Reduction and Elimination in Philosophy and the Sciences

Interpretability Relations of Weak Theories of Truth
Martin Fischer, Leuven, Belgium
1. Introduction
Axiomatic theories of truth are understood as extensions of
a syntactic base theory which is often taken to be Peano
Arithmetic, PA . One way to measure the strength of a
theory of truth is to take into account which formulas of
arithmetic it proofs. Weak theories in this respect are theories
that do not prove more than PA itself, theories that are
conservative extensions of PA . The concept of conservativity
has gained some interest in formalizing philosophical
criteria. This is also the case in the debate on truth, in
which conservativity is expected to explain the `no substance'
claim of deflationism. For theories of truth conservativity
over PA alone seems to be a very crude measure
since it does not differentiate between different conservative
theories of truth which have quite different properties
and prove different formulas containing the truth predicate.
A comparison of the truth-theoretic strength of
theories of truth is desirable. A direct comparison of the
truth-theoretic strength is the subset relation but it is only
a partial order so that not all theories can be compared.
Another measure of the strength of a theory of truth would
be their interpretability relations to other theories
especially their interpretability or noninterpretability in PA .
The most famous of these interpretability relations is
relative interpretability, introduced in (Tarski et al. 1953),
and it is a good measure for PA as base theory. On the
one hand the less restricted version of local interpretability
collapses in this case into relative interpretability. On the
other hand Tarskis theorem of undefinability of truth shows
that there is no definitional extension of PA by a one place
predicate τ , PA (τ ), so that PA (τ ) proves τ( ϕ)
↔ ϕ for
all sentences ϕ of the language of arithmetic.
2. Axiomatic theories
LA is the language of arithmetic and Lτ : = A {τ ∪ . The
arithmetical theories are as usual. For the interpretability
considerations take the arithmetic theories to be formulated
with predicate- instead of functionsymbols.
96
L }
( Ind P ) P( 0)
∧ ∀x(
P(
x)
→ P(
x + 1))
→ ∀x(
P(
x))
Q is Robinson Arithmetic and PA is Peano Arithmetic,
that is P A L Ind Q ∪ ) ⋅ ( , where P A L Ind ⋅ ) ( is the set of sentences
that result from replacing P in ( Ind P ) by a formula
of L A with at least one free variable. Accordingly,
IΣ k = Q ∪ ( IndP
) ⋅ Σk
.
Assume that L A contains the relevant syntactical
vocabulary: ‘ Ct ’ for closed term of L A , ‘ Sent ’ for
sentence of L A , ‘ Form 1’
for formula of LA with one free
variable, and so on, such that PA proves the relevant
syntactical theorems. Especially if m is the gödelnumber
of a formula ϕ (x)
with one free variable x and n of a
term t , then m (n)
is the gödelnumber of the substitution
τ
of the free variable x in ϕ (x)
by the numeral of t . PA is
PA formulated in the language L τ . A theory of truth T is
τ
a Lτ -theory with PA ⊆ T .
tot ( x)
: ⇔ Form1(
x)
∧ ∀y(
τ ( x(
y))
∨τ
( ¬ & x(
y )))
Disquotational theories of truth are formulated with a
scheme of T-biconditionals:
( TB P )
τ ( P) ↔ P
( UTB P ) ∀x(
τ ( P(
x&
) ) ↔ P(
x)).
Compositional axioms are the universally quantified versions
of the following formulas:
(C 1)
Ct ( x)
∧ Ct(
y ) → ( τ ( x = & y)
↔ val(
x)
= val(
y)).
(C 2)
Ct( x)
∧ Ct(
y)
→ ( τ ( x ≠&
y)
↔ val(
x)
≠ val(
y)).
(C 3)
Sent( x)
∧ Sent(
y)
→ ( τ( x ∧&
y ) ↔ τ(
x)
∧τ
( y)).
(C 4)
Sent ( x)
∧ Sent(
y)
→ ( τ ( ¬ & x ∧&
y)
↔ τ(
¬ & x)
∨ τ ( ¬ & y)).
(C 5)
Sent( ∀&
yx)
→ ( τ( ∀&
yx)
↔ ∀z(
τ ( x(
z)))).
(C 6)
Sent ( ¬ & ∀&
yx)
→ ( τ ( ¬ & ∀&
yx)
↔ ∃z(
τ ( ¬ & x(
z)))).
(C 7)
Sent( x)
→ ( τ ( ¬ & ¬ & x)
↔ τ ( x)).
(C 8)
Sent( x)
→ ( τ ( ¬ & x)
↔ ¬ τ ( x)).
The axiom of internal induction for total formulas is:
( It I)
∀ x( tot(
x)
∧τ
( x(
0))
∧ ∀y(
τ(
x(
y )) → τ(
x(
y + 1)))
→ ∀y(
τ(
x(
y )))).
The relevant theories are:
TB : = Q ∪ ( TBP
) ⋅ LA
∪ ( IndP
) ⋅ Lτ
UTB : = Q ∪ ( UTBP
) ⋅ LA
∪ ( IndP
) ⋅ Lτ
PT : PA ( C1)
( C7)
r
= ∪ −
−
PT : = PA ∪ ( C1)
− ( C7)
∪ ( ItI
)
PT : = PA ∪ ( C1)
− ( C7)
∪ ( IndP
) ⋅ Lτ
TC : PA ( C1)
( C8)
r
= ∪ −
−
TC : = PA ∪ ( C1)
− ( C8)
∪ ( ItI
)
r
TC is also known as PA (S)
and TC as T (PA)
.
3. Interpretability
Some basic results:
(i) TB ⊂ UTB ⊂ PT = TC
−
(ii) IΣ1 ∪ ( C1),
( C3),
( C5),
( C8)
∪ ( It
I)
= TC
−
(iii) IΣ1 ∪ ( C1)
− ( C7)
∪ ( It
I)
= PT
− −
(iv) PT , TC are finitely axiomatizable.
r r
(v) TB,
UTB,
PT , TC , PT,
TC are not finitely axiomatizable.
r r −
(vi) TB,
UTB,
PT , TC , PT are conservative extensions of PA .
−
(vii) TC , PT,
TC are nonconservative extensions of PA .
Definition
Let S, T be theories formulated in L S, LT
. Then
T is a pure extension of S iff T is an extension of S and L T = LS
.
T is reflexive iff T proves Con Δ for all finite Δ ⊆ T.
T is essentially reflexive iff all pure extensions of T are reflexive.
T has full induction iff for all formulas ϕ of L T : T proves
ϕ( 0)
∧ ∀x(
ϕ(
x)
→ ϕ(
x + 1))
→ ∀xϕ(
x)
.
Full induction and reflexivity are connected in the following
way, as shown for example in (Hájek/Pudlák 1993, p.189):
Lemma 1
If PA ⊆ T and T has full induction, then T is reflexive.