Reduction and Elimination in Philosophy and the Sciences

Structure of the Paradoxes, Structure of the Theories: A Logical Comparison of Set Theory and Semantics — Giulia Terzian
(cf. Potter p.34), it seems desirable to adopt a constructive
rather than a regressive strategy, to ensure the higher
reliability of the resulting theory.
However most accounts in the literature appear to
start from an analysis of the Liar paradox, and then
proceed to develop a suitable formalism which
accommodates this phenomenon as well as our intuitions
about truth. This is also clearly the case in Feferman 1984;
but by adopting such a plainly regressive strategy, the
resulting theory is also much more exposed to the danger
of misrepresenting our positive intuitions about truth.
One of the reasons for which the analogy account is
attractive is that it would allow for semantics to be informed
by set theory; but a genuine structural analogy should also
entail that semantics can inform set theory, so that settheoretic
norms can be imported over to semantics, and
vice versa.
A closer look at concrete theories shows that the
analogy is invariably left lopsided. For instance, the axiom
system proposed in Feferman 1984 is not only shown to
be an unsatisfactory account of truth, but it is also clear
that it is not inter-translatable with ZF, which leaves the
analogy account to stand on even shakier grounds.
5 Conclusion
Feferman and others 3 have shown that the set-theoretic
and semantic paradoxes can be reconstructed to follow a
common principle. On the other hand, the discussion in
Section 4 shows that set theory and semantics are based
on structurally mismatching guiding principles, and moreover
that their respective end-products – the axiom systems
– do not correspond to each other, as a genuine
structural similarity should guarantee.
Although the analogy account remains very
attractive, this paper should hopefully have shown that it
must be supported by a stronger argument if it is to resist
these problems.
3 E.g. Priest 1994.
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