Subsection 1.4.5 Bad Arguments
We hope that, by the end of this class, you’ll be able not just to construct good arguments but also to identify bad ones, which come in many flavors. We’ll mention just a few of them.
Existential vs. Universal Confusion.
The claim that there exist some things that possess some particular property is very different from the claim that all things possess that property.
Activity 1.4.12.
\(E\text{:}\) Some boys sure are stupid.
\(A\text{:}\) Wait, are you calling me stupid?
\(A\) has jumped to an unjustified conclusion. Notice that \(E\) has said only that some boys are stupid. That’s very different from saying that all of them (including \(A\)) are.
Necessary vs. Sufficient Conditions Confusion.
If we claim that P implies Q, we are saying that P is a sufficient condition for Q: P being true is sufficient to guarantee that Q is true, regardless of anything else. If we say that Q implies P, we are saying that P is a necessary condition for Q: Q cannot be true if P isn’t. These two claims are different and it’s possible for one to be true while the other is false. So when we reason with such claims, we must be careful.
Activity 1.4.13.
\(A\text{:}\) I really hate big cities.
\(E\text{:}\) So we’ll have a great time. Oshkosh is really small.
\(A\) has said that if a city is big, he’ll hate it. He has not said that if he hates a place, it’s necessarily big. It’s perfectly possible for him also to hate a smaller place.
Running a Proof Backwards.
A valid proof must start with premises and reason to conclusions. Sometimes, in searching for a proof, we start with our conclusion and look for things that would lead to it. But, in the end, we can’t start with the thing we’re trying to prove. If we assume some claim P, of course (trivially) we have a proof of it. But that tells us nothing.
Activity 1.4.14.
Assume the following premises:
If it’s raining, the sidewalks are wet.
If the sidewalks are wet, someone will slip and fall.
Jayce just slipped and fell.
Here’s a “proof” that it’s raining:
If it were raining, the sidewalks would be wet. Then someone would slip and fall. Jayce did just slip and fall. So it must be raining.
Our proof is flawed. We could correctly reason that if it were raining, someone would slip and fall. But that’s not what we’re trying to do. As we saw above in the big city example, we need different claims if we want to reason in the other direction (from slipping to wet sidewalks and from wet sidewalks to rain). And, in the real world, it’s simply not true that slipping implies that the sidewalks are wet. And wet sidewalks don’t necessarily imply rain. (Maybe the sprinklers just went off.)
Problems 1.4.15.
Mark True if \(B\)’s conclusion logically follows from \(A\)’s claims. Mark False if \(A\)’s claims are inadequate to support \(B\)’s conclusion.
(a)
\(A\text{:}\) It’s always hot in the summer.
\(B\text{:}\) Everyone in this picture is wearing shorts, so it must have been taken in the summer.
(b)
\(A\text{:}\) Some really crazy people go to Alaska in January.
\(B\text{:}\) I guess Jo is crazy because she went to Alaska over New Years last year.
(c)
\(A\text{:}\) Anyone who plays tennis in August is crazy.
\(B\text{:}\) Well, since I play tennis every day, I guess I must be crazy.