Subsection 8.13.1 Why It’s Hard
In everyday reasoning, we use all of the techniques that we’ve been discussing. But it turns out that the real world is complex and our understanding of it is incomplete.
Consider the claim:
x (Bird (x) CanFly(x))This seems straightforward enough. But, of course, it’s false.
Consider these:
Ostriches don’t fly and neither do penguins. And neither do baby birds or ones with clipped wings or ones with crude oil all over their wings.
In fact, there are very few universal claims that are true in the real world.
And there are other problems. Even coming up with predicates that capture everything we might like to be able to say and think about is hard. For example, a lot of real world concepts are squishy around the edges.
Consider the predicate CanFly that we’ve mentioned above. Do chickens fly? They can do something that is a little more powerful than “jump”. But they can’t fly across town the way other birds do. If x is a chicken, should we assert CanFly(x)?
Despite these problems, the sorts of logical reasoning that we’ve been studying are powerful tools in the real world. We can still use the argument structures that we’ve defined. And there are additional formal theories that can be used to solve some of the problems that we have with the straightforward theory that we’ve got.
Exercises Exercises
Exercise Group.
Let’s return to this version of Who Drives Me. Here are the premises:
J ∨ M John or Mary must drive me to the store.
J → L If John drives me to the store, he will be late for work.
¬L John cannot be late for work.
M → G If Mary must drive me to the store, she must buy gas.
G → Y If Mary must buy gas, she must have money.
Y → W If Mary must have money, then she must work at a paying job.
Part 1.
Focus on premise [5]. How have we simplified the world here? Are there other ways Mary could buy gas?
Part 2.
Focus on premise [6]. How have we simplified the world here? Are there other ways Mary could get money?