# How to translate a set of sentences into logic symbols

For example If there is justice in this life, then there is no need for a future life. If, on the other hand, there is no justice in our earthly life, then we have no reason to believe that God is righteous. But if we have no reason to believe that God is righteous, then we have no reason to believe that He will provide us with a future life. So either we don't need a future life, or we have no reason to believe that God will provide us with such a life.

If there is justice in this life-p

then there is no need for a future life.-q

If, on the other hand, there is no justice in our earthly life,- ~p

we have no reason to believe that God is righteous -r

we have no reason to believe that He will provide us with a future life - s

we don't need a future life -q

or we have no reason to believe that God will provide us with such a life. -s

I dont know how to connect the sentences here

ive got p->q

~p->r

r->s

q⩖(exclusive or) s


In the following, I use ~ whenever the text says "no," and I use multi-letter abbreviations as mnemonics. I think that makes the formulas a little more readable.

For example If there is justice in this life, then there is no need for a future life.

J -> ~NF (Justice -> no Need for Future life)

If, on the other hand, there is no justice in our earthly life, then we have no reason to believe that God is righteous.

~J -> ~RGR (no justice -> no Reason to believe God is Righteous)

But if we have no reason to believe that God is righteous, then we have no reason to believe that He will provide us with a future life.

~RGR -> ~RF (no reason to believe God is righteous -> no Reason to believe God will provide us with a Future life)

So either we don't need a future life, or we have no reason to believe that God will provide us with such a life.

~NF ∨ ~RF

Now why does the conclusion follow? It's based on a https://en.wikipedia.org/wiki/Constructive_dilemma . Since we had ~J -> ~RGR and ~RGR -> ~RF, we also have ~J -> ~RF. So we have both J -> ~NF and ~J -> ~RF. Now, we know that J ∨ ~J because this is a tautology, so using the constructive dilemma, we obtain the conclusion ~NF ∨ ~RF.