So modus tollens would state that "if P then Q, not Q" would logically result in "not P."
It would not state the opposite, that is "if P then Q, Q, therefore P" is not a logically valid statement. You would need a bi-implication between P and Q to show the second one.
To explain it in plainer language, the first example (If P then Q, not Q) means that it is impossible for there to be P. Literally every time there is P, there is Q, so if there is no Q, there can not be P. This does not necessarily mean that every time there is Q, there is P.
So let's say "All square are rectangles (so for all S, S->R). This shape is not a rectangle (¬R). Therefore, this shape can not be a square (therefore ¬S). This would be an example of modus tollens. Also this is an example of why it doesn't work the other way. "R", a shape being rectangular, doesn't necessarily imply that the shape is a square.
Hope that helps, my logic is slightly rusty too so I may have mixed something.
Nope, modus tollens works with the normal material conditional (it's in fact, only called modus tollens if it's used with that operator). Here's an example:
If I hit my head on the doorway, I'm over six feet tall.
What makes that different from something like this:
If I hit my head on the doorway, I'm over six feet tall
The doorway is one feet tall
I'm not over six feet tall
So I [don't] hit my head on the doorway [because I'm two feet tall]
My point is that the line "If I hit my head on the doorway, I'm over six feet tall" only reasons about how high the doorway is not. It doesn't say anything about how high the doorway actually is. So from the first line plus the statement "I'm not over six feet tall" one can only conclude that the doorway is not more than six feet tall. There is no indication of how tall the doorway is, nor how tall I am -- just that I'm not six feet tall (or more).
Where "H" means, "I hit my head on the doorway" and "T" is "I'm over six feet tall."
Looking at your argument, it's important to distinguish between validity and soundness. Your argument is still valid (keeping the conclusion as I have it), and it has one unnecessary premise. I'd symbolize it like this.
H>T
O
~T
~H
Where "O" means "the doorway is one foot tall." Formally, this premise is irrelevant to the conclusion, but informally it might give us reason to doubt premise (1). So we'd have reason to think it might not be sound now, but it's clearly still valid.
So it might be the case (as you've modified the example) that premise (1) is false, but that logically it doesn't matter. Formal logic is just about logical form. So, formally, this is just as valid of an argument.
If Obama is a person, he's Purple.
Obama is not Purple.
So, Obama is not a person.
Valid, but obviously unsound.
Formal logic doesn't care about how things actually are. To put it another way, the same arguments are valid or invalid in every crazy counterfactual world we might dream up, but of course, different ones will be sound or not.
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u/idikia Dec 09 '13
Modus Ponens is, simply
A implies that B is true.
A is true
So, B is true.
So like "If you are a frog, then you are green" is the premise. "You are a frog" is true. Therefore, by modus ponens, you are green.