Rewriting arguments using quantifiers, variables, and predicate logic

Question

There are two statements which sounds similar to me but their answers are different according to answer sheet.

1. All birds can fly

Answer: ∀ x [B(x) → F(x)]

2. Some men are genius

Answer: ∃ x [M(x) ^ G(x)]

In the first part implication is used but in second part AND is used. There is no mention of 'if then' or 'And' in statements so why the answers are different for similar sounding statements?

Answer 1

In the system you're studying (first-order logic), a language has a fixed "universe" (also called a "domain of discourse"), and the quantifiers $\forall$ and $\exists$ always range over (i.e. "apply to") all the objects in that universe.

If the universe is just birds (i.e. if our language is only for talking about birds and nothing else), then "all birds can fly" can be written as just $\forall x F(x)$, since the "birds" part is built into the $\forall$. Similarly, if the universe is just men (sad!), then "some men are genius" can be written as $\exists x G(x)$. However, if we want write both statements in the same language, then the domain of discourse needs to include both men and birds, so we think of the universe as just "objects" and introduce predicates $B$ and $M$ to identify the birds and men. The quantifiers of this language range over all objects.

It's fairly clear that "some men are genius" is equivalent to "there exists an object that is a man and is genius", so we can write it as $\exists x (M(x)\land G(x))$. "All birds can fly" is trickier: we want to say something about just birds, but $\forall$ is going to give us a statement about all objects. To make this work, we need a formula inside the $\forall$ that says $F(x)$ if $x$ is a bird but says nothing extra about $x$ if $x$ is not a bird. You should convince yourself that such a formula is $\lnot B(x)\lor F(x)$, which is equivalent to $B(x)\to F(x)$ by the definitions of the logical symbols. We think of $\forall x (B(x)\to F(x))$ as saying "for every object x, if x is a bird, then x can fly", and it works because $B(x)\to F(x)$ is true when $x$ is not a bird.

Answer 2

There are two ways to look at this. In the first statement about birds that can fly, the implication, B->F, just as any other implication can be written as F or -B, or else, -(B and -F). In the second statement about men geniuses, it concerns the existence of a man who is a genius. It is the same as stating that there is a genius who is a man. Here, being a man and being a genius have symmetric roles. In the first statement, being a bird and flying do not have symmetric roles and implication is not symmetric. For example, a single flying insect would invalidate the statement that all flying things are birds.