First-order logic representation

Question

I am having trouble translating these clauses to first order logic.

1) The only difference between a cat and a tiger is that a tiger kills. 2) If someone likes only people of the same sex then he is gay. 3) There is a group of people that is hostile towards "black" people while some of its members are "black". 4) Every year, there are some universities, that have some students, that pass all of the courses with good grades, without studying a lot.

Also can questions be translated to first order logic?

For example:

Who is that young man who plays the piano so well?

Answer 1

Several of these are the kind of sentences that people look at when they are interested in monadic second-order logic, which only allows quantification over unary predicates. These sentences have facets that makes them difficult to express in first-order logic with only equality (not set theory). This is closely related to the concept of plural quantification in philosophical logic. In mathematical logic, we normally just use set theory.

1) The only difference between a cat and a tiger is that a tiger kills.

Here the sentence appears to quantify over properties of cats and tigers. One somewhat natural way to express it, assuming $K$ is the property "kills", $t$ is "tiger", and $c$ is "cat", is $$ (\forall P)[ ((Pt \land \lnot Pc) \lor (Pc \land \lnot Pt)) \leftrightarrow P = K] $$ But that is not a "first-order sentence" because of the $(\forall P)$. (Of course, the sentence here is false, because another difference between cats and tigers is the one is a tiger and the other is not...).

2) If someone likes only people of the same sex then he is gay.

This one can be done in first order logic, assuming $L$ means "likes", $S$ means "same sex", and $G$ means "gay". $$ (\forall x)[ (\forall y)[ Lxy \to Sxy] \to Gx] $$

3) There is a group of people that is hostile towards "black" people while some of its members are "black".

This one also cannot be done in first order logic with equality. It is a classic-style example of plural quantification.

4) Every year, there are some universities, that have some students, that pass all of the courses with good grades, without studying a lot.

This one can be done in first-order logic with equality.

Answer 2

To get you started

1) The only difference between a cat and a tiger is that a tiger kills.

Let $D$ be the set of differences between a cat and a tiger, and let $k$ be the difference "a tiger kills".

$$\exists x \;(x\in D) \wedge \forall x \;(x \in D \implies x=k)$$

Alternatively: Assert that the universe of discourse is the differences between a cat and a tiger, and let $k$ be the difference "a tiger kills".

$$\exists x(x=k)\wedge\forall x(x=k)$$

Can you do the same for the rest?

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Also: Queries cannot be expressed in first order logic. The only expressions are declarative.