English to Propositional Calculus: M. If M and T then P. If L then T.

Question

I have the following collection of statements:

I have money. If I have money, and have time, I can buy a pizza. If I left home early enough, I have time.

The atomic sentences can be represented as the following symbols:

• M: I have money.
• T: I have time.
• P: I can buy pizza.
• L: I left home early enough.
• T: I have time.

How would I express the statements in terms of propositional calculus?

The part I'm confused about is how to join '$M$', '$M\land T\Rightarrow P$' and '$L \Rightarrow T$'.

Answer 1

Simply use the "$\land$" connective to join them, but note that since it has higher precedence than "$\to$" (by standard convention), you have to use brackets:

$M \land ( M \land T \to P ) \land ( L \to T )$".

Answer 2

I'm not sure what you mean by join.

You can convert ((M$\land$T)$\Rightarrow$P) and (L$\Rightarrow$T) using the law

(($\alpha$$\Rightarrow$$\beta$)$\iff$($\lnot$$\alpha$$\lor$$\beta$)).

Then using the law

(($\lnot$($\alpha$$\land$$\beta$))$\iff$(($\lnot$$\alpha$)$\lor$($\lnot$$\beta$)))

You can get rid of the $\land$ symbol.

Then since you have some disjunctions, you can cancel out atom pairs which have the form

$\alpha$ and ($\lnot$$\alpha$) via the method of resolution.

Only two atoms should remain if you do that, and one of them has $\lnot$ sign, the other does not.