Can anyone confirm if I have these correct? Or if not, where I am going wrong?
Translate these statements into English, where $K(x)$ is '$x$ is a Kangaroo' and $H(x)$ is '$x$ hops' and the domain consists of all animals.
- (a) – $\forall x(K(x)\rightarrow H(x))$ – "All animals that hop are Kangaroos."
- (b) – $\forall x(K(x) \land H(x))$ – "All animals are Kangaroos and hop."
- (c) – $\exists x(K(x) \rightarrow H(x))$ – "There exists a Kangaroo that hops."
- (d) – $\exists x(K(x) \land H(x))$ – "There exists an animal that is a Kangaroo and Hops."
Best Answer
If you can come up with good "test cases", you can use them to help you decide if your translations are correct.
Consider statement (a):
$$\forall x(K(x)\rightarrow H(x)).$$
This statement is satisfied in a universe in which the only animals are some kangaroos (every one of which hops), some frogs (every one of which also hops), and some snakes (which never hop). Let $x$ be any animal in this universe. Then either $x$ is a snake, so $K(x)\rightarrow H(x)$ is true because the antecedent is false; or $x$ is a frog, so $K(x)\rightarrow H(x)$ is true because the conclusion is true (also because the antecedent is false); or $x$ is a kangaroo, so $K(x)\rightarrow H(x)$ is true because the conclusion is true.
But it is not true in this universe that all animals that hop are kangaroos. Indeed, there are some frogs in this universe which are animals that hop but are not kangaroos.
An alternative way to translate $\forall x.P(x)$, where the domain of $x$ is "animals", is to start with the words "Every animal" and continue with an English phrase meaning "satisfies $P(x)$ where $x$ is that animal." The expression $K(x)\rightarrow H(x)$ signifies that if $x$ is a kangaroo, $x$ hops; so we might translate (a) as, "Every animal is such that if it is a kangaroo, it hops."
The preceding quoted sentence is very awkward English, however. An alternative wording is, "Every animal, if it is a kangaroo, hops." But most people, I think, would have more trouble parsing this sentence than it is really worth. Another alternative is to recognize that a common English translation of $K(x)\rightarrow H(x)$ is, "$x$ hops if $x$ is a kangaroo." Therefore the fully quantified statement in (a) could be translated, "Every animal hops if it is a kangaroo."
We can check that the universe of kangaroos, frogs, and snakes satisfies all of the actual sentences above that start with "Every animal."
On the other hand, if we introduce a kangaroo into our universe that cannot hop because it has no legs, the mathematical statement (a) is falsified and so are all of the "Every animal" translations of it. This of course does not prove that the translation is accurate, but it is a good test case to weed out inaccurate translations.
I would use similar techniques of modeling and translation for the other sentences, keeping in mind that sometimes a statement that starts with $\exists x$ can be translated into English starting with the words, "Some animal." Also keep in mind that good models for existential quantifiers are often minimalistic, for example, a universe with only a single frog and no other animals satisfies (c); so does a universe populated only by a single snake. But a universe whose entire animal population consists of legless kangaroos does not satisfy (c).