Reduction and Elimination in Philosophy and the Sciences

Let T I be the minimal theory of truth with full induction:
TI : = Q ∪ ( IndP
) ⋅ Lτ
.
Theorem 1
TI and every pure extension of T I is essentially reflexive.
Corollary 1
TB , UTB,
PT,
TC are essentially reflexive.
For other conservative theories with restricted induction it is
far more complicated to show that they are reflexive. One
example is Tarski's compositional theory with restricted
r
induction. In (Halbach 1999) the conservativity of TC over
PA is proved by a cut elimination proof. This proof has at
least two advantages in comparison to a model theoretic
proof along the lines of (Kotlarski et al. 1981). First it can
also be used for other base theories especially for all IΣ k
r
with k ∈ω
. ( IΣ
) : = IΣ
∪ ( C1)
− ( C8).
TC k k
Second it can be formalised in a way that makes it
provable in PA . So we get:
Theorem 2
For every k ∈ω
: PA proves
∀ x(
Sent(
x)
∧ Pr r ( x)
→ Pr ( x))
TC ( IΣ
) IΣk
k
r
With this it can be shown that TC proves the consistency
of all of its finite subtheories.
Theorem 3
r r
PT , TC are reflexive.
Proof:
Theorem 2 shows that for any k ∈ω
: PA proves
ConIΣ
→ Con r . Since PA is reflexive, all IΣ k are
k TC ( IΣk
)
finitely axiomatizable and
r
TC proves
Con r
TC ( IΣk
)
Interpretability Relations of Weak Theories of Truth — Martin Fischer
r
TC
PA ⊆ , for every k ∈ω
:
, which is enough to show that
r
r
TC is reflexive. A similar argument shows that PT is
reflexive. □
Theorem 4
− −
PT , TK are not reflexive.
Corollary 2
r r
PT , TC are not essentially reflexive.
For extensions of PA reflexivity and relative interpretability,
p , are connected by Π1 -conservativity in the following
way as shown for example in (Lindström 1997):
Theorem 5
Let PA ⊆ T . PA
conservative over PA .
This shows that:
T p iff T is reflexive and 1
Π -
Theorem 6
r r
TB , UTB,
PT , TC are relatively interpretable in PA .
On the other hand it is easy to see that theories that are
not reflexive or Π1 -conservative over PA are also not
interpretable in it.
Theorem 7
− −
PT , TC , PT,
TC are not relatively interpretable in PA .
Relative interpretability in PA implies reflexivity and Π1 -
conservativity over PA but it does not not imply conservativity
over PA .
Theorem 8
TB + ¬ Contb
is relatively interpretable in PA but
not conservative over PA .
4. Weak Theories of Truth
Considering the set TT of all theories of truth, that is theories
formulated in L τ and containing PA , there are the
two subsets with one criterion of weakness:
CTT : = { T ∈TT T is a conservative extension of PA } .
ITT : = T ∈ TT T p PA } .
{
The combination of these two criteria allow a more fine
grained picture of theories of truth, especially for weak
theories. In the preceding sections it was shown that
CTT, ITT , their complements and their combinations are
nonempty. There are four possibilities of combination.
Strong theories of truth not fulfilling either of both criteria
will not be investigated here.
The set ITT consists of theories that are
deductively weak not only in respect to their arithmetical
part but also in respect to their truth theoretic strength.
Relative interpretability is sometimes understood as a
relation of reduction. The interpretable theories of truth are
deductively too weak to be interesting as an explication of
a philosophical conception of truth besides a redundancy
conception. CTT ∩ ITT contains only theories that are
also weak in respect to their arithmetical part. Theories of
ITT ∩ CTT , interpretable but nonconservative theories,
are not as weak but they are quite artificial. Another
reason is that Π1 -conservativity is also a measure of the
arithmetical strength and therefore not directly connected
to truth-theoretic strength. Interestingly all the theories of
ITT are reflexive and not finitely axiomatizable and
therefore similar to PA .
Of more philosophical interest are the theories of
CTT ∩ ITT , conservative extensions of PA that are not
interpretable in PA . For deflationists conservativity over
the base theory is a positive aspect of a theory of truth. It
allows truth to be neutral and insubstantial. On the other
hand some deflationists claim that the truth predicate
fulfills an irreducible expressive function. So it would be an
advantage if a theory of truth is deductively strong in
respect to its truth-theoretic part. The noninterpretability of
a theory in PA would be an indicator that the truththeoretic
part of the theory cannot be ignored. The theories
of CTT ∩ ITT are also of help in extracting the essentials
of truth without influence of their arithmetical part.
The set CTT ∩ ITT is important for deflationism,
but not every theory that is an element of this set is as
good as any other. A further investigation which gives
more criteria would be of interest. None of the theories in
−
CTT ∩ ITT are reflexive and some of them like PT are
−
finitely axiomatizable. In this respect PT bears a
resemblance to ACA 0 . There is more than just a
resemblance, the two theories are equivalent in the
−
following sense: PT is a subtheory of a definitional
extension of ACA 0 and the other way around. This can be
− −
seen as an argument for the ‘naturalness’ of PT . PT is
also in other respects promising. Since it contains
compositional axioms and a form of induction for formulas
with the truth predicate the usual examples to show the
deductive weakness of deflationist theories do not obtain.
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