Hi, I am reviewing my lecture notes for an AI class and I have a question about a derivation that I am hoping someone on this board can help me clarify better than was done in the class. I have a feeling it is unclear because they want to use it as a test question -- so I want to be ready! So here I am converting from first-order predicate logic to clausal form through these 6 steps... 1. remove implications 2. move negations to atomic level 3. apply disjunctions to only connect atoms (keep conjuncts) 4. eliminate existential quantifier by using All Y Exists X ( S(y,x) ) = All Y ( S ( y, f(n))) substition (to apply sigma substitution set) 5. rename vars that have the same name and give specific instances to universal quantifier 6. re--express disjunctions as clauses. Here is the series of steps that I am questioning. Original statement: All X Exists Y ( person(x) -> person(y) & loyal_to(x,y) ) (REMOVE IMPLICATION) All X Exists Y ( !person(x) OR person(y) & loyal_to(x,y) ) (MOVE NEGATIONS TO ATOMIC LEVEL -- HERE IS WHERE I AM CONFUSED AT THEIR ANSWER All X Exists Y [ ( !person(x) OR person(y) ) & ( !person(x) OR (loyalto(x,y) ) ] Okay my question is this -- Where did the !person(x) come from in the second term? I don't see how distributing the disjunction turns it into that. Or is it because the conjunct got distributed? That seems to be true, however I don't know WHY they use & in stead of the ^ symbol. There are other places where they use the upside-down caret for OR so why do they then use the & here? Any clarification on this issue would be appreciated

ok, thanks! I have one other related question.. for the modus ponens inference rule, I want to mentally have a way of thinking of it that makes sense to me that jives with the boolean values represented.

The rule, X -> Y & X gives Y is true.

If you represent this as a truth table, then it doesn't seem to add up.

EDIT: I have really tried to get this to format properly and I cant' seem to do it -- but the values in column 4 should match those in column 2 if I understand it properly -- however, this may be exactly the point that I am understanding wrong. If so, this is what I would like to know what to understand differently.


X Y X->Y (X->Y) ^ X
0 0 1 0
0 1 1 0
1 0 0 1
1 1 1 1



I know that in terms of knowledge representation, it makes perfect sense -- right now X is true, and X implies Y therefore Y must be true. But why is this not consistent with the truth table representation? Please, I know this seems simple but I want to understand it properly.. Any clarification would be greatly appreciated.

because your truth table is wrong! check it again :)

metrix

ack! thanks, that's what I get for trying to look at this stuff too late at night. Here's the revised truth table, which does reflect that the rule holds:

X Y X->Y (X->Y) ^ X
0 0 1 0
0 1 1 0
1 0 0 0
1 1 1 1