Hello bmp05 Originally Posted by

**bmp05** Thanks Grandad, I guess, what I was wondering was how B(x, y) effects the proposition?

So does B(x, y) return True for 'x' and False for 'y'? I can't put my finger on it, but it seems strange that you can have a proposition, like $\displaystyle S(n) \Rightarrow B(x, y)$? S(n) implies there is 'something' better than it!

$\displaystyle S(n) \Rightarrow B(x, y) \Leftrightarrow \neg S(n) \vee B(x, y)$

Would you say this as "something (x) is better than Spy novels."

$\displaystyle B(x,y)$ is an example of a *two-place predicate*, since it takes two parameter values, $\displaystyle x$ and $\displaystyle y$. But it simply has a single Boolean value, True or False, like any other predicate, once you supply it with the correct number of parameters.

Another example of a two-place predicate might be:$\displaystyle L(x, y)$ meaning $\displaystyle x$ loves $\displaystyle y$.

So, for example, if $\displaystyle x =$ Grandad and $\displaystyle y =$ Curry, then $\displaystyle L(x,y)$ has the value True. But if $\displaystyle x$ = Grandad and $\displaystyle y$ = Pop Music, then $\displaystyle L(x,y)$ has the value False. (Sad, aren't I?)

Incidentally, I have realised that we don't need the $\displaystyle z$ in my expression. It works with:$\displaystyle \exists x\;\forall y \Big[M(x)\land\big(S(y) \Rightarrow B(x,y)\big) \land \big(N(y) \Rightarrow B(x,y)\big)\Big]$

which can be written in English as:There's at least one book, $\displaystyle x$, such that whenever we choose another book, $\displaystyle y$, $\displaystyle x$ is a mystery, and if $\displaystyle y$ a spy novel then $\displaystyle x$ is better than $\displaystyle y$, and if $\displaystyle y$ is a nonfiction book then $\displaystyle x$ is better than $\displaystyle y$.

In other words:There's at least one book, $\displaystyle x$, such that whenever we choose another book, $\displaystyle y$, $\displaystyle x$ is a mystery, and if $\displaystyle y$ a spy novel or a nonfiction book then $\displaystyle x$ is better than $\displaystyle y$.

which can be symbolised more simply as:$\displaystyle \exists x\;\forall y \Big[M(x)\land \Big(\big(S(y) \lor N(y)\big) \Rightarrow B(x,y)\Big)\Big]$

Grandad