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I wonder what "an A/B metaphor" means, for example, in the following quote from Wikipedia about formal language:

"In some applications, especially in logic, the alphabet is also known as the vocabulary and words are known as formulas or sentences; this breaks the letter/word metaphor and replaces it by a word/sentence metaphor."

In the example, what is used as a metaphor of what?

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In propositional calculus especially, there is a very small number of symbols available, and a very small number of ways they can combine. So it's tempting to think of this small stock of mostly special characters ( ¬ ⋀ ⋁ ⊃ ≡ p q r s ) as an alphabet.

In fact, however, these letters don't represent parts of words -- rather, they are words, and thus represent parts of sentences. For instance, ¬ means not, means and, and actual alphabetic letters like p, q, r, s are used to represent any proposition (i.e, sentence) at all. In other words, logic represents Sentences, and therefore its parts represent Words. That's the Word/Sentence metaphor.

If, however, one refers to the stock of symbols as an alphabet, then they are Letters and thus they make up Words, not sentences. This is the Letter/Word metaphor. They're not incompatible, but they do require one to switch assumptions, and to treat logic as mathematics (which uses formulae and letters) instead of semantics (which uses words and sentences).

For details, see my Logic Study Guide

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  • Thanks! In your next to last sentence, does "formulae" mean "words"? – Tim Jan 10 '12 at 17:30
  • No. It's just the Latin plural for formula. More formal, you know. – John Lawler Jan 10 '12 at 18:14
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There's the general question about 'A/B' and then there's the instance about formal languages. First the content:

Mathematical terminology is sometimes metaphorical. In the study of formal languages (not within a formal language), a 'language' is a 'set' of 'sequences' of 'elements'.

One mapping to more understandable things, that is a metaphor, is that a sequence corresponds to word, and element corresponds to an alphabet character, and the language is the set of acceptable words or vocabulary items, the letter/word metaphor.

Another metaphor is that a sequence is a sentence, the elements are the (finite set of) dictionary words. This is the word/sentence metaphor.

The wording of that example is unfortunate; it uses 'break' as though it's more meaningful when all that is really involved is a -different- metaphor.

As to 'A/B', It is an informal way in writing of signifying a pair. It is not a special way of giving any particular meaning to 'metaphor'. It just means involves a pair of things, A and B. That's all, nothing special.

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In logic the alphabet is also know as the vocabulary. A vocabulary is a collection of words that a human can use to form utterances. Utterances are typically sentences.

The components of logical formulas can be metaphorically referred to as words and sentences.

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