Subsection 5.1.9 Working with Universal Quantifiers: Arbitrary Elements
To make this work in the case of universally quantified expressions, we’ll introduce the idea of an arbitrary element. What we mean by “arbitrary” is that the element has no additional characteristics other than being an element of the universe.
To see how this helps, let’s return to the Breathes syllogism problem. Recall that we have:
[1] ∀x (Student(x) → Person(x)) All students are people.
[2] ∀x (Person(x) → Breathes(x)) All people breathe.
And we want to prove:
[3] ∀x (Student(x) → Breathes(x)) All students breathe.
We’ll assume a universe of living things. Let’s let c be a name for some “arbitrary living thing”. We could call our arbitrary living thing anything we want. We could call him/her/it supercalifragilisticone. The only thing that matters is that we don’t pick a name that we’re using anywhere else in our system. We don’t want to be able to pick up any extra information about our “arbitrary living thing” that we wouldn’t know of absolutely every living thing.
We know, from [1] above, that anyone who is a student is a person.
There ought to be an inference rule (and soon we’ll define one) that lets us apply this claim to the particular case of our arbitrary living thing c. That would give us:
[1a] Student(c) → Person(c)
Similarly, from [2] above, we have that anyone who is a person breathes.
The same rule that let us go from [1] to [1a] should let us go from [2] to [2a], which makes a particular claim about our arbitrary living thing c:
[2a] Person(c) → Breathes(c)
Now we’ve got two quantifier-free expressions. The Boolean hypothetical syllogism rule can be used to chain [1a] and [2a] together to produce another claim about our particular living thing c:
[3a] Student(c) → Breathes(c)
Now comes the biggie: Since c was an arbitrary living thing about whom we knew nothing except what we could derive from general statements about all living things, anything we know about c must generalize to the entire domain of living things. So we need a second new inference rule that will allow us to conclude that “any student breathes”:
[3] ∀x (Student(x) → Breathes(x))
The details will come soon, but let’s review the big picture of what we just did because this is the key:
We had a universal statement (actually two of them).
Using new-rule-to-come-1, we gave a name to an arbitrary element.
We argued (using Boolean logic) about the arbitrary element and came to some conclusion about it.
And then, since it was arbitrary, we expressed the conclusion, using new-rule-to-come-2, as a new universal statement.
Checkpoint 5.1.2.
1. Assume the following premises:
[1] ∀x (Phlobber(x) → Crazy(x)) All phlobbers are crazy.
[2] ∀x ((Green(x) ∧ Zamzow(x))→ Crazy(x)) All green zamzows are crazy.
[3] ∀x (Crazy(x) → Funny(x)) All crazy things are funny.
Let’s use the reasoning process that we just described. Let c be an arbitrary element of the universe (as we did above). Consider the following statements that we might like to derive:
[I] Phlobber(c) → Crazy(c) If c is a pholbber, c is crazy.
[II] Zamzow(c)→ Crazy(c) If c is a zamzow, c is crazy.
[III] Crazy(c) → Funny(c) If c is crazy, c is funny.
Which of the following statements is true:
Exactly one of these can be derived using the idea of an arbitrary element.
Just I and II can be derived using the idea of an arbitrary element.
Just I and III can be derived using the idea of an arbitrary element.
Just II and III can be derived using the idea of an arbitrary element.
All of them can be derived using the idea of an arbitrary element.