I do not understand Implication and Inference, I am going over the MIT Computer Science course and they have this part in their lecture notes, why is the second rule not a logical deduction? Can someone explain this in simpler terms?
The way I understand it in logical thinking p= Traffic light is green q = car starts driving.
Not(p) Implies Not(q) means that light is not green so the car does not drive, So why is the consequent saying that if the car is driving the light is green?
Rule 2.1.4. $$\dfrac{\operatorname{not}(P)\textrm{ implies } \operatorname{not}(Q)}{Q\textrm{ implies } P}$$ On the other hand:
Non Rule $$\dfrac{\operatorname{not}(P)\textrm{ implies } \operatorname{not}(Q)}{P\textrm{ implies } Q}$$ is not sound, if $P$ is assigned $\rm T$ and $Q$ is assigned $\rm F$ then the antecedent is true and the consequent is not.
Note: that a propositional inference rule is sound precisely when a conjunction (AND) of all its antecedents implies the consequent.
Here is a link to the whole chapter which it is in, starts at pg 2
The second "rule" isn't a deduction rule because it doesn't preserve truth. "If I'm Napoleon then you're not Napoleon" is trivially true, as I'm not Napoleon. But from that, you can't conclude that "If I'm not Napoleon then you're Napoleon": because, again, I'm not him, we could infer that you are in fact Napoleon — which I assume is false ;)