Predicate for a simple statement

Question

Today is my first day working with predicate logic. Here is what is given (I've written it down myself not sure if this is the correct way):

G(w) → w is a genre
L(x, G(w)) → x likes w genre
F(x, y) → x is from country y
S(m,g) → Suggest m mood song from genre g.
P(x,m) → x has m mood.

I want to write predicate logic for the statement:

All people from Jamaica likes Reggae genre.
If a person is from Jamaica and they are in a happy mood suggest a song from Reggae genre.

How can I write the above rule in predicate logic terms? I'm also not sure if L(x, G(w)) is the right way to describe x likes w genre

Answer 1

You need constant "raggae" (say "$r_0$") and "Jamaica" ("$j_0$"), such that $G(r_0)$ holds (Raggae is a genre) and $\exists x: F(x,j_0)$ (somebody comes from Jamaica , so it's an existing country; we are not given a predicate $C(x)$ that says $x$ is a country, or some such, while we do have one for "genre".

I think $L(x,w)$ means $x$ likes (genre) "$w$", you cannot have $G(w)$ as the second argument (as it is a Boolean predicate, not a member of the "universe" (which contains "people", "moods", "genres" and "countries".

All people from Jamaica like the genre of Raggae:

$\forall x: (F(x,j_0) \to L(x, r_0)$: for all things $x$, if $x$ comes from Jamaica (the constant we introduced above) (implicit is that $x$ is then a person?) then $x$ likes $r_0$, the constant for that genre.

"If people are from Jamaica and in a happy mood", can be expressed if we have a constant "happy", which must be a "mood". Say $h_0$ means happy and $P(x, h_0)$ means person $x$ is in a happy mood. So the second then can become:

$\forall x: (F(x,j_0) \land P(x,h_0)) \to S(x,r_0)$

Answer 2

I'm also not sure if L(x, G(w)) is the right way to describe x likes w genre

No. You are never allowed to use predicates inside predicates. Which makes sense: a predicate is a statement about objects, not about predicates.

So, you have to use $L(x,y)$ ... and this will translate into '$x$ likes $y$', rather than '$x$ likes genre $y$'.

The nice thing about this is that if $x$ and $y$ are both persons, then you can still use $L(x,y)$ to say that (person) $x$ likes (person) $y$.

However, if you want to say that '$x$ likes genre $y$', you'll have to make explicit that $y$ is a genre, i.e. you'd do $L(x,y) \land G(y)$. Indeed, there is nothing in $L(,y)$ that requires $x$ to be a person, so if you want $x$ to be a person, you'll have to explicitly add $P(x)$ as well, where $P(x)$ means '$x$ is a person'.

Likewise, I strongly suggest to make the following changes to your symbolization scheme:

$G(w)$: $w$ is a genre

$L(x,y)$: $x$ likes $y$

$F(x, y)$: $x$ is from $y$

$S(x,y)$: $x$ suggest $y$.

$H(x)$: $x$ is in a happy mood.

$P(x)$: $x$ is a person

$S(x)$: $x$ is a song

$j$: Jamaica

$r$: Reggea

With this:

All people from Jamaica likes Reggae genre.

becomes:

$\forall x ((P(x)\land F(x,j)) \to (L(x,r)\land G(r)))$

and

If a person is from Jamaica and they are in a happy mood suggest a song from Reggae genre.

which I assume is meant as:

If a person is from Jamaica and they are in a happy mood, then they suggest a song from Reggae genre.

translates as:

$\forall x (P(x) \land F(x,j) \land H(x)) \to \exists y (S(y) \land F(y,r))$

Note that you never really use the $G$ 'is a genre predicate', since both sentences referenced the specific genre of Reggae.