Can someone explain the reasoning behind the answers given? We reviewed these in class, but I wasn't able to grasp the logic. Especially c, d, and e.

The domain of possible values for variables X and Y is {Jim, Ann, Sal, Pat, Tom}. The following facts define the values for which the child predicate is true. The child predicate is false for all other cases.

Evaluate each expression using the domain values and predicates as defined and indicate if the expression is true or false.

child (Ann,Jim)

child (Sal,Jim)

child (Pat,Ann)

child (Tom,Sal)

a. $(\forall X)$ child$(X,Jim)$

b. $(\exists X)\neg $child$(X,Jim)$

c. $(\forall X)(\exists Y)($child$(X,Jim) \rightarrow $child$(Y,X))$

d. $(\exists Y)(\forall X)($child$(X,Jim) \rightarrow $child$(Y,X))$

e. $(\exists X)(\forall Y)($child$(X,Jim) \rightarrow $child$(Y,X))$

a. FALSE (Reasoning: Any case that make it false? Yes, so FALSE)

b. TRUE (Reasoning: Any case that makes it true? Yes, so TRUE)





Let me interpret $child(x,y)$ as "$x$ is a child of $y$", then:

(C) you have to check that every child of Jim has a child, Ann is a child of Jim and Pat is a child of Ann, good. Also Sal is a child of Jim and Tom is a child of Sal, nobody else is a child of Jim, so it is true.

(D) The interpretation of this one would be "there is someone who is a child of every child oj Jim". That someone can not be Jim because Jim is not a child of Ann (and Ann is a child of Jim), it can't 't be Ann because Ann is not a child of Sal, conversely it can't be Sal because Sal is not a child of Ann and finally it can't be Tom because he is not a child of Ann.

(E)The interpretation of this one would be "there is someone who verifies that if she/he is a child of Jim then everybody is a child of her/him", if yoy substitute Jim for your variable $x$ then the implication is true because the antecedent (Jim is a child of Jim) is false, so you found someone who verifies that, and so it is true.

Hope it is clear

  • $\begingroup$ I had to re-read your answer after sleeping on it, but your phraseology now makes more sense (even though I'm still having some difficulty). Thank you! $\endgroup$ – wad11656 Feb 24 '16 at 1:28
  • $\begingroup$ Yeah, probably i didn't express it quite well, my english is not so good, i need to improve it... Note that it always get harder to follow the logic structure of a proposition when you increases the number of alternating quantifiers (universal, existential, universal...), there is a whole arithmetic hierarchy behind which classifies the formulas based on their complexity. The typical example is the definition of limit of a function, most students find it hard to grasp at the beggining because we are not used to deal with propositions of our natural languages with three quantifiers alternated. $\endgroup$ – la flaca Feb 24 '16 at 2:15

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