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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Structure of <strong>the</strong> Paradoxes, Structure of <strong>the</strong> Theories: A Logical Comparison of Set Theory <strong>and</strong> Semantics — Giulia Terzian<br />
(cf. Potter p.34), it seems desirable to adopt a constructive<br />
ra<strong>the</strong>r than a regressive strategy, to ensure <strong>the</strong> higher<br />
reliability of <strong>the</strong> result<strong>in</strong>g <strong>the</strong>ory.<br />
However most accounts <strong>in</strong> <strong>the</strong> literature appear to<br />
start from an analysis of <strong>the</strong> Liar paradox, <strong>and</strong> <strong>the</strong>n<br />
proceed to develop a suitable formalism which<br />
accommodates this phenomenon as well as our <strong>in</strong>tuitions<br />
about truth. This is also clearly <strong>the</strong> case <strong>in</strong> Feferman 1984;<br />
but by adopt<strong>in</strong>g such a pla<strong>in</strong>ly regressive strategy, <strong>the</strong><br />
result<strong>in</strong>g <strong>the</strong>ory is also much more exposed to <strong>the</strong> danger<br />
of misrepresent<strong>in</strong>g our positive <strong>in</strong>tuitions about truth.<br />
One of <strong>the</strong> reasons for which <strong>the</strong> analogy account is<br />
attractive is that it would allow for semantics to be <strong>in</strong>formed<br />
by set <strong>the</strong>ory; but a genu<strong>in</strong>e structural analogy should also<br />
entail that semantics can <strong>in</strong>form set <strong>the</strong>ory, so that set<strong>the</strong>oretic<br />
norms can be imported over to semantics, <strong>and</strong><br />
vice versa.<br />
A closer look at concrete <strong>the</strong>ories shows that <strong>the</strong><br />
analogy is <strong>in</strong>variably left lopsided. For <strong>in</strong>stance, <strong>the</strong> axiom<br />
system proposed <strong>in</strong> Feferman 1984 is not only shown to<br />
be an unsatisfactory account of truth, but it is also clear<br />
that it is not <strong>in</strong>ter-translatable with ZF, which leaves <strong>the</strong><br />
analogy account to st<strong>and</strong> on even shakier grounds.<br />
5 Conclusion<br />
Feferman <strong>and</strong> o<strong>the</strong>rs 3 have shown that <strong>the</strong> set-<strong>the</strong>oretic<br />
<strong>and</strong> semantic paradoxes can be reconstructed to follow a<br />
common pr<strong>in</strong>ciple. On <strong>the</strong> o<strong>the</strong>r h<strong>and</strong>, <strong>the</strong> discussion <strong>in</strong><br />
Section 4 shows that set <strong>the</strong>ory <strong>and</strong> semantics are based<br />
on structurally mismatch<strong>in</strong>g guid<strong>in</strong>g pr<strong>in</strong>ciples, <strong>and</strong> moreover<br />
that <strong>the</strong>ir respective end-products – <strong>the</strong> axiom systems<br />
– do not correspond to each o<strong>the</strong>r, as a genu<strong>in</strong>e<br />
structural similarity should guarantee.<br />
Although <strong>the</strong> analogy account rema<strong>in</strong>s very<br />
attractive, this paper should hopefully have shown that it<br />
must be supported by a stronger argument if it is to resist<br />
<strong>the</strong>se problems.<br />
3 E.g. Priest 1994.<br />
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