New Essays on the Knowability Paradox by Salerno Joe;

Author:Salerno, Joe;
Language: eng
Format: epub
Publisher: Oxford University Press, Incorporated
Published: 2009-01-15T00:00:00+00:00

That’s a mistake. I can assert that Fermat’s last theorem is unprovable even though everything I assert logically commits me to the truth of that theorem (since it has been proven). Antirealists are wont to appeal to idealizations, so it wouldn’t be surprising to find some appealing here to the concept of what is assertible by an ideal rational agent, substituting for the concept of assertion the concept of what a logically omniscient being is committed to in virtue of what s/he asserts. Too much idealizing, methinks. The being would have to be quite unlike us, capable of knowing an uncountably infinite number of things and propositions with uncountably infinite components. If we want to speak of God here, theists like myself will have no problem with the discourse, but to think of such a being in terms of some finite extension of our own abilities and capacities is intolerable.

These same points hold for the concept of what is written. Everything written is inscribed, but sometimes only a string of morphemes is inscribed and sometimes the writing expresses a proposition as well. In my terminology to inscribe a sentence is the scribal form of uttering a string of phonemes, and writing relates to a proposition in scribal form in the way asserting relates in a vocal form to a proposition. As before, we’ll have the same reasons to focus on the concept of writing rather than inscribing, since what is inscribed is not itself a bearer of truth-value; but when we consider the writing operator W, we find that it is distributive only if the operator includes a reference to the logical consequences of what is written, and then the operator is not that of writing. For it is one thing to write down a claim, and it is another thing for what one has written to commit one logically to some further claim.

This problem about the W operator is not likely to detain the proof-theoretician for long. For one thing, there is no reason we can’t interpret the W operator as “logically implied by what is written.” Such an operator would purportedly show the falsity of the intuitive idea that anything true can be logically implied by something written truthfully on my blackboard.

Even so, there are costs to the syntactic generalization strategy. The W operator, on this understanding, is now logically complex, requiring reference to some correct logic for its interpretation. By contrast, in the knowability paradox, the rules of inference are intrinsic to the formal shorthand for the ordinary concept of knowledge. The more complex the operator, the more tempting it is to attribute the perplexing result of the proof to the complexity of the operator and the difficulty in processing this complexity. That is, the temptation is to treat it like we do barber sentences (“there is a barber who shaves all and only them who do not shave themselves”): once we see the implications, we relieve our perplexity by simply reminding ourselves of the logical complexity of the sentence, so that the appearance of possibility is misleading.

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