## Notes to Curry's Paradox

1.
Where *Tr(x)* is our given truth predicate and [*A*] an
appropriate name of *A*, the *Release rule* for truth
sanctions the inference (or step, or etc) from *Tr([A])* to
*A*, for all sentences *A*. (The corresponding Capture
rule goes the other direction.) In so-called naive truth theory, one
not only has Capture and Release in rule form, but one also has the
corresponding biconditionals (often called 'T-biconditionals').

2.
One might think that semantical properties just are sets, and so think
that USP and UA are the same. One reason for doubt comes from
Russell's paradox.
It is reasonable to
axiomatize away Russell's paradox for *sets*, where sets are
whatever mathematics needs them to be. If mathematics doesn't require
*sets* (or, for that matter, *classes* or whathaveyou)
corresponding to every meaningful predicate, then so be it. Semantics,
on the other hand, seems to be different, where a principle like USP
is harder to give up. (Notably, Gödel apparently drew a
distinction between sets and semantical properties, thinking that
Russell's paradox is really a paradox for the latter but not the
former. See Myhill's opening remarks in Myhill 1975.)

3.
An alternative approach, recently championed by Alan Weir (2005), in
effect gives up on the general transitivity of implication (or
validity). (The approach gives up the so-called *Cut* rule.)
Weir's approach falls into neither the paracomplete nor paraconsistent
camps. (Author's note: I hope to add discussion of Weir's approach in
a future edition of this entry.)

4. Lukasiewicz's continuum-valued conditional is suitable in the given sense, and has the attraction of being a familiar operator on values (‘truth-functional’, in a familiar sense). As Restall (1992) and, in more generality, P. Hajek et al. (2000) showed, this conditional gives rise to omega-inconsistency in Peano arithmetic. As a result, both paraconsistent and paracomplete truth theorists have looked elsewhere.

5.
Stronger such logics may be achieved via the ternary relation familiar
from (link: relevant logics), due to Routley and Meyer (1973), Priest
and Sylvan (1992), and also Restall (1993). For a
(‘transparent’) truth theory utilizing one such logic, see
Beall (forthcoming). [NB: The logics discussed by Restall (1993) are
*too* strong for purposes of truth theory; they don't give a
suitable conditional in the going sense, since they yield
Curry-generating contraction.]

6. I should note that Field's relevant work spans from about 2000 forward. Instead of citing the individual papers, I point only to his chief work on the subject (viz., 2008), which cites the earlier work but presents the overall framework in a fuller context. My aim here is not to present the full philosophical motivation or theory involved in Field 2008, but only sketch one way of thinking about Field's suitable conditional. Again, see Field 2008 for full details.

7.
Actually, there is something strange in Priest's suggestion
(1992). Unlike Field (or, relatedly, Beall forthcoming), Priest thinks
of the given (suitable) conditional as *entailment*, a
connective that, as Priest suggests, is supposed to express laws of
logic. But, as Colin Caret (in conversation) has emphasized, this
seems to be the wrong idea for what is going on. In particular, while
Priest's semantics (either the given approach above or more
sophisticated approaches) validates Modus Ponens in rule form, it
fails (as it must, due to Curry) what Restall calls *pseudo modus
ponens*, namely,

A& (A→B) →B

But, then, the given conditional fails to express the given ‘law’ concerning the validity of Modus Ponens. So, Priest's suggestion seems to be problematic. But I leave this for future debate.