Your notation would be frowned upon by a professor of logic, but you have just demonstrated modus tollens (denying the consequent) and the formal logical fallacy of denying the antecedent quite nicely. Your car example is very much like my one about drying clothes. One of the classic ones from my first semester of doing this shit was something like this:
Take the proposition "if you drink poison then you die" and assume that it is 100% true. The two standard syllogisms we can take from this are modus ponens (affirming the antecedent):
P1. If you drink poison then you die.
P2. I just drank some poison.
therefore
C. I will now die.
...and modus tollens (denying the consequent):
P1. If you drink poison then you die.
P2. I am not dead.
therefore
C. I have not drunk any poison.
The two common formal fallacies that are similar to these, one of which was committed by Bazilla (Bazilla... committed... HA!), don't have fancy Latin names. They are affirming the consequent:
P1. If you drink poison then you die.
P2. John Lennon is dead.
therefore
C. John Lennon drank poison. (A certain Mr Champman may disagree...)
...and denying the antecedent:
P1. If you drink poison then you die.
P2. John Lennon has not drunk poison.
therefore
C. John Lennon is not dead. (Again, Mr Chapman may wish to argue the point...)
Most of the standard syllogisms are not as fun as these. They are stuff like "I will eat the fish or I will eat the steak. I did not eat the fish, therefore I ate the steak" and the mind-numbingly obvious "Dogs have four legs. Dogs bark. Therefore, dogs have four legs and bark."
As for set theory (Americans are human, Rush Limbaugh is American, therefore Rush Limbaugh is human) this is getting into the fun world of predicate logic, a wonderful little subset of deductive logic, in which you get such amazing things as universal propositions not affecting the existence of substantives and such... lots of fun...
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