Predicate Logic (family relationships problem)

[Knark]

Homework Statement

Formalize (in PL) the relations/predicates stated in (a)-(e) using just these relations/predicates:

1) Pxy: x is a parent of y
2) Fx: x is a female
3) Sxy: x is a sibling of y

(a) x is an uncle of y
(b) x is a great-aunt of y
(c) x is an aunt
(d) x is a great-uncle
(e) x is an uncle of a great-aunt of y

Homework Equations

The above relations/predicates (1, 2 and 3) are the only ones we are allowed to use. Furthermore, in brackets the question has: A great-aunt/uncle is the sister/brother of a parent. You can assume that you're working within a universe of discourse consisting just of human beings.)

The Attempt at a Solution

There are two aspects of this problem that I am unsure of which can be illustrated by what I have tried so far.

(a) ~Fx & (Pzy & Sxz)
(b) Fx & (Pwz & (Pzy & Swx)
(c) Fx & (Pzy & Sxz)

I haven't tried (d) and (e) since I ran into some confusion doing these first three. I'm wondering how I differentiate between (a) and (c). In my answers they are identical except for the ~Fx to show that an uncle is not a female. Yet I believe they should be distinct because (a) asks for "x is an uncle of y" but (c) asks only for "x is an aunt" and not "x is an aunt of y". How do I formalize the fact that they are different in this way.

Additionally, I'm wondering whether I am within the bounds of the question to include the predicates z and w. I just don't see any other way to formalize a relationship that includes an uncle or great-uncle without using a third and fourth predicate.

 

[Billy Bob]

"x is an aunt" means "there exists y such that x is an aunt of y."

 

[Knark]

So I would have to add in (∃y)?

 

[bfbruin]

You must remember to define your scopes.

In order for each sentence to be a proper sentence of PL, you must bind all your variables. For example, my answer for a):

(∃x)(∀y)[~Fx&(∃z)(Sxz&Pzy)]

I'm definitely going to have to check it over, but I believe this to be the gist.

 

[Knark]

How would you contrast your answer for a) with the answer for c).

Would you switch up the quantifiers into something like:

(∃y)(∀x)[~Fx & (∃z)(Sxz & Pzy)]

How exactly would you show that "x is an aunt" and not "x is an aunt of y".

 

[bfbruin]

I was thinking:

(∃x)(∃y)[~Fx & (∃z)(Sxz & Pzy)]

 

[Knark]

I like it.

So for b) and d) something like:

b: (∃x)(∀y)[Fx & (∃w)(Swx & (∃z)(Pwz & Pzy)]
d: (∃x)(∃y)[~Fx & (∃w)(Swx & (∃z)(Pwz & Pzy)]

 

[bfbruin]

It's what I put. I'm not sure it's correct, but it seems to work. =P

 

[Knark]

Woops, in c) we have "~Fx" but it should actually be "Fx" since x is an aunt. What did you get for e)?

 

[bfbruin]

E) is just an extension. Think of it as (x is not female, and x is the sibling of the parent of the parent of the sibling of the parent of y), and for all things y.

 

[Knark]

Does this seem similar to your answer?

(∃x)(∀y)[~Fx (∃u)(Sxu & (∃v)(Puv & (∃w)(Pvw & (∃z)(Swz & Pzy)]

I'm worried because I had to use u, v, w, x, y, z to fill in all the needed people but I seem to remember something about only being allowed to use w, x, y, z. Did you use 6 variables also or am I missing something.

 

[bfbruin]

Remember to close up your parentheses.

You can use w-z, so I added on w₁

 

[Knark]

(∃x)(∀y)[~Fx (∃w1)(Sxw1 & (∃z1)(Pw1z1 & (∃w)(Pz1w & (∃z)(Swz & Pzy))))]

Did you only use w1? My answer needs 6 variables so I added z1 to account for that. Good catch on the parentheses, are they properly closed now?

 

[bfbruin]

I used w2 as well, sorry about that.

And the parentheses appear correct. By the by, check your private messages.

 

[bestgym]

are you sure you need to bind all variables? it doesn't ask for a sentence, it asks for a formula

 

[Knark]

That's what I wasn't sure of, hence my first attempts included no use of Universal or Existential Quantifiers. I believe it actually wants just a formula and not a sentence since question #3 of our homework is the question that specifically states "Formalize the two SENTENCES". It makes sense that it would proceed from simpler formula's into more complex sentences. Just a theory though because without the Quantifiers I have no idea how to display the difference between (a) and (c).

Btw: Are you from our class also?

 

[bfbruin]

It says to formalize these sentences in PL. I would imagine that you should try and keep the sentence structure, if only because adding x-quantifiers will show the difference between the cases where a specific y is mentioned, and where it is only assumed.

Having said this, I have no clue. I only put them in there because of the above reason, and I can be persuaded otherwise.