One may indeed conclude that:
1) All Cats are black.
2) Sophie is a cat.
3) Therefore, Sophie is black.
BUT, not with any of the laws of Logic priorly taught. For symbolically, one can see that this wouldn't follow in any modus ponens kind of way.
Obviously, this is an invalid argument. That's why we need Quantification!
This is not a new form of logic, it's just a bit of the science behind "Quantified Statements". In reality, Quantified statements turn out to be a form of the rules taught earlier.
Universally Quantified Statements
Universally Quantified Statements are statements about all or none of a group. These are actually, "if, then" statements disguised as "all x is y". For example, if I say, "All cats are black", I'm basically saying, "If anything is a Cat, it is black". Or if I say, "No dog is yellow", I'm saying, "If anything is a dog, it is not yellow".
"x" is used as the Variable to represent "Anything" in order to symbolize Quantified statements.
So, when I say, "All dogs are confused", I say, for any x, if x is a dog, it is confused.
To symbolize this, we say, "(x)(Dx->Cx). (x) represents the "For any X". "Dx" represents the antecedent clause, "If x is a dog", and "Cx" represents the Consequent clause, "x is confused".
This way, we can put a subject in as x:
(x)(Dx->Cx) (For any x, if x is a Dog, x is confused)
Let's say Sam is a Dog. Sam, then, becomes the subject for the argument to apply to.
2) Ds (Sam is symbolized as "s" attaching to the D "Dog")
3) Ds->Cs (Sam is symbolized as "s" attaching to the C "confused")
The only way you can learn these is through examples, so try some of these out:
1) Every car is oily.
2) Sally is a car.
3) If Sally is a car, Sally is oily.
3) Therefore, Sally is Oily.
1) (x)(Cx->Ox)     (For any x, if x is a car, x is oily)
2) Cs     (Sally is a car)
3) Cs->Os    (If Sally is a car, Sally is Oily)
4) Os     (Therefore, Sally is Oily)
1) No humans have wings.
2) Sam is a human.
3) If Sam is a human, Sam does not have wings.
4) Therefore, Sam does not have wings.
1) (x)(Hx-> ¬Wx)     (For any x, if x is a human, x does not have wings)
2) Hs     (Sam is a human [Note how "H" represents the human and "s" represents Sam])
3) Hs -> ¬Ws     (If Sam is a human, then Sam does not have wings)
4) ¬Ws     (Sam doesn't have wings)
1) All Christians are morally reprobate.
2) Evan is a Christian.
3) If Evan is a Christian, then Evan is morally reprobate.
4) Therefore, Evan is morally reprobate.
1) (x)(Cx->Mx)     (For any x, if x is a Christian, x is morally reprobate)
2) Ce     (Evan is a Christ [Evan is symbolized by "e"])
3) Ce->Me     (If Evan is a Christian, Evan is morally reprobate)
4) Me     (Therefore, Evan is morally reprobate.
One last and slightly more complicated example:
1) All bears have claws.
2) Everything that has claws can kill.
3) Bobby is a bear.
4) If Bobby is a bear, then Bobby has claws.
5) If Bobby has claws, then Bobby can kill.
6) Therefore, if Bobby is a bear, Bobby can kill.
8) Therefore, Bobby can kill.
1) (x)(Bx->Cx)    (For any x, if x is a bear, x has claws)
2) (x)(Cx->Kx)    (for any x, if x has claws, x can kill)
3) Bb     (Bobby is a bear [Bobby symbolized by "b"])
4) Bb->Cb     (If Bobby is a bear, then bobby has claws)
5) Cb->Kb     (If Bobby has claws, then Bobby can Kill)
6) Bb->Kb     (Therefore, if Bobby is a bear, Bobby can Kill)
7) Kb     (Therefore, Bobby can Kill)
If you have any questions, please comment on this post or send me an email. I'm more than willing to help, as I know these posts aren't as clear as a text book would have them.