑ i)⇔At ℜ i .
The e(L) set of formulas so defined satisfies the
conditions 2.4) i) -iv).
Let us now consider the new signature S e ={e(Pi)|Pi
∈S} generated by the images of primitive predicates in S
and apply to it the usual definition of formula, generating
the set L e of formulas having the following properties:
2.6) i) the set of L e formulas coincides with that
generated by S.
ii) the set of atomic formulas generated by S e is different
to that generated by S.
iii) the set of sentences generated by S e is the same,
modulo logical equivalence, as that generated by S.
An immediate consequence of definition 2.5) is:
2.7) i) e(H 1
ℑ)⇔H 1
ℜ
ii) e(H 1
ℜ)⇔H 1
ℑ.
So for every sentence G:
2.8) H 1
ℜ⇒G if and only if e(H 1
ℑ )⇒ G;
H 1
ℑ⇒ G if and only if e(H 1
ℜ )⇒ G .
Let us now consider any primitive predicate Pi. We
have, for 2.4) iv):
2.9) i) H 1
ℜ⇒Pix if and only if e(H 1
ℜ)⇒e(Pix);
ii) H 1
ℑ⇒Pix if and only if e(H 1
ℑ)⇒e(Pix).
The Fraissè Theorem and Goodman’s Paradox - Tiziano Stradoni
Hence e(H 1
ℑ ) and H 1
ℑ are satisfied by the same
structure, as is e(H 1
ℜ) and
H 1
ℜ . In particular:
2.9) i) ℜ⇒ e(H 1
ℜ);
ii) ℑ⇒ e(H 1
ℑ).
We thus arrive at the conclusion that the two
structures ℜ, ℑ are elementarily equivalent:
2.10) For each sentence F: ℜ⇒F if and only if ℑ⇒F.
Therefore ℑ≡ℜ.
Proof:
ℜ⇒F if and only if H 1
ℜ ⇒F, for the theorem on
disjunctive normal forms;
H 1
ℜ⇒ F if and only if e(H 1
ℑ)⇒F, since H 1
ℜ ⇔e(H 1
ℑ );
e(H 1
ℑ)⇒F if and only if ℑ⇒F, for 2.9) and the theorem
on normal disjunctive forms.
By assumption 2.0) however, the two structures are
neither finitely isomorphic nor isomorphic, contradicting the
Fraissè theorem.
More generally it can be demonstrated that:
2.11) Two monadic f.o. structures are elementarily
equivalent if and only if they are Boolean isomorphic.
Finally we can introduce the following corollary:
2.12) Given a monadic f.o. structure ℜ such that
H 1
ℜ⇒¬∃ x( H 0
ix) for at least one H 0
ix, always there is at
least one structure elementarily equivalent but not
isomorphic to it.
3) An Application: the Goodman Paradox.
For the sake of simplicity in this section we will only
consider finite structures: in this case in fact isomorphism
and finite isomorphism coincide and the Fraissè theorem is
resolved by the assertion that two S-structures are
elementarily equivalent if and only if they are isomorphic.
Given the signature S={R,T} , let us consider the
two structures not isomorphic between each other:
3.0) Φ=
3.1) Θ=
are:
The two Hintikka S-formulas satisfied by Φ and Θ
3.2) H 1
Φ⇔∃ x(Rx ∧Tx) ∧∃ x(Rx ∧¬Tx) ∧¬∃ x(¬Rx ∧Tx)
∧¬∃ x(¬Rx ∧¬Tx)
3.3) H 1
Θ⇔∃ x(Rx ∧Tx) ∧¬∃ x(Rx ∧¬Tx) ∧¬∃ x(¬Rx ∧Tx)
∧∃ x(¬Rx ∧¬Tx)
Let us now consider an arbitrary 5 measure function
μ(T, Ev) of inductive support offered by the evidence Ev to
the generalisation T, such as to satisfy the following
conditions:
5 But in the Hintikka tradition: see [5]
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