Transcription

Back to natural language We’ve already done a fair bit of work on simplifying and formalizing natural languages and some of the main things to look out for. But it’s worth returning to the topic now we’ve mastered some of the basic ideas of formal sentence logic.

English connectives In particular, it is worth spending some time thinking about how to translate English connectives into logical connectives. We know that our three connectives (∧, ∨ and ~) are supposed to correspond to the English connectives ‘and’, (inclusive) ‘or’, and ‘not’.

English connectives But English is full of connectives - many more than just these three! There are plenty of commonplace ones (exclusive ‘or’, ‘but’, ‘yet’, ‘unless’, ‘neither… nor…’ and so on). So how do we translate these? How would we put an English sentence like ‘Geoff is neither intelligent nor stupid’ into sentence logic?

English connectives There is no straightforward way of answering these sorts of questions. The best way to go about it is to think carefully about what the sentence means, and under what conditions it would be true or false. And remember – our sentence logic transcription of the English sentence may not preserve the original meaning perfectly. That’s ok, so long as we don’t forget that we are dealing with rough approximations, not perfect translations.

An example Let’s look at an example. ‘Geoff is neither intelligent nor stupid.’ How should we transcribe this into sentence logic?

An example Well, let’s think. Under what conditions is the sentence ‘Geoffrey is neither intelligent nor stupid’ true? What would have to be the case for that to be true? It seems like it would have to be the case that (1) Geoffrey is not intelligent; and (2) Geoffrey is not stupid. Only then would it be true that Geoffrey is neither intelligent nor stupid.

An example So it must be false that Geoffrey is intelligent and it must be false that Geoffrey is stupid. Let’s start to move towards formalization: it is not the case that (Geoffrey is intelligent) and it is not the case that (Geoffrey is stupid).

An example So, with the following transcription guide P = Geoffrey is intelligent and Q = Geoffrey is stupid, we get the sentence logic expression: ~P ∧ ~Q

Checking for adequacy of transcription So, this seems like a good bet for a good transcription. But now we have it we can check it, using a couple of helpful methods from Teller’s textbook.

First Transcription Test The first transcription test is just to translate the proposed transcription back into English as literally as possible, and check whether the new English sentence seems to mean the same thing as the old one. Let’s do this with our ~P ∧ ~Q. The most literal translation into English would be: it is not the case that Geoffrey is intelligent and it is not the case that Geoffrey is stupid.

First Transcription Test Does this seem to mean the same thing as ‘Geoffrey is neither intelligent nor stupid’? Well, yes! That just seems to be a more natural way of expressing the same thing. So it looks like we passed the first test.

Second Transcription Test Teller’s second transcription test is to try to imagine a case in which your proposed transcription is true, but the original sentence is false. If you can think of such a case, you probably don’t have a good translation. But if you can’t, you’re in business! So let’s try this one.

Second Transcription Test Can we imagine a case where ~P ∧ ~Q would be true but the English sentence ‘Geoffrey is neither intelligent nor stupid’ would be false? (Or vice versa?)

Second Transcription Test I don’t think we can! Because ~P ∧ ~Q is true if and only if Geoffrey isn’t intelligent and isn’t stupid. And that’s also exactly when the English sentence ‘Geoffrey is neither intelligent nor stupid’ is true too! So we’re not going to be able to pull them apart – they are true in exactly the same cases.

Logical equivalence (briefly) So it looks like we have an adequate transcription! The methods we’ve just gone through touch on the idea of logical equivalence. We will expand on this in future sessions so there’s no need to go into it too much now. But briefly: we say that two sentences are logically equivalent if they are true and false in exactly the same cases. In sentence logic, we say that two sentences ‘say the same thing’ if they are logically equivalent.

Let’s try some more examples 'Sadie likes books but she’s not boring.’ ‘Either James got the job, or Doug did. (They can’t both have got it!)’ ‘Unless we are very much mistaken, the earth goes around the sun.’