The Liar paradox is intuitive. Having explained the Liar to my elder children some years ago, I asked them to come up with versions of their own. My son, Leif, then 13, suggested:

A policeman asks a suspect whether he is lying, and the criminal just says ‘Yes’.

One can see how this would work, and it is similar to a known version of the Liar originating with L. Jonathan Cohen.

After some time, my elder daughter, Veronique, then 11, devised the Pinocchio paradox. Pinocchio says ‘My nose will be growing’. Pinocchio’s nose grows, so the story tells us, whenever he tells a lie. The use of a future tense ties in with when Pinocchio’s nose is supposed to grow – after telling a lie. Philosophers will naturally want to know how soon afterwards. (There is an interesting version of the Epimenides though, if one does not restrict how soon Pinocchio’s nose should grow. Imagine Pinocchio says ‘My nose will grow’ but everything else Pinocchio says is true. Then, Pinocchio’s nose will grow if and only if it does not.)

Veronique had in mind a nasal event specifically related to this particular statement by Pinocchio. I think this is clear enough, at least for most audiences. Nevertheless, we can tweak the scenario if we wish to use the present tense.

Pinocchio’s nose grows if and only if (iff) what he is stating is false, and Pinocchio says ‘My nose is growing’. So, Pinocchio’s nose is growing iff it is not growing. It is clearly a version of the Liar. Indeed, it has an obvious Truth-teller variant. Its distinctive feature though is that it moves away from using a synonym for truth in the Liar statement. Having one’s nose grow is a facial, not a semantic feature. Moreover, although Pinocchio’s nose grows just when he is telling an untruth, the relationship is not semantic. It might be supposed to be causal or of some other nature, but it is not semantic. If Pinocchio’s nose is growing it is because he is saying something false; otherwise, it is not growing; and ‘because’ stands here for a non-semantic relation. The facts are that ‘is growing’ is not a synonym for ‘is not true’ and the Pinocchio story is intelligible without pretending that ‘is growing’ has a different meaning.

The Pinocchio paradox is, in a way, a counter-example to solutions to the Liar that would exclude semantic predicates from an object-language, because ‘is growing’ is not a semantic predicate. Tarski’s analysis of the source of pathology of which the Liar is symptomatic led him to conclude it arose from free use of semantic predicates in the object-language. Tarski’s solution was to restrict such predicates strictly to the metalanguage. Intuitively, predicates like ‘is growing’ are typical of just the sorts of predicates one wants in a useful object-language. If empirical predicates like ‘is growing’ need to be restricted in the object-language to avoid versions of the Liar, the intuitive bounds on which predicates need to be restricted in the object-language to avoid Liar-like paradoxes have been breached.

Tarski’s breakthrough approach has been criticized in other ways and improved on by other theories. The chief pragmatic issue with the strict metalanguage-hierarchy solution to the Liar is that too many determinate uses of ‘is true’ have to use the metalanguage. The strict avoidance of ‘is true’ in the object language is too broad a stricture: it cuts out many non-pathological uses of the truth predicate to avoid those that generate paradoxes. Under Kripke’s application of the truth-value gap approach, a non-strict metalanguage-hierarchy is used to address the Liar, so that non-paradoxical uses of ‘is true’ can be meaningfully made in the object language (Kripke 1975). As a theory of truth, Kripke addresses all those versions of the Liar which are paradoxical statements involving the truth predicate. The same also holds true for versions using synonyms for the truth predicate – the Liar paradox is so-named after all.

Kripke’s solution restricts the interpretation of the truth predicate to non-pathological uses. These uses are safe, because the truth value of any object-language sentence that receives one is grounded on the values of atomic sentences that do not use semantic predicates. The Liar sentence, ‘The Liar sentence is not true’, is in the object-language but is not in the extension or anti-extension of its truth predicate. This truth-predicate is a partially defined semantic predicate. (Non-semantic predicates are interpreted in the standard way, as they are assumed to be fully defined.)

Kripke’s solution still relies on a metalanguage hierarchy. Thus, we may say in the metalanguage that the object-language Liar sentence is not true. And, we may say in the meta-metalanguage that the metalanguage Liar sentence is not true. Kripke’s approach can endorse Tarksi’s analysis of the Liar while addressing the criticism that Tarksi’s solution is too broad a stricture on consistent usage of the truth predicate in the object-language.

The Pinocchio paradox raises a purely logical issue for any metalanguage-hierarchy solution, strict or liberal. The Pinocchio scenario is not going to arise in our world, so it is not a pragmatic issue. It seems though that there could be a logically possible world in which Pinocchio’s nose grows if and only if he is saying something not true. However, there cannot be such a logically possible world wherein he makes the statement ‘My nose is growing’. A metalanguage hierarchy approach cannot explain this based on Tarski’s analysis, and therefore cannot solve the Pinocchio paradox, which is a version of the Liar.

Suppose, per impossible, that we seek to apply a Kripkean solution for such a world where Pinocchio utters this inconsistency. What Pinocchio says uses a non-semantic predicate, which is fully defined; this forces a truth value for his statement. Consequently, the biconditional has a truth value and an inconsistency will be forced by the interpretation. Here is an example of how this would work. Suppose there is a logically possible world where Pinocchio’s nose grows just as he says something untrue and Pinocchio says ‘My nose is growing’. If, on the one hand, Pinocchio’s nose is growing, what he says is in the extension of truth – it involves only non-semantic predicates, which are fully defined. So then ‘ “Pinocchio’s nose is growing” is true’ is also in the extension of truth, so the biconditional ‘Pinocchio’s nose is growing iff “Pinocchio’s nose is growing” is not true’ is false, which contradicts what was supposed to be true of this world. If, on the other hand, Pinocchio’s nose is not growing, his statement will be in the anti-extension of truth. But if Pinocchio’s statement is not true, his nose is growing, by the supposition about this world.

Personally, I think this version of the Liar is not necessarily an issue for a theory of truth about the truth predicate, but is an issue for a theory of the sort of truth that is preserved by validity. Ideally, though, they would be the same theory.

Reference

Kripke
S
Outline of a theory of truth
Journal of Philosophy
 , 
1975
, vol. 
72
 (pg. 
690
-
716
)