## Translating English to SL (Chapter 2 and Section 3.1)

There are a few general comments to keep in mind when translating between natural language and the formal languages we have.

### Other Formal Translations of Language

Overall warning: This translation to SL does not capture the full meaning of the original text. There are many types of sentences in language which require other types of formal languages:
• Quantifiers: for all, some, none, etc.
These will be covered in later sections on Predicate Logic. (Chapter 6)
• Modal termns: possibly, necessarily, never ...
These are terms which are translated in Modal logic (covered in the Philosophy course on Deviant Logic but also used in computer science).
• Timing statements: eventually, within 10 minutes, never (in the sense of time)
These are terms in Temporal Logic - also used in Computer Science to analyse time dependent contral systems etc.
• Second Order: for all sequences, for arbitrary finite strings...
These statements which involve infinite collections of our basic objects, belong to higher order logic, which we will not cover. However they are used all the time in Mathematics (and Computer Science).
• Space statements: At the workshop on Thinking with Diagrams (which I missed some classes to attend) people described a number of systems (including modified modal systems) which correspond to parts of 'visual reasoning' as it effects, say, design of computer interfaces, or even learning of diagrams for logic!
My point is - our representations in SL are crude. Therefore there is some room for some disagreements about how words are used in specific contexts. We will concentrate on words as they are used in Mathematics, Computer Science, etc. and there are pretty solid agreements within these communities about the correspondences between 'natural language' and the formal symbols we use.

### Confusing translations from 3.1

Remember from Chapter 2 that there are two forms of or :
(a) inclusive or which we symbolize as 'A v B'; A or B (or both);
(b) exclusive or A or B (but not both) which requires a complicated translation [A v B] & ~[A&B] or equivalently [A v B] & [~A v ~B] or equivalently A \equiv ~B

A number of confusing translations can also be viewed as variants of this issue. In particular, the distinction between inclusive or and exclusive or rests on one line of the truth table:
(T,T) | T vs (T,T) | F If we look at the table of the conditional A > B, which is logically equivalent to the inclusive ~A or B, this issue rests on the line of the truth table for (F,T). We have said this is (F,T) ... T. People who anticipate that this line for conditional would be (F,T) F, are giving a truth table for the biconditional A \equiv B Thus the distinction between conditional and biconditional is has the same roots as the distinction between inclusive and exclusive or . We want these two tables to be different and to represent distinct words in natural language. Perhaps this comparison will clarify what is happening in some confusing translations. Consider the confusion in class of translating A unless B Logically equivalent translations of this include: ~B > A, A v B and ~A > B. However if you read the or of 'unless' as an exclusive OR, the equivalent forms include:
[A v B] & [~A v ~B] and [~B > A] & [ A > ~B] and ~B <> A .
In all cases, the convention in mathematics is thart or is inclusive. If we mean exclusive or , we will say that. If we mean biconditional rather than conditional, we will say that as well.

### Using logical equivalence

Switching among logically equivalent sentences in SL is usefull way to analyse translations. There is no guarantee that two correct translations will be identical in form - only that they will be logically equivalent (have the same truth tables).

In general, I find it very useful, when puzzling about a sentence in SL to consider other logically equivalent forms. I recommend building up your repetoire of quick exchanges among logically equivalent forms. The lists inside the from cover of the text are a good place to start.