Formal Logic 101 – Part 18: Proofs in Predicate Logic (Part 1)

Universal Instantiation (UI)

So we’ve learned how to symbolize statements in predicate logic. Now how do we prove something with them? Let’s consider an argument from way back at the beginning of the course:

1. All poodles are dogs.            P
2. Max is a poodle.                    P
3. Therefore, Max is a dog.       From 1 and 2


Even at a glance this argument is obviously valid when viewed in plain English. That said, let’s look at what happens when we symbolize it. Let’s have Px be “x is a Poodle”, Dx be “x is a dog” and “m” be Max. That gives us:
1. (x)(Px→Dx)                    P
2. Pm                                   P – Prove Dm
3. ???

Step 3 should be a sticking point. After all, inference rules like Modus Ponens can’t be used on parts of lines, so that “Px→Dx” being inside of parentheses is messing things up. Also, even if we could get it out or parentheses, we don’t have “Px” we have “Pm”, and that’s not the same thing. What do we do here?
 
This brings us to the first of our four inference rules: Universal Instantiation (UI). UI essentially states “If it’s true of everything, then it must be true of any one thing.” If the statement “If it’s a poodle then it’s a dog” applies to everything, surely it must apply to Max. So we can complete the proof as follows:

3. Pm→Dm                1 UI
4. Dm                          2,3 MP

Do note that you can pick any letter you like when using UI. I could just as easily have used UI on line 1 to conclude Pc→Dc, Pj→Dj, or Pq→Dq. The main problems with doing that, however are 1) I haven’t defined c, j, or q, so I don’t know what those statements now say, and 2) using any letter other than m doesn’t get me closer to my intended conclusion. As such, UI is definitely one of those rules where it’s best to have a solid reason why you’re using it before you use it.

Also each time we use UI we must change all instances of the variable (in this case “x”) to the same constant (in this case, “m”). I could NOT use UI to conclude (x)(Pm→Dx) or something like that. Nor could I use it to conclude Pa→Dm. Every instance of “x” must be replaced by the same letter.


Existential Generalization (EG)

Existential Generalization is another of those rules that allows us to take a specific statement and make it a little bit more vague. Consider our previously prove statement: Max is a dog. (Dm). Let’s say I’m looking at Max, he’s sitting right in front of me, he’s obviously a dog. From this alone, I can conclude “There is at least one dog” (∃x)(Dx). EG let’s us take any specific instance and use an Existential quantifier (∃x) to walk it back in this same way. So from any specific instance we can conclude an Existential statement, as follows:

1. Dm                   P
2. (∃x)Dx             1 EG

Unlike with UI, we do not necessarily need to change all instances of a given letter. For instance “Max is a dog and is brown” would be

1. Dm ^ Bm                 P
So using EG could give us
2. (∃x)(Dx ^ Bx)         1 EG
But it could also give us:
2. (∃x)(Dx ^ Bm)        1 EG
Both of these sentences are true, just one is a bit more vague than the other.

That said, if there is a sentence that indicates that there may be multiple individuals (e.g. “Max is a dog and Phil is a cat”) we will need to use a different letter in our Existential quantifier each time. For instance, using “Max is a dog and Phil is a cat” as our example, we would symbolize that as:

1. Dm ^ Cp                      P
Using EG once would give us
2. (∃x)(Dx ^ Cp)             1 EG

Using it a second time would give us
3. (∃x)(∃y)(Dx ^ Cy)        2 EG (“there is at least one x and one y, such that x is a dog, and y is a cat”)

(Note: This last statement, (∃x)(∃y)(Dx ^ Cy) leaves open the possibility that x and y are separate entities, but it does not necessarily mean that x and y are two separate entities. Much as we use the Inclusive “Or” where “Or” means “at least one, but possibly both”, our use of multiple existential quantifiers means that we are speaking about at least one entity, and possibly multiple entities.)


After looking over my textbooks and notes, I cannot find any examples of proofs that use UI or EG, that don’t also use our next two rules (EI and UG), so I’m going to pass on having examples this time and just use all four rules next time.

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