29.1 Simple Informal Proofs
We reason with quantifiers in English all the time. For example,
1.Every animal in the shelter is a dog or a cat.
2. An animal is ready for adoption if and only if it’s been cleared.
3. All cats must be checked for fleas to be cleared.
4. All dogs must be washed to be cleared.
5. Floyd is a new animal in the shelter, who hasn’t been washed or checked for fleas.
6. There’s an animal not ready for adoption.
Here’s what a proof would look like:
Proof. Since Floyd is an animal in the shelter (prem 5), Floyd must be cleaned before he’s ready for adoption (prem 2). Now, Floyd is either a dog or a cat (prem 1). Next, we do proof by cases using that disjunction. Case 1: Say he’s a dog. Then he must be washed in order to be cleared (prem 4). But he hasn’t been washed (prem 5), so he’s not ready for adoption. Case 2: Say he’s a cat. Then he must be checked for fleas in order to be cleared (prem 3). But he hasn’t been checked for fleas (prem 5), so he’s not read for adoption. Thus we know Floyd’s not ready for adoption, and therefore there’s an animal not ready for adoption. Done.
That’s a long proof, but hopefully every step of it is obvious.
Our goal is to understand how the steps with quantifiers work.
The first place we use a quantifier is the first step of the proof, where we infer: “Floyd must be cleaned before he’s ready for adoption (prem 2).”
Premise 2 is a universal quantifier: “Every animal must be cleared before it’s ready for adoption.”
Since it is a universal quantifier, it holds for every object in the domain. That means it holds for Floyd.
So we used the universal quantifier by instantiating it for a particular object in the domain: Floyd.
You might be wondering, how do we know Floyd is an object in the domain? Because that is how names work: they must pick out an object from the domain. If the name didn’t do that, then it wouldn’t be functioning as a name in the language.
When we reason from a universal quantifier like this, it’s called Universal Instantiation. We instantiate the general claim in the quantifier for a specific claim about a particular object.
Besides the universal quantifiers in the premises, the other quantifier in this argument is the existential conclusion.
Notice how we proved that existential: we first proved that Floyd is not ready for adoption, which allowed us to infer that some animal is not ready for adoption.
That move is called Existential Generalization., because it moves from a particular to a general claim.
Reasoning from universals and reasoning to existentials are the two easy proof methods for quantifiers.
They are both so obvious we don’t require you to cite the name, as long as you refer to the premises correctly, like we did above.
Your turn to practice. Provide a proof of this argument:
1. All humans are mortal.
2. All mortal beings feels existential angst.
3. Socrates is human.
4. Someone feels existential angst.