Reduction and Elimination in Philosophy and the Sciences
Reduction and Elimination in Philosophy and the Sciences
Reduction and Elimination in Philosophy and the Sciences
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Interpretability Relations of Weak Theories of Truth<br />
Mart<strong>in</strong> Fischer, Leuven, Belgium<br />
1. Introduction<br />
Axiomatic <strong>the</strong>ories of truth are understood as extensions of<br />
a syntactic base <strong>the</strong>ory which is often taken to be Peano<br />
Arithmetic, PA . One way to measure <strong>the</strong> strength of a<br />
<strong>the</strong>ory of truth is to take <strong>in</strong>to account which formulas of<br />
arithmetic it proofs. Weak <strong>the</strong>ories <strong>in</strong> this respect are <strong>the</strong>ories<br />
that do not prove more than PA itself, <strong>the</strong>ories that are<br />
conservative extensions of PA . The concept of conservativity<br />
has ga<strong>in</strong>ed some <strong>in</strong>terest <strong>in</strong> formaliz<strong>in</strong>g philosophical<br />
criteria. This is also <strong>the</strong> case <strong>in</strong> <strong>the</strong> debate on truth, <strong>in</strong><br />
which conservativity is expected to expla<strong>in</strong> <strong>the</strong> `no substance'<br />
claim of deflationism. For <strong>the</strong>ories of truth conservativity<br />
over PA alone seems to be a very crude measure<br />
s<strong>in</strong>ce it does not differentiate between different conservative<br />
<strong>the</strong>ories of truth which have quite different properties<br />
<strong>and</strong> prove different formulas conta<strong>in</strong><strong>in</strong>g <strong>the</strong> truth predicate.<br />
A comparison of <strong>the</strong> truth-<strong>the</strong>oretic strength of<br />
<strong>the</strong>ories of truth is desirable. A direct comparison of <strong>the</strong><br />
truth-<strong>the</strong>oretic strength is <strong>the</strong> subset relation but it is only<br />
a partial order so that not all <strong>the</strong>ories can be compared.<br />
Ano<strong>the</strong>r measure of <strong>the</strong> strength of a <strong>the</strong>ory of truth would<br />
be <strong>the</strong>ir <strong>in</strong>terpretability relations to o<strong>the</strong>r <strong>the</strong>ories<br />
especially <strong>the</strong>ir <strong>in</strong>terpretability or non<strong>in</strong>terpretability <strong>in</strong> PA .<br />
The most famous of <strong>the</strong>se <strong>in</strong>terpretability relations is<br />
relative <strong>in</strong>terpretability, <strong>in</strong>troduced <strong>in</strong> (Tarski et al. 1953),<br />
<strong>and</strong> it is a good measure for PA as base <strong>the</strong>ory. On <strong>the</strong><br />
one h<strong>and</strong> <strong>the</strong> less restricted version of local <strong>in</strong>terpretability<br />
collapses <strong>in</strong> this case <strong>in</strong>to relative <strong>in</strong>terpretability. On <strong>the</strong><br />
o<strong>the</strong>r h<strong>and</strong> Tarskis <strong>the</strong>orem of undef<strong>in</strong>ability of truth shows<br />
that <strong>the</strong>re is no def<strong>in</strong>itional extension of PA by a one place<br />
predicate τ , PA (τ ), so that PA (τ ) proves τ( ϕ)<br />
↔ ϕ for<br />
all sentences ϕ of <strong>the</strong> language of arithmetic.<br />
2. Axiomatic <strong>the</strong>ories<br />
LA is <strong>the</strong> language of arithmetic <strong>and</strong> Lτ : = A {τ ∪ . The<br />
arithmetical <strong>the</strong>ories are as usual. For <strong>the</strong> <strong>in</strong>terpretability<br />
considerations take <strong>the</strong> arithmetic <strong>the</strong>ories to be formulated<br />
with predicate- <strong>in</strong>stead of functionsymbols.<br />
96<br />
L }<br />
( Ind P ) P( 0)<br />
∧ ∀x(<br />
P(<br />
x)<br />
→ P(<br />
x + 1))<br />
→ ∀x(<br />
P(<br />
x))<br />
Q is Rob<strong>in</strong>son Arithmetic <strong>and</strong> PA is Peano Arithmetic,<br />
that is P A L Ind Q ∪ ) ⋅ ( , where P A L Ind ⋅ ) ( is <strong>the</strong> set of sentences<br />
that result from replac<strong>in</strong>g P <strong>in</strong> ( Ind P ) by a formula<br />
of L A with at least one free variable. Accord<strong>in</strong>gly,<br />
IΣ k = Q ∪ ( IndP<br />
) ⋅ Σk<br />
.<br />
Assume that L A conta<strong>in</strong>s <strong>the</strong> relevant syntactical<br />
vocabulary: ‘ Ct ’ for closed term of L A , ‘ Sent ’ for<br />
sentence of L A , ‘ Form 1’<br />
for formula of LA with one free<br />
variable, <strong>and</strong> so on, such that PA proves <strong>the</strong> relevant<br />
syntactical <strong>the</strong>orems. Especially if m is <strong>the</strong> gödelnumber<br />
of a formula ϕ (x)<br />
with one free variable x <strong>and</strong> n of a<br />
term t , <strong>the</strong>n m (n)<br />
is <strong>the</strong> gödelnumber of <strong>the</strong> substitution<br />
τ<br />
of <strong>the</strong> free variable x <strong>in</strong> ϕ (x)<br />
by <strong>the</strong> numeral of t . PA is<br />
PA formulated <strong>in</strong> <strong>the</strong> language L τ . A <strong>the</strong>ory of truth T is<br />
τ<br />
a Lτ -<strong>the</strong>ory with PA ⊆ T .<br />
tot ( x)<br />
: ⇔ Form1(<br />
x)<br />
∧ ∀y(<br />
τ ( x(<br />
y))<br />
∨τ<br />
( ¬ & x(<br />
y )))<br />
Disquotational <strong>the</strong>ories of truth are formulated with a<br />
scheme of T-biconditionals:<br />
( TB P )<br />
τ ( P) ↔ P<br />
( UTB P ) ∀x(<br />
τ ( P(<br />
x&<br />
) ) ↔ P(<br />
x)).<br />
Compositional axioms are <strong>the</strong> universally quantified versions<br />
of <strong>the</strong> follow<strong>in</strong>g formulas:<br />
(C 1)<br />
Ct ( x)<br />
∧ Ct(<br />
y ) → ( τ ( x = & y)<br />
↔ val(<br />
x)<br />
= val(<br />
y)).<br />
(C 2)<br />
Ct( x)<br />
∧ Ct(<br />
y)<br />
→ ( τ ( x ≠&<br />
y)<br />
↔ val(<br />
x)<br />
≠ val(<br />
y)).<br />
(C 3)<br />
Sent( x)<br />
∧ Sent(<br />
y)<br />
→ ( τ( x ∧&<br />
y ) ↔ τ(<br />
x)<br />
∧τ<br />
( y)).<br />
(C 4)<br />
Sent ( x)<br />
∧ Sent(<br />
y)<br />
→ ( τ ( ¬ & x ∧&<br />
y)<br />
↔ τ(<br />
¬ & x)<br />
∨ τ ( ¬ & y)).<br />
(C 5)<br />
Sent( ∀&<br />
yx)<br />
→ ( τ( ∀&<br />
yx)<br />
↔ ∀z(<br />
τ ( x(<br />
z)))).<br />
(C 6)<br />
Sent ( ¬ & ∀&<br />
yx)<br />
→ ( τ ( ¬ & ∀&<br />
yx)<br />
↔ ∃z(<br />
τ ( ¬ & x(<br />
z)))).<br />
(C 7)<br />
Sent( x)<br />
→ ( τ ( ¬ & ¬ & x)<br />
↔ τ ( x)).<br />
(C 8)<br />
Sent( x)<br />
→ ( τ ( ¬ & x)<br />
↔ ¬ τ ( x)).<br />
The axiom of <strong>in</strong>ternal <strong>in</strong>duction for total formulas is:<br />
( It I)<br />
∀ x( tot(<br />
x)<br />
∧τ<br />
( x(<br />
0))<br />
∧ ∀y(<br />
τ(<br />
x(<br />
y )) → τ(<br />
x(<br />
y + 1)))<br />
→ ∀y(<br />
τ(<br />
x(<br />
y )))).<br />
The relevant <strong>the</strong>ories are:<br />
TB : = Q ∪ ( TBP<br />
) ⋅ LA<br />
∪ ( IndP<br />
) ⋅ Lτ<br />
UTB : = Q ∪ ( UTBP<br />
) ⋅ LA<br />
∪ ( IndP<br />
) ⋅ Lτ<br />
PT : PA ( C1)<br />
( C7)<br />
r<br />
= ∪ −<br />
−<br />
PT : = PA ∪ ( C1)<br />
− ( C7)<br />
∪ ( ItI<br />
)<br />
PT : = PA ∪ ( C1)<br />
− ( C7)<br />
∪ ( IndP<br />
) ⋅ Lτ<br />
TC : PA ( C1)<br />
( C8)<br />
r<br />
= ∪ −<br />
−<br />
TC : = PA ∪ ( C1)<br />
− ( C8)<br />
∪ ( ItI<br />
)<br />
r<br />
TC is also known as PA (S)<br />
<strong>and</strong> TC as T (PA)<br />
.<br />
3. Interpretability<br />
Some basic results:<br />
(i) TB ⊂ UTB ⊂ PT = TC<br />
−<br />
(ii) IΣ1 ∪ ( C1),<br />
( C3),<br />
( C5),<br />
( C8)<br />
∪ ( It<br />
I)<br />
= TC<br />
−<br />
(iii) IΣ1 ∪ ( C1)<br />
− ( C7)<br />
∪ ( It<br />
I)<br />
= PT<br />
− −<br />
(iv) PT , TC are f<strong>in</strong>itely axiomatizable.<br />
r r<br />
(v) TB,<br />
UTB,<br />
PT , TC , PT,<br />
TC are not f<strong>in</strong>itely axiomatizable.<br />
r r −<br />
(vi) TB,<br />
UTB,<br />
PT , TC , PT are conservative extensions of PA .<br />
−<br />
(vii) TC , PT,<br />
TC are nonconservative extensions of PA .<br />
Def<strong>in</strong>ition<br />
Let S, T be <strong>the</strong>ories formulated <strong>in</strong> L S, LT<br />
. Then<br />
T is a pure extension of S iff T is an extension of S <strong>and</strong> L T = LS<br />
.<br />
T is reflexive iff T proves Con Δ for all f<strong>in</strong>ite Δ ⊆ T.<br />
T is essentially reflexive iff all pure extensions of T are reflexive.<br />
T has full <strong>in</strong>duction iff for all formulas ϕ of L T : T proves<br />
ϕ( 0)<br />
∧ ∀x(<br />
ϕ(<br />
x)<br />
→ ϕ(<br />
x + 1))<br />
→ ∀xϕ(<br />
x)<br />
.<br />
Full <strong>in</strong>duction <strong>and</strong> reflexivity are connected <strong>in</strong> <strong>the</strong> follow<strong>in</strong>g<br />
way, as shown for example <strong>in</strong> (Hájek/Pudlák 1993, p.189):<br />
Lemma 1<br />
If PA ⊆ T <strong>and</strong> T has full <strong>in</strong>duction, <strong>the</strong>n T is reflexive.