Lesson on Formal Logical Notation -Epps Discrete Mathematics With Applications
Note: I am not confused about how to negate these statements but just about how to initially translate them into formal notation. The negation will follow once I understand that. Thanks for your help!
Trying to understand the proper way to parse informal statements into formal logical ones, and this section has me confused. First, is this even the proper way to form these statements? From the examples I've seen here on Stack Overflow, it seems to me that most people place all ∃ and ∀ symbols before the parentheses, whereas Epps places these symbols within the parentheses in a confusing manner (to me).
Second, this section is very brief and I don't understand how these types of statements are to be translated. Especially statement C, I don't understand where that $\to$ comes from. Again from examples on here I would have thought the statement should be initially translated as
$\forall x, \exists y ($Circle$(x), $Square$(y) \wedge $SameColorAs$(x,y))$ or something like that.
Any help would be appreciated.
This kind of interpretation takes a lot of practice.
As a general rule, statements like "For all circles $x$,..." will have a conditional inside a universal quantifier, while statements like "There exists a circle $x$..." will have a conjunction inside an existential quantifier.
This is confusing, so let's go through specific examples.
(As for notation, my preferred notation for quantifiers is $(\forall x)(...)$ and $(\exists x)(...)$. If that confuses you, I'm happy to edit to try to adapt to something else. Notation will vary a bit in different sources, as you've noticed.)
Consider "All circles are round." Semi-formalizing, we have "For all circles $x$, $x$ is round." Now when we write $\forall x$, we'll be quantifying over all objects $x$, and we need a way to get the circle condition in there. Let $Cx$ mean "$x$ is a circle" and $Rx$ mean "$x$ is round." If we wrote a conjunction $(\forall x)(Cx \& Rx)$, this would mean that for every object $x$, $x$ is a circle and $x$ is round, i.e., all objects are round circles. This is clearly much stronger than what we wanted to say: we wanted to say something about circles, but not about any other object. This is nicely done with the conditional $(\forall x)(Cx \implies Rx)$. Now if $x$ is not a circle, $Cx \implies Rx$ comes out true vacuously, and if $x$ is a circle, $Cx \implies Rx$ tells us $x$ is round. This seems to capture what we want to say by "All circles are round."
On the other hand, consider "There is a purple circle," or semi-formally, "There is a circle $x$ such that $x$ is purple." Let $Cx$ mean "$x$ is a circle" and $Px$ mean "$x$ is purple." A conditional $(\exists x)(Cx \implies Px)$ would be too weak because all it would give would be the existence of some object $x$ such that $Cx \implies Px$ holds; any non-circle $x$ would do, since $Cx \implies Px$ would hold vacuously, whereas we want our statement to give the existence of a circle. So the conjunction $(\exists x)(Cx \& Px)$ is the way to go.
Now try applying this rule (conditionals with universals, conjunctions with existentials) to the examples you have.