To build on VideCorSpoon's point of the purpose of translating english into logic say you have the very hairy looking argument made by a lawyer in a court room:
"If my client is guilty, then the knife was in the drawer. Either the knife was not in the drawer or Jason Pritchard saw the knife. If the knife was not there on October 10, it follows that Jason Pritchard did not see the knife. Furthermore, if the knife was there on October 10, then the knife was in the drawer and also the hammer was in the barn. But we all know that the hammer was not in the barn. Therefore, ladies and gentlemen of the jury, my client is innocent." *
Although for some the previous argument at first glance would make perfect sense, but for the rest of us it hurts to read. However, logic comes in to save the day. By logically translating the english statements into logical ones you can systematically prove that the previous argument is indeed sound.
So lets go through it and translate and prove that it is sound. First we need to assign variables.
G = Client is guilty
K = Knife was in the drawer
J = Jason Pritchard saw the knife
O = Knife was there on October 10
H = Hammer was in the barn
Now that we have variables to represent everything we can now translate:
[(G -> K) AND (K' OR J) AND (O' -> J') AND (O -> (K AND H)) AND (H')] -> G'
Ok so this is the logic sentence of the lawyers argument. Using Equivalence and Inference rules of logic we can now check to see if his argument is sound. First lets write down what we know (hypothesis).
1. G -> K (hyp)
2. K' OR J (hyp)
3. O' -> J' (hyp)
4. O -> (K AND H) (hyp)
5. H' (hyp)
And we are trying to prove G'
So lets do it.
6. O' OR (K AND H) (Implication 4)
7. (O' OR K) AND (O' OR H) (Distribution 6)
8. (O' OR K), (O' OR H) (Simplification 7)
9. O -> H (Implication 8)
10. K -> J (Implication 2)
11. G -> J (Hypothetical Syllogism 1,10)
12. J -> O (Contraposition 3)
13. G -> O (Hypothetical Syllogism 11,12)
14. G -> H (Hypothetical Syllogism 9,13)
15. G' (Modus Tollens 5,14)
And there we are. We just logically stepped through and proved the lawyer's argument to be true.
*this argument was taken from Mathematical Structures for Computer Science: A Modern Approach to Discrete Mathematics, 6E
by Judith L. Gersting.