# Predicate Logic (family relationships problem)

• Knark
In summary, the formal relations/predicates stated in (a)-(e) can be formalized as follows:1) Pxy: x is a parent of y2) Fx: x is a female3) 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
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.

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

So I would have to add in (∃y)?

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.

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".

I was thinking:

(∃x)(∃y)[~Fx & (∃z)(Sxz & Pzy)]

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)]

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

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

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.

(∃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.

Remember to close up your parentheses.

You can use w-z, so I added on w1

(∃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?

I used w2 as well, sorry about that.

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

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

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?

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.

## 1. What is Predicate Logic?

Predicate Logic is a formal mathematical system used to represent and reason about relationships between objects, individuals, or propositions. It is based on the use of predicates, which are statements that describe a property or relationship, and quantifiers, which specify the extent to which the predicate applies.

## 2. How is Predicate Logic used in the context of family relationships?

Predicate Logic can be used to represent and reason about various family relationships, such as parent-child, sibling, and grandparent-grandchild relationships. It allows us to make precise statements about these relationships and use logical rules to draw conclusions about them.

## 3. What are some common symbols used in Predicate Logic for family relationships?

Some common symbols used in Predicate Logic for family relationships include ∀ for "for all" or "every", ∃ for "there exists" or "some", → for "implies", and & for "and". These symbols are used to represent quantifiers and logical connectives in statements about family relationships.

## 4. Can Predicate Logic be used to solve real-world problems related to family relationships?

Yes, Predicate Logic can be used to solve real-world problems related to family relationships, such as determining who is eligible for inheritance or making deductions about family medical history. It provides a formal and rigorous way to represent and reason about relationships, making it a powerful tool for problem-solving in this context.

## 5. What are some limitations of using Predicate Logic for family relationships?

One limitation of Predicate Logic is that it is a highly formal and abstract system, which may make it difficult for non-experts to understand and use. Additionally, it may not be able to capture the complexities and nuances of real-life family relationships, as it relies on simplifying assumptions and does not account for emotions or personal experiences. Finally, as with any logical system, it is only as accurate as the premises and rules it is based on, so incorrect or incomplete information can lead to incorrect conclusions.

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