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, Bristol, England, UK
1. Introduction
F. Ramsey famously argued that the “logical” and the “semantical”
paradoxes should be studied separately. Those
of the first kind “involve only logical or mathematical terms
such as class and number, and show that there must be
something wrong with our logic or mathematics”. On the
other side are those contradictions which “cannot be
stated in logical terms alone, for they all contain some
reference to thought, language, or symbolism, which are
not formal but empirical terms.” (1925, p.353)
With the development of modern set theory and
semantics over the 20 th century, many have rejected this
classification, arguing that there is a unique shared
structure underlying most of the known paradoxes, and
that therefore a joint solution is also to be expected.
Arguments to this effect can be found for instance in
Herzberger 1970, Feferman 1984, Priest 1994; in the next
section, we will devote some attention to the second of
these papers.
2. A simple sophisticated story
In 1984, S. Feferman carries out a parallel reconstruction
of Russell’s Paradox and the Liar Paradox, to show that
both derive from the combination of three features in the
background theory:
1. The language has enough syntactical resources
to allow self-reference;
2. Classical logic is assumed;
3. The following unrestricted schemes are respectively
assumed as basic principles (for φ a formula
of the language):
(CA) ∃x∀y(y ∈ x ↔ φ[y])
(TA) T( φ ) ↔ φ.
Restriction of each of these accordingly corresponds to a
possible solution strategy for the paradoxes. Russell and
Tarski pursued the first option, developing typed formalisms
which “were early recognized to be excessively restrictive”
(1984, p.75). To test the second strategy, Feferman
constructs a common formalism for set theory and
semantics, making use of three-valued logical schemes
(the primary references in semantics are of course Martin
and Woodruff 1975, Kripke 1975). However the resulting
theory is argued to be again too restrictive, insofar as
“nothing like sustained ordinary reasoning can be carried
out in [three-valued] logic.” (p.95) But it should also be
noted that restriction of logic does not constitute a standard
option for set theorists, and this on its own diminishes
the prospects of obtaining a parallel solution of the paradoxes.
The rest of the paper is then devoted to the strategy
of restricting basic principles: since it has been successful
in ZF theory, where Russell’s paradox is blocked, the aim
is to prove a similar result for semantics. Although this
conjecture is not supported by a direct argument in the
paper, this can be made explicit as follows:
(1) Set-theoretic and semantic paradoxes bear a
structural similarity.
(2) ZF set theory is both a faithful account of set
theorists' notion of set membership, and it successfully
deals with the set-theoretic paradoxes.
Therefore:
(C) Any adequate solution to the semantic paradoxes
is to be expected to bear a structural similarity
to ZF set theory.
A solution strategy for the paradoxes is a particular application
of the theoretical framework of set theory and semantics;
thus acceptance of (C) is presumably dependent
on the soundness of a more general argument which
should show that a structural similarity holds between set
theory and semantics themselves. The upshot of the resulting
account (hereafter analogy account) would be that
set theory can inform semantics in the choice of the normative
principles which would underlie a successful (axiomatic)
theory of truth.
Is the analogy account sound? If the paradoxes do
have a joint solution and thereby constitute evidence that a
deeper analogy holds, then a structural analogy ought
surely to hold at the level of the foundations of the
theories. Hence the first condition on accepting the
analogy account is that there exist semantic counterparts
of the normative principles underlying the choice of the ZF
axioms.
3 Why the ZF axioms?
The literature is unanimous in identifying two conceptions
which enshrine the pre-theoretic intuitions concerning the
notion of set, and which moreover acted both as motivations
and normative constraints in the development of
modern set theory.
These are limitation of size and the iterative
conception:
Remarks:
(LIM) The axioms ought to entail that the setforming
operation applies to a collection of objects if
and only if the collection is small enough; or more
formally, if and only if its objects are not in one-one
correspondence with all the objects of the universe
of sets.
(IT) The axioms ought to entail that a collection of
objects is a set if and only if it is produced in a process
of the following sort: at stage 0 we have the
empty set Ø; at stage 1, Ø and its singleton set {Ø};
and so on, into the transfinite; crucially, every set
appears at some stage of this cumulative hierarchy.
In the standard formalization:
V0 = Ø; Vα+1 = P (Vα); Vγ = Uα