More Vague Propositions than Sentences

There is another reason not to try to reduce propositional vagueness to sentential vagueness: there are more statements of propositional vagueness to be analysed than statements of sentential vagueness to provide that analysis.

In the parallel modal debate, David Lewis once objected to linguistic theories of modality that identified possible worlds with certain kinds of sets of sentences on

propositional variant of the T-schema.

But this argument seems in many ways parallel to the following flawed argument. Surely it is possible for Martha to uniquely have the following favourite number: the successor ofMartha's favourite number. Again one gets a contradiction, but this time from purely numerical facts and no propositional T-schema. I think in this case it is clear, even if Martha goes about declaring ‘my favourite number is the successor of Martha's favourite number, that she hasn't succeeded in making this her favourite number.

¹⁷One would have to be careful about how one formulates the Barcan principle, given the points made in section 4.5. See McGee [102], for one way of making this precise if you are in the language of arithmetic. the grounds that there are simply more possible worlds than sets of sentences.^{^[54]}A related point applies here. It certainly seems like there are more vague propositions than vague sentences, and there certainly appear to be vague propositions which are not expressed by any sentence of English.

To see this imagine that we encounter an alien race who have evolved on a planetary system orbiting a white dwarf.

Consequently the range of light that stimulates their visual system is in the ultraviolet spectrum. Like us they have different names for different segments of that region depending on how things appear to them. For example, just as we're unsure whether to apply the term ‘red' to the region of the visual spectrum that is borderline red, there are corresponding regions of the ultraviolet spectrum where the aliens are unsure whether to apply their terms. It seems very natural to say that there is something the aliens can say which we can't. These things aren't synonymous with the precise claims we can make about the ultraviolet spectrum in our language, and therefore encode pieces of information not expressible in English. If the vague propositions exist independently of us and the way we happen to speak, just as precise propositions do, the vague propositions we described would exist even if there weren't any aliens speaking this way. It seems particularly hard to imagine how the unknowability of these vague propositions could be explained by any sentence in English being borderline.

We can massage this point into a much more general worry about the prospects of giving a reductive account of propositional borderlineness in terms of sentential borderlineness. The problem is that there are simply not enough sentences to account for all of the vague propositions. There are, for example, at least uncountably many different vague propositions. Take the colour spectrum and divide it into three adjacent connected regions such that (i) the two outermost regions are each roughly the same width as the range of colours that we are unable to know are red, and (ii) the inner region is roughly the same width as the range of colours we are able to identify as red. There are uncountably many such divisions, but it is natural to think that each of these could be the range of unknowability and knowability of some other vague property much like the property of being red except shifted. There are simply not enough sentences of any spoken language to account for all such properties.^{^[55]}

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Source: Bacon Andrew. Vagueness and Thought. Oxford University Press,2018. — 361 p. — (Oxford Philosophical Monographs). 2018

More Vague Propositions than Sentences

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