Paradoxes of Expression

I would like to thank an anonymous referee, Johannes Korbmacher, Tobias Martin, Graham Priest, Stewart Shapiro, and Niko Strobach for helpful discussions and comments.

I use underlining to indicate a name-forming operator.

I call ‘E(x : p)’ not an ‘expressibility’ but an ‘expression’ predicate-operator because, given the intended meaning of ‘E(x : p)’, the notion of expressibility is captured much better by the defined operator ‘\(\exists x \ E(x:p)\)’; cf. (D2).

In my discussion of the paradoxes of expression, I will consider and reject an objection to (E) that is based on the observation that some sentences are ambiguous.

However, the conditional that obeys modus ponens would still be needed for the formulation of the Liar and Curry sentences.

This way of formalizing a Liar sentence via propositional quantification and negation is not original. Cf., e.g., Prior [14]. But to my knowledge, it has not yet been transferred to Curry’s paradox.

Note that, as ‘l’ is an individual constant that can only stand in the position of a singular term, ‘\(l \wedge \lnot l\)’ is ill-formed and cannot be used to express the result of the Liar reasoning. ‘\(r_{0}\)’, in contrast, is a propositional constant so that ‘\(r_{0} \wedge \lnot r_{0}\)’ is well-formed.

This is witnessed by line 7 of the Liar derivation and line 7 and 8 of the Curry derivation.

One could (and I personally would) draw an entirely different moral from the paradoxes of expression and all other paradoxes that are based on self-referential expressions, namely that the required kind of self-referential expressions which attribute semantic properties to themselves do not exist, after all; cf. Pleitz ([8]). But in the present context of a discussion within the horizon of those paradox-solvers who have taken the path of logical revision, the existence of the problematical self-referential expressions is of course a given.

Priest makes a brief remark about an expression ‘E(x, p)’, which he calls a “binary predicate” (towards the end of Sect. 5.2 of Priest [12]), that is similar in intended meaning to our expression predicate-operator ‘E(x : p)’. He proposes to define a truth predicate from it as ‘\(\exists p(p \wedge E(x, p))\)’ or as ‘\(\forall p \ (E(x, p) \supset p)\)’. I note that these are not equivalent, because in contrast to the first, the second of these two open formulas will be vacuously satisfied by anything that does not express something.

With a view of recovering the paradoxes in the system discussed by Priest, we could put the predicate-operator ‘E(x : p)’ to the side, and work in its stead with a unary expression-operator for each one of the problematic sentences, e.g., ‘L(p)’ with the intended meaning that the specific sentence l expresses that p, and ‘\(C_{n}(p)\)’ with the intended meaning that the specific sentence \(c_{n}\) expresses that p. In parallel to the stipulations (L) and (C) above, we could lay down that \(L(\forall p(L(p) \rightarrow \lnot p)\), that \(C_{n}(\forall p(C_{n}(p) \rightarrow (p \rightarrow q_{0})))\), and so on, and would thus guarantee that l would be a Liar sentence, \(c_{n}\) would be a Curry sentence, and so on. Now, given principles much like the respective instances of (E) for each one of the sentences associated with these unary operators – e.g., \(\forall p \forall q(L(p) \wedge L(q) \rightarrow (p \leftrightarrow q))\) for the operator ‘L(...)’ that concerns sentence l, – counterparts of the above derivations would be valid. I would like to thank an anonymous referee for alerting me to this possibility. But I am not convinced that much would be gained for our discussion of Priest’s system by the ensuing proliferation of unary operators, each governed by its own specific principle. In view of the dialectic of the debate and the aim of testing the system discussed by Priest, it is important to introduce resources that capture a detaching notion of expression in a way that is motivated independently of the paradoxes. And, in contrast to the principles governing the specific operators needed for the paradoxical derivations, there are general considerations that provide such independent motivation in the case of principle (E).

With a view of recovering the paradoxes, we could put the predicate-operator ‘E(x : p)’ to the side, and work in its stead with an operator ‘I : p’ with the intended meaning ‘I now unambiguously say that p’,  governed by the principle \(\forall p \forall q(I:p \ \wedge \ I:q \rightarrow (p \leftrightarrow q))\), that would here be justified already by the intended unambiguousness. Now we would get something like a Liar sentence and a Curry sentence by saying (!) ‘\(I:\forall p(I:p \rightarrow \lnot p)\)’ and ‘\(I:\forall p(I:p \rightarrow (p \rightarrow q_{0}))\)’, respectively. However, the ‘I say now that’-variant of Curry’s paradox is likely to be less harmful because one can say only so many things, and thus it would be more difficult to trivialize the system.