Thread: Inference questions

I have this question

Three person, Ben, gabriel and Alvin are suspects for housebreaking. They testify under oath, making the following assertions:

ben said: gabriel is guilty and Alvin is innocent.
gabriel said: If ben is guilty then so is alvin.
Alvin said: I am innocent and at least one of the two others are guilty.

Assume that everyone is telling the truth. Apply propositional logic based inference rules to deduce who is/are innocent, who is/are guilty. Use the following notations: B to indicate that Ben is innocent, G to indicate that Gabriel is innocent, A to indicate that Alvin is innocent, and likewise, ~B to indicate that Ben is guilty, and so on.

I derived out my working
~G ^ A
~B => ~A
A ^ (~B v ~G)
And how should I continue from here?

The first formula gives the answer for Gabriel and Alvin. The second formula is equivalent to A => B by contraposition, so this gives the answer for Ben as well.

To get help with a more formal derivation please say which inference rules you are supposed to use, as stipulated in this sticky thread.

Sorry I don't understand. Can explain it to me?

By assumption, the first formula ~G ^ A is true. The truth tables for conjunction and negation imply that A is true and G is false. Further, ~B => ~A is equivalent to A => B (i.e., these formulas are either both true or both false for any truth values of A and B). Since A is true, the truth table for implication implies that B us true. One can check that under the assignment A = B = True and G = False, the third formula is true.

To help with applying "propositional logic based inference rules," we need to know the inference rules you are using. There are many sets of inference rules used to construct formal derivations, similar to how there are many programming languages.

Which inference rules should I apply?

This depends on your course and/or textbook. Imagine that somebody asked you to write a program that, given a number n, would compute the smallest prime greater than n. You can write this program in many languages, but if your course is using Java and you write it in Haskell, you probably wouldn't get full credit.

I have to apply propositional logic based inference rules to deduce who is/are innocent, who is/are guilty.

Here is a derivation of A and B in Fitch-style natural deduction.

Code:

0  ~G /\ A	Given
1  A		1 Conjunction elimination
2    ~B		Assumption
3    ~B => ~A	Given
4    ~A		2, 3 Modus Ponens
5    False	1, 4 Negation elimination
6  ~~B		2, 5 Negation introduction
7  B		6 Double-negation elimination

~G is derived from the given ~G /\ A in one step using conjunction elimination.

I have to apply propositional logic based inference rules

I have already said in post #4 that there are many sets of inference rules for propositional logic, and some of them look very different from each other. The only way to find out which rules you need to use is to go back to your textbook/lecture notes or ask your instructor. Mathematics starts by positing precise definitions of objects it studies (in this case, inference rules). Without definitions, there is no math.