I have the following question that I am trying to figure out:
'All swans are white'
Identify a natural domain of discourse for this sentence, the variable and the predicate.
I am trying to study this on my own, and this was my own practice question from Coursera. I believe the statement itself is the predicate, the domain of discourse , "all swans", and the variable is "swan". How should I think about this statement in regards to each of these definitions?
I see two ways of formalizing this sentence which give different answers to the questions. One is to have the domain of discourse be all swans. The predicate is "is white" or "are white"-predicates do not care how many things satisfy them but English does. You can write $x$ is white as $W(x)$ and the whole sentence $\forall x~W(x)$. The variable would be one swan, which we refer to as $x$.
Another approach is that swans are things in your domain of discourse, but not the whole thing. This statement picks out the swans and says they are all white. A natural domain of discourse would be all birds, all animals, all things on earth, or something similar. Now you have two predicates, $S(x)$ says $x$ is a swan and $W(x)$ says $x$ is white. You would then write the sentence $\forall x~(S(x)\implies W(x))$
Incidentally, the statement is false. I have seen black swans in the wild, but don't have a photo handy.