Tutorial Contents Tutorial Two: Propositional Calculus: Language - Truth Functors - The Propositional Language - "Implicate" - Stand-alone sentences - Scope More Tutorials One Three Four Five Six Seven

#### [jÙy] (conjunction)

[jÙy] has the following truth-table:

 j  y [jÙy] T  T T T  F F F  T F F  F F

One can read it as "j and y". Notice the brackets: these are part of the truth-functor, and should not be omitted.

Using it one could translate "John and David are Welsh" as "[John is Welsh Ù David is Welsh]". This is a pretty good translation, since both the original sentence and the translation will be true just if John is Welsh and David is Welsh.

One should not, however, translate "John and David are brothers" (taken in the most obvious way) as "[John is a brother Ù David is a brother]", since the latter will be true if John is Mary's brother and David is Robert's brother but David is not John's brother.

One could translate "John is a Welsh rugby player" as "[John is Welsh Ù John is a rugby player]". But one could not translate "John is a good rugby player" as "[John is good Ù John is a rugby player]", since either could be true without the other one being true.

What about "John is English but John is a rugby player"? Can one translate this as "[John is English Ù John is a rugby player]"? Certainly the translation leaves something out, because the original implies that there is some sort of contrast between being John's being English and his being a rugby player. But this difference does not mean that the two sentences differ in truth-value. And in fact, if John is English and John is a rugby player, one would concede that the original was true, even if one thought that there was in fact no contrast. For our purposes, what matters is that the truth-value of the translation should always agree with the truth-value of the original - and sometimes, as we noticed in the case of "John isn't Welsh" we shall be content with a bit less even than that.

Similarly we can translate "Although she was poor, she was honest" as "[She was poor Ù she was honest]", and will be prepared to reckon it true if she was both poor and honest, however much we might deprecate the implied contrast.

Before passing on to our remaining three truth-functors, let us look a little further at the sense of "imply" in the last two paragraphs.