Curry's Paradox - Discussion

Discussion

Curry's paradox can be formulated in any language meeting certain conditions:

1. The language must contain an apparatus which lets it refer to, and talk about, its own sentences (such as quotation marks, names, or expressions like "this sentence");
2. The language must contain its own truth-predicate: that is, the language, call it "L", must contain a predicate meaning "true-in-L", and the ability to ascribe this predicate to any sentences;
3. The language must admit the rule of contraction, which roughly speaking means that a relevant hypothesis may be reused as many times as necessary; and
4. The language must admit the rules of identity (if A, then A) and modus ponens (from A, and if A then B, conclude B).

Various other sets of conditions are also possible. Natural languages nearly always contain all these features. Mathematical logic, on the other hand, generally does not countenance explicit reference to its own sentences, although the heart of Gödel's incompleteness theorems is the observation that usually this can be done anyway; see Gödel number. The truth-predicate is generally not available, but in naive set theory, this is arrived at through the unrestricted rule of comprehension. The rule of contraction is generally accepted, although linear logic (more precisely, linear logic without the exponential operators) does not admit the reasoning required for this paradox.

Note that unlike the liar paradox or Russell's paradox, this paradox does not depend on what model of negation is used, as it is completely negation-free. Thus paraconsistent logics can still be vulnerable to this, even if they are immune to the liar paradox.

The resolution of Curry's paradox is a contentious issue because resolutions (apart from trivial ones such as disallowing X directly) are difficult and not intuitive. Logicians are undecided whether such sentences are somehow impermissible (and if so, how to banish them), or meaningless, or whether they are correct and reveal problems with the concept of truth itself (and if so, whether we should reject the concept of truth, or change it), or whether they can be rendered benign by a suitable account of their meanings.

Linear logic disallows contraction and so does not admit this paradox directly, but one must remove its exponential operators, or else the paradox reappears in a modal form.