The Propositional Language

We are now going to meet five truth-functors which belong to the language of the propositional calculus. It is sometimes useful to use this language to translate English sentences, because the language is subject to very precise rules and is not prone to ambiguity. This means that it is sometimes easier to determine the logical properties of the new sentences. There can however be some disadvantages, arising from the fact that the translations do not always capture the exact meaning of the originals; indeed they do not always exactly capture their logical properties.

¬j (negation)

¬j has the following truth-table:

One can read it as "not j".

Using it one could translate "It isn't raining" as "¬it is raining"; or "There aren't any oranges" as "¬there are some oranges".

Could one translate "John isn't Welsh" as "¬John is Welsh"? The answer is that it is not an exact translation. Consider what would happen if "John" failed to refer to anyone - if there were no such person. In that case "John is Welsh" would not be true; but neither, surely, would "John isn't Welsh" be true. On the other hand "¬John is Welsh" would be true. (It must be, because "John is Welsh" would be false - on the assumption, at any rate, that if it is not true, it is false.) Nonetheless, one can often safely assume in such cases that there is such a person as John (or whoever it may be). So it will often be safe enough to use the inexact translation.