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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offer a solution to <strong>the</strong> bad company problem) <strong>and</strong> show<br />
that both abstraction pr<strong>in</strong>ciples satisfy it.<br />
A famous worry for <strong>the</strong> Neo-Fregean project, called<br />
<strong>the</strong> ‘bad company’ problem, relates to <strong>the</strong> fact that not all<br />
abstraction pr<strong>in</strong>ciples are consistent. A famously<br />
<strong>in</strong>consistent abstraction pr<strong>in</strong>ciple is Frege’s notorious<br />
Basic Law V:<br />
Benacerraf <strong>and</strong> Bad Company (An Attack on Neo-Fregeanism) — Michael Gabbay<br />
∀F∀G [ εx.Fx = εx.Gx ↔ ∀x(Fx ↔ Gx) ] (14)<br />
We can use (14) to derive Russell’s paradox. An argument<br />
of Heck [Heck 1992] shows that <strong>the</strong>re are many<br />
undesirable abstraction pr<strong>in</strong>ciples. For example, <strong>the</strong>re are<br />
many Φ for which <strong>the</strong> abstraction pr<strong>in</strong>ciple:<br />
∀F∀G [ εx.Fx = εx.Gx ↔ Φ ∨ ∀x(Fx ↔ Gx) ] (15)<br />
entails that Φ. It is not hard to f<strong>in</strong>d plenty of second order<br />
sentences Φ (some of which conta<strong>in</strong> F <strong>and</strong> G) that are<br />
entailed by an abstraction pr<strong>in</strong>ciple like (15) which we<br />
would certa<strong>in</strong>ly th<strong>in</strong>k ought not to be true. Fur<strong>the</strong>rmore,<br />
different abstraction pr<strong>in</strong>ciples can be <strong>in</strong>compatible with<br />
each o<strong>the</strong>r, although <strong>in</strong>dividually consistent; this is strange,<br />
as abstraction pr<strong>in</strong>ciples are supposed to be analytic <strong>and</strong><br />
so ought to be true, <strong>and</strong> hence compatible, <strong>in</strong> any context.<br />
There is <strong>the</strong>n a question whe<strong>the</strong>r some pr<strong>in</strong>ciple can be<br />
given to discern <strong>the</strong> acceptable abstraction pr<strong>in</strong>ciples from<br />
<strong>the</strong> unacceptable ones (see [Weir 2003] for many<br />
examples of unacceptable abstraction pr<strong>in</strong>ciples). I now<br />
present such a pr<strong>in</strong>ciple.<br />
Let λ be any <strong>in</strong>f<strong>in</strong>ite card<strong>in</strong>al, <strong>the</strong>n <strong>the</strong> consistency<br />
constra<strong>in</strong>t for λ is <strong>the</strong> condition that any abstraction<br />
pr<strong>in</strong>ciple should have <strong>the</strong> form:<br />
∀F∀G [ εx.Fx = εx.Gx ↔ Ψλ(F, G) ∨ ∀x(Fx ↔ Gx) ] (16)<br />
Where<br />
i. Ψλ(F, G) is a second order sentence conta<strong>in</strong><strong>in</strong>g<br />
no free variables o<strong>the</strong>r than F <strong>and</strong> G, <strong>and</strong> also<br />
does conta<strong>in</strong> <strong>the</strong> ‘new’ abstraction operator εx.<br />
ii. Ψλ is a transitive <strong>and</strong> symmetric relation on<br />
unary predicates. That is:<br />
- Ψλ(F, G) implies Ψλ(G, F)<br />
- Ψλ(F, G) <strong>and</strong> Ψλ(G, H) implies Ψλ(F, H)<br />
iii. For any model M of card<strong>in</strong>ality λ, <strong>the</strong>re are at<br />
most λ many valuations σ such that σ(Ψλ(F, G))<br />
= ⊥ . 1<br />
Note that <strong>the</strong> familiar examples of ‘bad’ abstraction<br />
pr<strong>in</strong>ciples (e.g. <strong>in</strong> [Weir 2003]) violate this condition. For<br />
example <strong>in</strong> Frege’s Basic Law V has <strong>the</strong> form<br />
∀F∀G [ εx.Fx = εx.Gx ↔ ⊥ ∨ ∀x(Fx ↔ Gx) ]<br />
which clearly violates this condition for any λ. Note also<br />
that Benacerraf’s pr<strong>in</strong>ciple <strong>and</strong> Hume’s pr<strong>in</strong>ciple satisfy <strong>the</strong><br />
consistency constra<strong>in</strong>t for any <strong>in</strong>f<strong>in</strong>ite λ. 2 Now we can show<br />
that any abstraction pr<strong>in</strong>ciple satisfy<strong>in</strong>g <strong>the</strong> consistency<br />
constra<strong>in</strong>t for λ can be <strong>in</strong>terpreted <strong>in</strong> any second order<br />
model of card<strong>in</strong>ality at least λ. Let Mλ be a model (that can<br />
<strong>in</strong>terpret <strong>the</strong> language of Ψ) with doma<strong>in</strong> |Mλ| of card<strong>in</strong>ality<br />
λ. Let R be a relation on properties (i.e. a relation on<br />
subsets of |Mλ|) such that<br />
1 This says that <strong>the</strong> (second order) property represented by Ψλ groups <strong>the</strong><br />
properties of <strong>the</strong> doma<strong>in</strong> <strong>in</strong>to at most λ many different equivalence classes. In<br />
o<strong>the</strong>r words, Ψλ is only allowed to dist<strong>in</strong>guish up to extensionality, all properties<br />
(i.e. subsets of |M| ) of card<strong>in</strong>ality < λ . Ψλ must be unable to dist<strong>in</strong>guish all<br />
but λ of <strong>the</strong> 2λ properties of card<strong>in</strong>ality λ.<br />
2 We must view Hume’s pr<strong>in</strong>ciple as: ∀F∀G [ εx.Fx = εx.Gx ↔ F1~1G ∨<br />
∀x(Fx ↔ Gx) ]<br />
R(P, Q) iff σ(Ψλ(F, G)) = T<br />
for any valuation σ such that σ(F) = P <strong>and</strong> σ(G) = Q. 3 In<br />
o<strong>the</strong>r words, R is <strong>the</strong> <strong>in</strong>terpretation of Ψλ <strong>in</strong> <strong>the</strong> model Mλ.<br />
S<strong>in</strong>ce Ψλ(F, G) conta<strong>in</strong>s no free first or second order<br />
variables o<strong>the</strong>r than F <strong>and</strong> G, R does not depend on σ.<br />
If P ⊆ |Mλ| <strong>the</strong>n let PR = {Q: R(P, Q) or P = Q}.<br />
Clearly, PR is an equivalence class. Now consider <strong>the</strong> set<br />
A = {PR: P ⊆ |Mλ|} <strong>and</strong> let μ be its card<strong>in</strong>ality. If μ > λ, <strong>the</strong>n<br />
<strong>the</strong>re would be more than λ many valuations σ that falsify<br />
Ψλ(F, G) (at least one for each of <strong>the</strong> μ-many pairs of<br />
different equivalence classes <strong>in</strong> A). This would violate<br />
condition (iii) of <strong>the</strong> consistency constra<strong>in</strong>t for λ. So μ ≤ λ,<br />
i.e. <strong>the</strong> card<strong>in</strong>ality of A is less than or equal to <strong>the</strong><br />
card<strong>in</strong>ality of |Mλ|. It follows <strong>the</strong>n, that <strong>the</strong>re is an <strong>in</strong>jection f<br />
from{PR: P ⊆ |Mλ|} <strong>in</strong>to |Mλ|.<br />
We may use f to identify elements eP ∈ |Mλ|:<br />
eP = f(PR) (17)<br />
It is now a straightforward matter to check that<br />
eP = eQ iff R(P, Q) or P = Q<br />
It follows that we can extend any second order model Mλ of<br />
card<strong>in</strong>ality λ, with an abstraction pr<strong>in</strong>ciple satisfy<strong>in</strong>g <strong>the</strong><br />
consistency constra<strong>in</strong>t for λ: we def<strong>in</strong>e eP as <strong>in</strong> (17) <strong>and</strong><br />
<strong>the</strong>n extend Mλ to <strong>in</strong>terpret <strong>the</strong> new language us<strong>in</strong>g (18):<br />
σ(εx.Fx) = e{m: m∈|M λ | <strong>and</strong> σ [x/m] (Fx) = T}<br />
(18)<br />
Let me describe this <strong>in</strong>terpretation <strong>in</strong> English: Ψλ forms<br />
equivalence classes of properties; <strong>the</strong> conditions on Ψλ<br />
guarantee that <strong>the</strong>re is a one-one function f from <strong>the</strong>se<br />
equivalence classes <strong>in</strong>to <strong>the</strong> doma<strong>in</strong> |Mλ| of Mλ; we<br />
<strong>in</strong>terpret <strong>the</strong> referent of εx.Fx under valuation σ as <strong>the</strong><br />
element e which <strong>the</strong> function f assigns to <strong>the</strong> equivalence<br />
class of properties that Ψλ forms from <strong>the</strong> extension of F.<br />
We now have <strong>in</strong> (16) a general criterion for <strong>the</strong><br />
legitimacy of abstraction pr<strong>in</strong>ciples. This criterion<br />
legitimates Benacerraf’s pr<strong>in</strong>ciple as well as Hume’s<br />
pr<strong>in</strong>ciple. The only difference between <strong>the</strong>m be<strong>in</strong>g that<br />
extend<strong>in</strong>g a model to validate Benacerraf’s pr<strong>in</strong>ciple is<br />
slightly more straightforward than Hume’s pr<strong>in</strong>ciple. Say<br />
that an abstraction pr<strong>in</strong>ciple is almost analytic if it satisfies<br />
<strong>the</strong> consistency constra<strong>in</strong>t for any <strong>in</strong>f<strong>in</strong>ite λ. It is now a<br />
matter of dispute whe<strong>the</strong>r <strong>the</strong> fact that λ has to be <strong>in</strong>f<strong>in</strong>ite<br />
detracts from <strong>the</strong> analyticity of Hume’s pr<strong>in</strong>ciple <strong>and</strong><br />
Benacerraf’s pr<strong>in</strong>ciple. A po<strong>in</strong>t <strong>in</strong> favour of <strong>the</strong> Neo-<br />
Fregean programme is that we can give an <strong>in</strong>dependently<br />
motivated formal reasons for treat<strong>in</strong>g Hume’s pr<strong>in</strong>ciple as<br />
analytic <strong>and</strong> rule out pr<strong>in</strong>ciples like Basic Law V. However,<br />
a po<strong>in</strong>t aga<strong>in</strong>st <strong>the</strong> Neo-Fregean programme is that<br />
Benacerraf’s pr<strong>in</strong>ciple also comes out as analytic, <strong>and</strong> <strong>the</strong><br />
argument of Section 3.3 st<strong>and</strong>s.<br />
4.2 The concept of number<br />
Perhaps some argument relat<strong>in</strong>g to our concept of number<br />
will differentiate between <strong>the</strong> two pr<strong>in</strong>ciples.<br />
The possibility of any such argument is extremely<br />
doubtful, if anyth<strong>in</strong>g but for <strong>the</strong> fact that nei<strong>the</strong>r Hume’s<br />
pr<strong>in</strong>ciple nor Benacerraf’s pr<strong>in</strong>ciple make good analyses of<br />
our number concepts. Wright acknowledges this:<br />
3 σ assigns elements of Mλ to first order variables <strong>and</strong> subsets of Mλ to second<br />
order variables; σ[x/m] is a valuation that agrees with σ on all variables<br />
except that it maps <strong>the</strong> variable x to m.<br />
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