Translating Propositional Functions

Question

I have been given the following translating propositional logic question:

Let the universe of discourse be all people

Let F(x) = x is a farmer

Let C(x) = x has a cow

Let H(x) = x has a horse

The desired sentence is: Some farmers have cows but do not have horses

The main area of struggle lies in specifying the universe of discourse. I know that:

∃x denotes a uniqueness quantification

which could be used to denote some farmers. Thus:

∃xF(x) ≡ some farmers

Additionally:

C(x) ∧ ¬H(x) ≡ have cows but do not have horses

however I am unsure how to combine these statements so that the logic means for some farmers that have cows but do not have horses and not: for some human that might also be a farmer that has cows but do not have horses

Logical symbols

Answer 1

You really should not try to split up the sentence the way you do!

$\exists x F(x)$ does not mean 'some farmers'. Rather, it means that 'there are some farmers'

$C(x) \land \neg H(x)$ does not mean 'have cows but not horses', but rather '$x$ has cows but not horses'

So, what you should do instead is to think: ok, I want to make a claim about some farmers, and what I claim about those farmers is that they have cows but not horses. How do I do that? Well, it's a claim about some farmers, so I will need an existential, and thus the basic format becomes:

$\exists x ('x \text{ is a farmer' } \land 'x \text{ has cows but no horses'})$

Now, to say that '$x$ is a farmer', we can of course just use $F(x)$, and to say that '$x$ has cows but no horses' we can use, as we saw, $C(x) \land \neg H(x)$. OK, so plug those in, and we get:

$\exists x (F(x) \land C(x) \land \neg H(x))$

So, my advice to you is: learn some of the basic common patterns, such as 'all $P$'s are $Q$'s' becomes $\forall x (P(x) \rightarrow Q(x))$, and 'no $P$'s are $Q$'s' becomes $\forall x (P(x) \rightarrow \neg Q(x))$, etc. ... and ask yourself 'what things is the claim about?' (i.e. What is the subject term?), and what do I want to claim about those things? (i.e. What is the predicate term?). If you do that, and make those distinctions, then it becomes a matter of divide and conquer. Good luck!

Answer 2

$\vee$ means "OR" and $\wedge$ means "AND".

You should use "AND" over here rather than "OR", since the desired sentence means:
There EXISTS some x, they are farmers AND they have cows AND they do not have hourses

Therefore, the propositional logic formula will be: $$ \exists x(F(x) \wedge C(x) \wedge \neg H(x))$$

Tips: normally we use $\exists$ with $\wedge$, and use $\forall$ with $\Rightarrow$.