# Logic: wff into symbols

All birds cannot fly.

If $p(x)=x$ is a bird and $q(x)=x$ can fly, then the translation would be $$\forall x(p(x)\to\lnot q(x))$$ or $$\forall x(p(x)\land\lnot q(x))$$?

Both make sense to the sentence, both seem correct but they are not equivalent. Which is the correct?

I think it's the second one.

• There is an ambiguity in the English language when you say "All X is not Y". For example, "all that glitters is not gold" should more strictly be "not all that glitters is gold". So "all birds cannot fly" could be interpreted as "not all that is a bird can fly". Commented Nov 19, 2017 at 11:14
• @DanielV but "all that glitters is not gold" is different of "not all that glitters is gold", in the first one means that all of them it's not... and in the second one means that there might be some of them that it's not... Commented Nov 19, 2017 at 19:24
• Tell that to Shakespeare : nfs.sparknotes.com/merchant/page_90.html Commented Nov 19, 2017 at 23:14
• @DanielV You make a good point. Interpreting the original statement in that way might make more sense because "not all birds can fly" is actually a true statement. I will update my answer to account for this interpretation. Commented Nov 20, 2017 at 17:54

The first one is correct. It all comes down to the universe of discourse: if $x$ can be anything, then $x$ could be something that is not a bird, e.g. a bee. Obviously, the original statement says nothing about bees, so this case should be ignored. The first translation takes care of this, but the second one doesn't.

If you try to translate the statements back into English, perhaps this becomes more clear:

1. $\forall x(p(x)\to \neg q(x))$ For everything you give me, if it's a bird, it can't fly.
2. $\forall x(p(x)\land \neg q(x))$ Everything is a bird and doesn't fly.

The second is obviously not what the original statement is saying.

EDIT: Everything so far has been under the interpretation that by "all birds cannot fly" it is meant that "no bird can fly." If, however, you are using the interpretation analogous to that of "all that glitters is not gold", then the original statement is actually saying "not all birds can fly", i.e. there exist flightless birds. In this case, you would represent the statement like so: $$\exists x(p(x)\land \neg q(x))$$

• I didn't know the difference of this translations, now I understand more thank you! Commented Nov 19, 2017 at 4:41

$\forall x~(p(x)\to\neg q(x))$ says "all things cannot fly if they are a bird".

$\forall x~(p(x)\wedge\neg q(x))$ says "all things are a bird and cannot fly".

So, no, they are not the same claim and only one of them is the one you want.

The English language is tricky. An intended logical way to write "All birds cannot fly" could be

$$\{x \mid \text{Birds(x)} \} \ne \{x \mid \text{Fly(x)} \}$$

Similarly to how someone would say "everyday is not your birthday" or "all that glitters is not gold". And given that such a phrase would probably only be used to identify the existence of penguins or ostriches, if you heard such a phrase casually, a person would almost certainly mean "there are birds which don't fly".