Semantic Indecision

The first putative example of semantic indecision is adapted from Brandom [19], although I shall be following Field's presentation in Field [51].

Field begins by inviting us to imagine that at some point before the sixteenth century, a community of English speakers had been separated from the rest of us and that they had then been reintegrated with us at the present day. Upon reintegration, we find out that their mathematicians had also discovered the complex numbers. We also discover that their dialect of English is much like ours, except that in introducing names for the two square roots of —1 they used the symbols ‘/’ and ‘\', instead of ⁱi^, and i^,.We may suppose that the two dialects of English become integrated, but the four names for the two square roots of — 1 remain. Now it is easy to show that i and — i are the only two square roots of — 1, and since / refers to a square root of — 1, so we can prove that either ‘/’ refers to i or ‘/’ refers to — i. But which of these two possibilities obtains? The dominant view seems to be that there is no fact of the matter; the names ‘/’ and ‘\' are referentially indeterminate (symmetrical reasoning suggests a similar conclusion for ‘i’ and ‘—i’). This is because there is an automorphism of the complex numbers which maps i to —i, so there is no ‘/’-free sentence that the isolated community could utter that would distinguish / from i without it also distinguishing it from —i.

To turn this example of referential indeterminacy into an example of a semantically undecided sentence, consider the sentence:

/ is positive

where an imaginary number is taken to be positive if it is a positive real multiple of i, and negative if it is a negative real multiple of i.

We assume that ‘positive' will not be a notion the separated community will have until they are reintegrated. This sentence presumably either means that i is positive (making it true) or that — i is positive (making it false), but there is nothing that settles which of these two things it might mean.

The second putative example of genuine semantic indecision arises from considering incomplete definitions. Here is an example taken from Fine [56]. Suppose that I stipulate that a natural number is nice* if it is less than 15 and that it's not nice* if it is larger than 15, and that is all that I stipulate about the meaning of the word nice*. Assuming that this stipulation is good, we can go on to say many contentful things about being nice*. For example, we may say that there are nice* primes, or that there are at least 10 nice* numbers. It seems that being nice* is a perfectly legitimate property, which some numbers have and others don't. The difficulty concerns cases like the following:

15 is nice*.

What is the status of this sentence? Of course, one could question whether the stipulation succeeded at all, and that our uses of‘nice*' ever express a property. However, it is common practice to make use of a word without there being explicit necessary and sufficient conditions for its application in all cases, and the stipulation we made seems similar enough to this phenomenon to make it hard to see how it might differ. For example, even in mathematics, we freely make use of exponentiation even though there are many functions, f, which agree with our use of exponentiation but differ over the value off (0,0). As before, this looks like a case of semantic indecision. The candidate properties ‘nice*’ might denote appear to be the property of being at most 15 and the property of being at most 14. Both properties are determinate, and the indeterminacy is just a matter of which of the two properties the word ‘nice*’ picks out; the indeterminacy appears to arise from semantic indecision stemming from our incomplete definition of the word ‘nice*’.

The last putative example of semantic indecision arises in relation to questions about what to make of scientific terms after our scientific theories have undergone some kind of radical change.

In Field [47], Field considers the example of the word ‘mass’ before and after Newtonian mechanics was replaced by special relativity. The theory of special relativity reveals that there are in fact two closely related properties, relativistic mass and proper mass, that, at ordinary speeds, play pretty much exactly the same role. Before the discovery of special relativity, however, it seemed as though it was indeterminate which property people were using the word ‘mass’ to refer to— as Field [47] puts it, ‘there are two physical quantities that each satisfy the normal criteria for being the denotation of the term’. Assertions like ‘mass is conserved in all interactions’ appear to be indeterminate depending on which of the properties we take ‘mass’ to mean.

In each case, the phenomenon bears a family resemblance with vagueness, yet it is hard to imagine that we could explain these phenomena without appealing to the way that language is used.

17.2

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

Semantic Indecision

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