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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48<br />
Diagonalization. The Liar Paradox, <strong>and</strong> <strong>the</strong> Appendix to Grundgesetze: Volume II — Roy T Cook<br />
It is worth not<strong>in</strong>g that we can also prove:<br />
Corollary 3: In <strong>the</strong> Grundgesetze – BLV, for any predicate<br />
Φ(x), <strong>the</strong>re is a sentence G such that:<br />
G → Φ(G)<br />
is a <strong>the</strong>orem.<br />
This result is obta<strong>in</strong>ed by replac<strong>in</strong>g our def<strong>in</strong>ition of “Diag”<br />
above with:<br />
Diag(x, y) = (∀Z)(y = §Z → x = Z(y))<br />
The trick is that without BLV we cannot prove that <strong>the</strong>re is<br />
a s<strong>in</strong>gle sentence G such that both:<br />
<strong>and</strong>:<br />
Φ(G) → G<br />
G → Φ(G)<br />
Thus, we can prove an analogue of Gödel’s diagonalization<br />
lemma with<strong>in</strong> <strong>the</strong> Grundgesetze, <strong>and</strong> restricted versions<br />
of diagonalization hold <strong>in</strong> <strong>the</strong> consistent sub-system<br />
not conta<strong>in</strong><strong>in</strong>g BLV. The reader might wonder why we<br />
have made so much of <strong>the</strong>se results. After all, we already<br />
knew that <strong>the</strong> Grundgesetze (<strong>in</strong>clud<strong>in</strong>g BLV) was <strong>in</strong>consistent,<br />
so <strong>the</strong> news that one can construct <strong>the</strong> Liar paradox<br />
as well as Russell’s paradox with<strong>in</strong> Frege’s system is not<br />
exactly earth-shatter<strong>in</strong>g (although <strong>the</strong> ‘naturalness’ of <strong>the</strong><br />
construction of <strong>the</strong> Liar paradox <strong>in</strong> <strong>the</strong> Grundgesetze is<br />
somewhat surpris<strong>in</strong>g, at least to <strong>the</strong> author). In addition,<br />
<strong>the</strong> corollaries that follow for <strong>the</strong> consistent subsystem<br />
Grundgesetze–BLV are trivial <strong>in</strong> any system of sufficient<br />
expressive strength – just let G be any tautology <strong>in</strong> Corollary<br />
2, <strong>and</strong> any contradiction <strong>in</strong> Corollary 3.<br />
The <strong>in</strong>terest of <strong>the</strong>se results lies <strong>in</strong> <strong>the</strong>ir connection<br />
to Frege’s attempted fix of <strong>the</strong> Grundgestze <strong>in</strong> <strong>the</strong><br />
appendix to Volume II, to which we now turn.<br />
2. Diagonalization <strong>and</strong> <strong>the</strong> Appendix to<br />
Grundgesetze<br />
A quick exam<strong>in</strong>ation of Theorem 1 reveals that <strong>the</strong> full<br />
strength of BLV is not required <strong>in</strong> order to prove <strong>the</strong> full,<br />
biconditional form of diagonalization. Instead, we merely<br />
need <strong>the</strong> resources to <strong>in</strong>fer l<strong>in</strong>e (2):<br />
Φ(F(§F))<br />
from l<strong>in</strong>e (5):<br />
(∃Z)(§F = §Z ∧ Z(§F) = Z(§F) ∧ Φ(Z(§F))<br />
In order to get from (5) to (2), we do not need it to be <strong>the</strong><br />
case that concepts with <strong>the</strong> same value-range are always<br />
co-extensive. Instead, we merely need concepts to agree<br />
on <strong>the</strong>ir shared value-range. Thus, we can recapture<br />
Theorem 1 by replac<strong>in</strong>g BLV with <strong>the</strong> (prima facie weaker)<br />
Fixed-Po<strong>in</strong>t Pr<strong>in</strong>ciple for value-ranges:<br />
FPP: (∀X)(∀Y)(§(X) = §(Y) → (X(§X) = Y(§X)))<br />
If FPP holds, <strong>the</strong>n we can move from:<br />
§F = §Z<br />
to:<br />
F(§F) = Z(§F)<br />
<strong>and</strong> thus from:<br />
Φ(Z(§F))<br />
to:<br />
Φ(F(§F))<br />
Thus, any pr<strong>in</strong>ciple meant to replace BLV <strong>and</strong> provide<br />
identity conditions for value ranges cannot, on pa<strong>in</strong> of Liar<strong>in</strong>duced<br />
contradiction, imply FPP.<br />
Surpris<strong>in</strong>gly, <strong>in</strong> response to <strong>the</strong> detection of Russell’s<br />
paradox, <strong>and</strong> without any (apparent) knowledge that<br />
<strong>the</strong> Liar paradox could also be derived with<strong>in</strong> <strong>the</strong><br />
Grundgesetze, Frege isolated FPP as exactly <strong>the</strong> problematic<br />
consequence of BLV.<br />
In <strong>the</strong> appendix of Volume II of <strong>the</strong> Grundgesetze,<br />
Frege beg<strong>in</strong>s his discussion of Russell’s paradox by dist<strong>in</strong>guish<strong>in</strong>g<br />
between <strong>the</strong> two ‘directions’ of BLV:<br />
BLVa: (∀X)(∀Y)((∀z)(X(z) = Y(z)) → §X = §Y)<br />
BLVb: (∀X)(∀Y)(§X = §Y → (∀z)(X(z) = Y(z)))<br />
He notes that, if we are to <strong>in</strong>dividuate concepts extensionally<br />
(an assumption he is unwill<strong>in</strong>g to give up), <strong>the</strong>n BLVa<br />
cannot be <strong>the</strong> problem – after all, any function ƒ from concepts<br />
to objects will satisfy:<br />
(∀X)(∀Y)((∀z)(X(z) = Y(z)) → ƒX = ƒY)<br />
So BLVb must be where <strong>the</strong> problem lies, <strong>and</strong> Frege sets<br />
out to discover exactly what goes wrong with this pr<strong>in</strong>ciple.<br />
He outl<strong>in</strong>es his strategy as follows:<br />
We shall now try to complete our <strong>in</strong>quiry by reach<strong>in</strong>g<br />
<strong>the</strong> falsity of (Vb) as <strong>the</strong> f<strong>in</strong>al result of a deduction,<br />
<strong>in</strong>stead of start<strong>in</strong>g from (Vb) <strong>and</strong> thus runn<strong>in</strong>g<br />
<strong>in</strong>to a contradiction. (1893, p. 288 <strong>in</strong> <strong>the</strong> Frege<br />
Reader)<br />
Thus, <strong>in</strong> order to underst<strong>and</strong> exactly what it is about BLVb<br />
that causes <strong>the</strong> problem, we need to f<strong>in</strong>d a direct proof of<br />
its negation, <strong>and</strong> not rely merely on a reductio of it via<br />
Russell’s construction. In o<strong>the</strong>r words, Frege requires a<br />
direct proof of:<br />
(∃X)(∃Y)(§X = §Y ∧ (∃z)(X(z) ∧ ¬Y(z)))<br />
In search<strong>in</strong>g for such a proof, Frege discovers that he can<br />
obta<strong>in</strong> a stronger result, which I have elsewhere (Cook [<strong>in</strong><br />
progress]) called:<br />
Frege’s Little Theorem: For any function ƒ from concepts<br />
to objects one can prove:<br />
(∃X)(∃Y)(ƒ(X) = ƒ(Y) ∧ X(ƒ(X)) ∧ ¬Y(ƒ(X)))<br />
So, given any function from concepts to objects, <strong>the</strong>re exist<br />
two concepts such that <strong>the</strong> function maps both concepts to<br />
<strong>the</strong> same object, yet <strong>the</strong> concepts differ on that very object.<br />
Here is <strong>the</strong> rub: The <strong>in</strong>stance of Frege’s Little Theorem<br />
obta<strong>in</strong>ed by substitut<strong>in</strong>g <strong>the</strong> <strong>the</strong> value range operator<br />
“§” for “ƒ” is <strong>the</strong> negation of FPP! In o<strong>the</strong>r words, <strong>the</strong> pr<strong>in</strong>ciple<br />
that Frege identifies as caus<strong>in</strong>g Russell’s paradox is<br />
exactly <strong>the</strong> pr<strong>in</strong>ciple that is needed to turn <strong>the</strong> proofs of our<br />
corollaries <strong>in</strong>to proofs of <strong>the</strong> diagonalization.<br />
The proof runs as follows (see Frege 1893, pp.<br />
285 – 288 <strong>in</strong> <strong>the</strong> Frege Reader, for Frege’s orig<strong>in</strong>al proof):