Formal Logic 101 – Part 21: Relational Predicate Logic (Part 2)

Sentences with multiple quantifiers

Clear back in part 18, we briefly touched on the possibility of sentences with multiple quantifiers Stating that “there is a dog and a cat” could be symbolized as (∃x)(∃y)(Dx ^ Cy). (There is an x and a y such that x is a Dog and y is a Cat.) Today, we’re going to dive a bit deeper into that topic, especially as it relates to the Relational Predicate Logic that we covered last time around.

For our discussion, let’s look at the phrase “x loves y”. Using a single quantifier, we could make a handful of sentences. For instance:
(∃x)Lxj – someone loves John
(x)Lxj – Everyone loves John
(∃x)Ljx – John loves someone
(x)Ljx – John loves everyone

But we can also create sentences using multiple quantifiers such as:
(x)(y)Lxy – for all x and all y, x loves y (Everyone loves everyone).
In total, there are eight possible configurations that we can form using two quantifiers:
(x)(y)Lxy and (y)(x)Lxy – These two are functionally identical and comes out to “everyone loves everyone” and “everyone is loved by everyone” respectively.
(∃x)(∃y)Lxy and (∃y)(∃x)Lxy – These two are also functionally identical and come out to “There is an x and a y, such that x loves y” (someone loves someone) and “There is a y and an x such that x loves y” (someone is loved by someone) respectively.
The other four permutations have unique meanings, so care should be taken to phrase them appropriately:
(x)(∃y)(Lxy) – For all x, there is a y such that x loves y (everyone loves someone.)
(∃x)(y)(Lxy) – There is an x such that, for all y, x loves y (there is one specific person who loves everyone.)
(y)(∃x)(Lxy) – For all y, there is an x such that x loves y. (everyone is loved by someone)
(∃y)(x)(Lxy) – There is a y such that, for all x, x loves y (everyone loves this one specific person.)
Obviously, we can use three quantifiers if need be, but I’m not getting that far into the tall grass on facebook. But yeah, you can do it if you get that far in.

Using “Not” with Multiple Quantifiers

Next, I think it would be good to touch on the way that Not works in sentences with more than one quantifier. Consider the phrase:

~(∃x)(y)(Lxy) – There is not even one x who, for all y, x loves y (“No one loves EVERYONE!”)
We can use quantifier negation on this sentence, just like any other. If we use QN, we end up with the following:

1. ~(∃x)(y)(Lxy)         P
2. (x) ~(y)(Lxy)         1 QN – For all x, it’s not the case that, for all y, x loves y (“everyone does not love someone or other.)

Using QN again, we can move to:
3. (x)(∃y)~(Lxy)         2 QN – For all x, there is a y such that x does not love y (“everyone has someone who they don’t love.)

So, long story short, if you’re using QN on sentences with multiple quantifiers, just move the ~ one spot to the left, or one spot to the right, and use it as many times as needed to get where you need to go.

Sentences with complex subjects and predicates

Last thing before we call it a day. Shifting gears a bit and moving away from who loves who, let’s look at the phrase “Every dog that wears a collar has an owner.”
We can rephrase this to say “If x is a dog and wears a collar, then x has an owner” while retaining our original meaning.

Now, let’s start with just the front half of that sentence: “x is a dog and wears a collar”. We could try symbolizing that as
(x)(∃y)[Dx ^ (Cy^ Wxy)]
where Dx = x is a dog, Cy = y is a collar, and Wxy = “x wears y”.

However, let’s translate that back to English: For all x, there is a y such that x is a dog, y is a collar, and x wears y. (Or, more simply: All dogs wear collars.) Plugging that back into our original sentence we’re left with something like “if all dogs wear collars, then all dogs have owners”. So that doesn’t quite retain our original meaning.

Instead, what we can do is phrase it as:
(x)[Dx ^ (∃y)(Cy ^ Wxy)]
Which comes to “for all x, x is a dog, and there is a y such that y is a collar and x wears y”, or “x is a dog and wears a collar.” Which is exactly what we need to plug into our "if/then" statement to keep the meaning consistent.

To finish up the if/then statement, we could let Ozx= z owns x, and craft the statement:
(x){[Dx ^ (∃y)(Cy ^ Wxy)]→ (∃z)Ozx}
Which comes to “For all x, if x is a dog, and there is a y such that y is a collar, and x wears y, then there is a z such that z owns x.” (or, “if x is a dog, and wears a collar then there is someone who owns x”).

The takeaways here are: 1) Not every sentence needs to have its quantifiers on the far left of the statement. 2) Make a habit of translating the sentence back to English to make sure it still means what you intended it to mean. And 3) when in doubt, translate a smaller portion of the sentence first, and then work your way out from there.

Back to Part 20            Continue to Part 22