# Thread: Predicates And Quantifiers Hell

1. ## Predicates And Quantifiers Hell

I'm REALLY struggling with this subject. I think I'm there or there abouts but have this feeling I'm missing something out (which is driving me crazy). Any help really appreciated.

Predicates:
Set Family is members of a family.
Set Ages is integers giving ages of members of Family

BROTHER(p, q): p is a brother of q
MARRIED(p, q): p and q are married
TALL(p): p is over six feet tall
AGE(p, n): p is n years old.

Express each natural language statement formally using usual notation of formal logic, predicates, sets mentioned above and == for equality of strings.

A. "Jackie" is twelve years old and is not over six feet tall.
B. Everyone is the family who is over six feet tall is married.
C. The only member of the family who is both over six feet tall and thirty years old is "Jason", who is a brother of "Sue".

My attempts are below.

A.

$
AGE("Jackie", 12) \wedge \neg TALL("Jackie")
$

B.

$
\forall p in Family[TALL(p) \rightarrow MARRIED(p, q)]
$

C.

$
\exists p in Family[ ((TALL(p) \wedge AGE(p, 30)) \rightarrow (p == "Jason")) \wedge BROTHER(p, "Sue")
$

With B i'm not sure about q?

2. In B, $q$ should be existentially quantified. I.e., For every $p$ ... there exists a $q$ such that $p$ and $q$ are married.

3. Proposition C seems to assert both that such person (30 years old and tall) exists and that he is Jason and a brother of Sue. I believe it should be split into two conjuncts. The first says that some person has all these qualities (or that Jason is 30, tall and a brother of Sue) and the second says that every person who is 30 and tall is Jason (and a brother).

Right now your formula in C is true even for a family without Jason. Indeed, there exists a $p$, namely, "Jackie", for whom the premise is false and so the whole formula is true.

4. Ever been waiting for that lightbulb moment but it never came...

Here's my attempt at fixing b.

$
\forall p in Family[TALL(p) [\exists q \rightarrow MARRIED(p, q) \wedge MARRIED(q, p)]]
$

5. $\exists q\to\dots$ is not a well-formed formula (a programming language compiler would say "syntax error"). Also, it is not necessary to assert that q is married to p, in addition to p being married to q. The sentence simply says "everyone ... is married" (to somebody).

Your original attempt at B is almost right. You only need to properly introduce q. Your formula says that every tall member of the family is married to q. This is not a proposition because it may have different truth values depending on q. For it to become a proposition, q must be introduced, and variables are introduced only by two quantifiers: "for all" and "exists". So, instead of saying that every tall p (note that p is properly introduced using "for all") is married to q, say that every tall p is married to somebody, i.e., there exists a q such that p is married to him/her.

6. B.

$
\forall p in Family[\exists q[TALL(p) \rightarrow MARRIED(p, q)]]
$

Hows that looking?

7. Almost perfect. This formula is equivalent to $\forall p\in\mbox{Family}.\,(\mbox{TALL}(p)\to\exists q\,\mbox{MARRIED}(p,q))$, which is preferable.

The first variant can be put into English like this: For every member of the family p (even short ones) there exists a q such that if p is tall, then p is married to q. The second variant is closer to the original: For every tall member of the family p there exists a q such that p is married to q.

The reason they are equivalent is that in the first variant, for short people p one can specify, for example, q = Alexander the Great. Since the premise TALL(p) is false, the whole implication is true and so such p does not matter.

8. Below is my revised attempt for C.

$
\forall p in Family[ ( TALL(p) \wedge AGE(p, 30) \wedge BROTHER(p, "Sue") ) \rightarrow (p == "Jason") ]
$

9. C. The only member of the family who is both over six feet tall and thirty years old is "Jason", who is a brother of "Sue"
Yesterday 05:03 PM
This is not right because the proposition asserts, under certain conditions, that some person is a brother of Sue. On the other hand, in your formula, being a brother is one of the conditions. There is a difference between "If a person is tall and ..., then he is a brother of Sue and ..." and "If a person is a brother of Sue and ..., then ...".

What do you think of my earlier post?
Proposition C seems to assert both that such person (30 years old and tall) exists and that he is Jason and a brother of Sue. I believe it should be split into two conjuncts. The first says that some person has all these qualities (or that Jason is 30, tall and a brother of Sue) and the second says that every person who is 30 and tall is Jason (and a brother).

10. Here is my revised C.

$
\exists p in Family[ TALL(p) \wedge AGE(p, 30) \wedge BROTHER(p, "Sue") ]
$

$
\forall p in Family[ ( TALL(p) \wedge AGE(p, 30) ) \rightarrow (p == "Jason")
$

$
\wedge BROTHER("Jason, "Sue") ]
$

This seems to make more sense but I'm not sure of the layour of it. At the moment its two formal statements I think when it should be one. Should I remove the ] at the end of the first part to join the two together?

11. I think this is equivalent to the English sentence. You need to join both formulas using $\land$.

By the way, you can write \in for $\in$.

12. Thanks for all the help, this has really helped me get this subject