What does linguistic under-determinacy mean? and why are irony, metaphor, metonymy, hyperbole, simile, understatment and indirect answer cases of of under-determinacy?

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Let's make a comparison between formal languages and ordinary (or natural) languages. In the former, you say exactly what you utter. In the latter your communicative intention can differ from the exact wording of your statement.

So, for example, a mathematical expression like a+b consistently and predictably means "the sum of a and b", and nothing else (once the basic conventions on symbols have been fixed).

In the ordinary language, on the other hand, the senses of the utterance highly depend on the speaker's intentions. Such philosophers as Strawson, Grice and others have pointed out that the so-called speaker's meaning can differ from the literal meaning of an expression. For example, when you say I think Mr. Johnson is a good administrator your intended meaning can vary a lot, ranging from literal interpretation to irony and metaphor. In case you were asked whether Mr. Johnson is a good physicist, your answer results elusive, which implies that you are ironically communicating a negative answer to the original question (to know more on this topic read something about Grice's conversational implicatures, starting from Wikipedia).

The important thing is that language allows under-determinate utterances, i.e. such that you cannot determine their meaning a priori, consistently and conventionally.

The meaning is, thus, the result of a sort of free negotiation between the speaker and the hearer. At the end of this negotiation an agreement is reached on how a certain referential value is assigned to certain words. Such assignment can be more or less unconventional. The more it's unconventional, the more metaphorical is your expression. But, notice, it's a matter of degree, not of kind.

This amounts to say that there is no such thing as "the right/exact word to say one particular meaning"; in theory, you can use whatever word to communicate whatever meaning, provided that your hearer gets that meaning (in case s/he does not, you have to reformulate).

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