Introducing Universal Generalization

Universal Generalization is a rule of predicate logic that lets you go from a statement about an individual to a generalization, but there are restrictions on how it can be used. I will begin by demonstrating its valid use, then go on to show how it cannot be used.

Consider this argument:

All calicos are felines.
All felines are animals.
∴ All calicos are animals.

This is a valid argument, but we need a new rule to show its validity. This rule is Universal Generalization, and it lets us go from a premise about an individual to a universal conclusion. Here is how it is used to prove the validity of the argument just given above.

1. (∀x)(Cx ⊃ Fx)
2. (∀x)(Fx ⊃ Ax)
∴ (∀x)(Cx ⊃ Ax)
3. Ca ⊃ Fa // 1 Universal Instantiation
4. Fa ⊃ Aa // 2 Universal Instantiation
5. Ca ⊃ Aa // 3,4 Hypothetical Syllogism
6. (∀x)(Cx ⊃ Ax) // 5 Universal Generalization

Although it is valid here, there are limitations on the use of Universal Generalization. When we make claims about actual individuals, it is invalid to infer a generalization. Consider this argument, which is superficially similar to the previous one.

All calicos are felines.
If Holly is a feline, then Holly is an animal.
∴ All calicos are animals.

1. (∀x) (Cx ⊃ Fx)
2. Ch ⊃ Fh
∴ (∀x) (Cx ⊃ Ax)
3. Ch ⊃ Fh // 1 Universal Instantiation
4. Ch ⊃ Ah // 1,3 Hypothetical Syllogism
5. (∀x) (Cx ⊃ Ax) // 4 Universal Generalization?

In this one, the inference on line 5 is invalid. The difference between these two arguments is that the latter refers to a specific individual, Holly, while the former refers to an arbitrary individual whom we know nothing about except that the claim in line 1 is supposed to be true of it. This arbitrary individual could be anything or anyone, such as a cat, a dog, a flea, a mite, a paramecium, or whatever. The specific details about this arbitrary individual are unknown and don't matter. That makes this arbitrary individual indistinct from any other individual we could instantiate line 1 to, and by remaining indistinct from any of the others, we are free to generalize whatever we deduce about this individual in our argument.

The same kind of thing is done in a geometry proof. We may identify something as a triangle without specifying its angles or the lengths of its sides. So we don't know whether it is a right triangle, an isosceles triangle, or whatever. Since we don't know anything about it that distinguishes it from other triangles, anything we can prove about it can be generalized to all triangles.

Here is another argument to more concisely make the point that you cannot generalize from claims about specific individuals. Just because Holly is a calico, it certainly does not follow that anything at all is a calico.

1. Ch
∴ (∀x)(Cx)
2. (∀x)(Cx) // 1 Universal Generalization

An arbitrary subject can be introduced by Universal Instantiation, as we've already seen, or by an assumption, which we learned about in the lesson Introducing Conditional Proof. Here is an example of a valid argument in which an arbitrary individual is introduced with an assumption. This argument proves a predicate logic version of the Law of Excluded Middle.

Prove: (∀x)(Px ∨ ~Px)
1. | Pa // Assumption
2. Pa ⊃ Pa // 1-1 Conditional Proof
3. ~Pa ∨ Pa // 2 Material Implication
4. Pa ∨ ~Pa // 3 Commutation
5. (∀x)(Px ∨ ~Px) // 4 Universal Generalization

Since the individual a is introduced in an assumption, we know a is not a known individual, and we know nothing else in particular about a. This allows us to treat a as an arbitrary individual and generalize from what we can prove about a.

But there is an exception to this. Universal Generalization cannot be used within the scope of an assumption from a premise whose subject appears within the same scope of the same assumption. Without this restriction, we could use Universal Generalization to prove this obviously invalid argument

Not everything is wonderful. Therefore, nothing is wonderful.

1. ~(∀x)(Wx)
∴ (∀x)(~Wx)
2. | Wa // Assumption
3. | (∀x)(Wx) // 2 Universal Generalization
4. Wa ⊃ (∀x)(Wx) // 2-3 Conditional Proof
5. ~Wa // 1,4 Modus Tollens
6. (∀x)(~Wx) // 5 Universal Generalization

I have striked out line 3, because Universal Generalization is not actually allowed here. Look at line 4, the line that follows by conditional proof. This says that if a is wonderful, then everything is wonderful. Individual a may be an arbitrary individual, but it certainly isn't true that if some arbitrary individual is wonderful then everything is wonderful. And when we deny that everything is wonderful, as line 1 does, it does not imply that some arbitrary individual is not wonderful. So we should not be allowed to get line 5, which says that some arbitrary individual is not wonderful. The use of Universal Generalization on line 6 is valid, but we should have never been able to derive line 5 in the first place. Because of the invalid use of Universal Generalization in line 3, this fails to prove this argument valid.

Here is a quick synopsis of this rule:

Φν
∴ (∀μ)(Φμ)

In this and the previous lesson, we have covered Universal Instantiation and Universal Generalization. In the next lesson, we will look at the rules of Existential Instatiation and Existential Generalization. These are similar to their universal counterparts, but restrictions are different.