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Humanities Libertexts

2.1: Grouping in Transcription

As we saw in chapter 1, for many English sentences we can find corresponding sentences of sentence logic. For example, if 'A' stands for the sentence 'Adam loves Eve.' and 'B' for the sentence 'Adam is blond.' then 'Bv~A' corresponds to 'Either Adam is blond' or he does not love Eve.'

   Many logicians use the word 'translation' to describe the relation between a sentence of English and a corresponding sentence of logic. I think that 'translation' is the wrong word to use. If a first sentence translates a second, the two sentences are supposed to have exactly the same meaning. But the correspondence between English and logic is often looser than having the same meaning, as the next examples show.

   Consider the sentence 'Adam loves Eve, but he left her.' This English sentence is a compound of two shorter sentences, 'Adam loves Eve.', which we will transcribe with the sentence letter 'A', and 'He left her.' (that is, Adam left Eve), which we will transcribe with the sentence letter 'L'. These two sentences have been connected in English with the word 'but'. So we can get a partial transcription into logic by writing 'A but L'. We are still not finished, however, because 'but' is a word of English, not logic. What in logic corresponds to 'but'?

   If I assert the sentence 'Adam loves Eve but he left her.', what am telling you? Well, first of all, I assert that Adam loves Eve. In asserting the original sentence, I am also telling you that Adam left Eve. In other words, so far, I seem to be saying: 'Adam loves Eve and he left her.'

   What's the difference between 'Adam loves Eve but he left her.' and Adam loves Eve and he left her.', that is, between 'A but L' and 'A&L'? Not much. In English, we tend to use the word 'but' when we want to assert two things (a conjunction), but the first thing asserted may well lead one to expect the opposite of the second thing asserted. 'But' functions much as do the words '. . . and, contrary to what I just said would lead you to expect. . . .': 'Adam loves Eve, and, contrary to what I just said would lead you to expect, he left her.'

   Logic has no way of expressing the idea of 'contrary to what the first conjunct would lead you to expect.' So we simply transcribe 'but' as '&'. In sentence logic we can't improve upon 'A&L' as a transcription of 'Adam loves Eve but he left her.' Several other English words function very much like 'but', and should likewise get transcribed as '&': 'however', 'nevertheless', 'although', and 'despite (the fact that)'.

   Perhaps you are starting to see why I want to talk about transcribing, instead of translating, English into logic. 'A&L' isn't a very good translation of 'Adam loves Eve but he left her.' If it were a good translation, we would have to say that 'and' means the same thing as 'but', which it clearly does not. However, '&' is the closest we have to 'but' in logic, so that's what we use.

A Transcrpition of an English sentence into sentence logic is a sentence of sentence logic which expresses, as closely as possible, what the English sentence expresses.

   Logicians sometimes use the words 'paraphrasing' or 'symbolizing' for what I am calling 'transcribing' English sentences in logic.

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