The sentence to the left can be read either literally or metaphorically. My analysis in this entry is concerned with a literal reading only.
1. If nothing is written in stone, then no sentence is written in stone. But the sentence to the left is written in stone. Therefore, it is not the case that nothing is written in stone. Therefore, the sentence to the left, if true, is false. And if it is false, then of course it is false. (Our sentence is not like the Liar sentence which, if true is false, and if false is true.) Therefore, whether the stone sentence is true or false, it is false. Therefore, it is necessarily false, and its negation -- 'Something is written in stone' -- is necessarily true. (Bivalence is assumed.)
But this is paradoxical! For while it is the case that the sentence is false it could have been true. For it is possible that nothing ever have been written in stone. Therefore, it is not the case that the sentence in question is necessarily false. Something has gone wrong with my analysis. What has gone wrong, I think, is that I have failed to observe a distinction I myself have drawn in earlier entries between propositional self-refutation and performative self-refutation.
2. Consider 'There are no true propositions.' This is a proposition and it is either true or false. If true, then false. And if false, then false. So necessarily false. This is a clear example of propositional self-refutation. The proposition refutes itself by itself. No human act or performance comes into the picture. 'There are no assertions' is quite different. This is either true or false. And we know it is false as a matter of contingent fact. But it is not self-refuting because if it were true it would not follow that it is false. It does not refute itself by itself. For if it were true that there are no assertions, then it would be true that there are no assertions. (Compare: if it were true that that there are no true propositions, then it would be false that there are no true propositions.)
All we can say is that 'There are no assertions,' while it can be asserted, cannot be asserted with truth. For the performance of assertion falsifies it. We thus speak here of performative inconsistency or performative self-refutation. The truth of 'There are no assertions,' if it is true, is assertively inexpressible. It is impossible that I, or anyone, assert, with truth, that there are no assertions; but it it does not follow that it is impossible that there be no assertions.
'I do not exist' is another example of performative self-refutation. I cannot assert, with truth, that I do not exist. For I cannot make the assertion without existing. Indeed, I can't even think the thought *I do not exist* without existing. But the impossibility of my thinking this thought does not entail the necessity of my existence. Necessarily, if I think, then I exist. But the necessity of the consequence does not transfer to the consequent. Both of the following are true and thus logically consistent: I cannot think without existing; I exist contingently. I cannot use the Cartesian cogito to show that I am a necessary being. (Nor can you.)
And similarly with 'Nothing is written in stone' inscribed in stone. The 'performance' of inscribing in stone falsifies the sentence while 'verifying' its negation: if I inscribe in stone 'Something is written in stone,' I provide a concrete instance of the existentially general sentence. (Am I punning on 'concrete'?)
My point, then, is that our lapidary example is not an example of strictly propositional self-refutation but of performative self-refutation where the performance in question is that of inscribing in stone. But why is this so interesting?
3. One reason is that it raises the question of inexpressible propositions. Interpreted literally, though perhaps not charitably, our stone sentence expresses a proposition that cannot be expressed salva veritate in stone. For if we try to express the proposition by producing an inscription in stone, we produce a sentence token whose existence falsifies the proposition. This holds in every possible world. In no world in which nothing is written in stone can this proposition be expressed in stone.
But the proposition expressed by the stone sentence can be expressed salva veritate in speech. Consider a possible world W in which it is literally true that nothing is written in stone, i.e., a world in which there are no stone inscriptions, in any language, of any declarative sentence. If a person in W assertively utters the sentence 'Nothing is written in stone,' he expresses a proposition true in W.
'There are no sayings' cannot be expressed salva veritate in speech but it can be expressed in stone.
I conclude that there are possibly true propositions which, while they are expressible, are not expressible in all media. The proposition expressed by our stone inscription above is true in some possible worlds but not expressible by stone inscriptions in any possible world.
Note also that there are actually true propositions that cannot be expressed in some media. In the actual world there is no ink that is compounded of the blood of Irishmen, 5W30 motor oil, and the urine of my cat, Max Black. So it is actually true that there is no such ink. This truth, however, cannot be expressed in writing that uses the ink in question.
A really interesting question is whether there are true propositions or possibly true propositions that are inexpressible salva veritate in every medium. I mean inexpressible in principle, not inexpressible due to our finite resources.
Buddhists typically say that all is empty and all is impermanent. Could it be true that all is empty despite the fact that this very thesis must be empty and therefore devoid of a determinate sense and a determinate truth value? Could it be true that all is impermanent despite the fact that this very thesis is impermanent?