Formal Logic 101 – Part 16: Predicate Logic Symbolization Part 1 (All and Some)

Last time we discussed the need for predicate logic, as some sentences such as “War and Peace is a novel” simply cannot be usefully symbolized in propositional logic. For predicate logic, we use a capital letter for our predicate, followed by a lower case letter for our subject. So “War and Peace is a novel” becomes “Nw”. “Angela is Happy” becomes “Ha” where “a” is Angela and “H[ ]” is “[ ] is Happy”.

As a slightly more complex example: “Max is neither a fish, nor a bird” can be symbolized as
~Fm ^ ~Bm
F[ ] = [ ] is a fish
B[ ] = [ ] is a bird
m = Max

Hopefully this is looking similar enough to what we’ve been doing this whole time that denoting the subject and predicate of our propositions isn’t throwing anyone too much, but if you find yourself a bit lost, remember, the comments are your oyster!

Now, predicate logic also introduces two other ideas into our logic vocabulary: “Some” and “All”.

All (Universal Quantifier)

Let’s say that I’m dealing with a sentence where I want to make a general statement about the universe. For instance, “Everything that exists is material.”

In terms of meaning, this is the same as saying “For all things, if it is exists, then it is material”. We symbolize that “For all things” as (x). (In later proofs as our sentences get a bit more complex, we may also need to use (y) , and (z), but for right now (x) is all we need concern ourselves with).

So our fully symbolized statement becomes:
(x)(Ex→Mx)

In this instance, think of “x” as a sort of fill-in-the-blank letter. If it’s helpful, think of the phrase as
(______)(E______ → M______)
And read it as “For whatever you happen to fill in the blank with, if it exists, then it’s material.” (Bearing in mind all three blanks need to be filled with the same thing.)

So our original statement translates roughly to “For whatever you replace ‘x’ with, if it exists, then it’s material.”

(Note: Some systems use the term ∀ for the universal quantifier instead of (x). Again, I was taught (x) and I find it more intuitive so I’m teaching it that way, but if you really like ∀ instead, there’s certainly nothing wrong with it.)

(Note 2: To make things easier, it is possible to specify what is called a “universe of discourse”, which basically says “we’re only talking about this specific set of things”. For instance, if I say, “everyone failed the test” or “(x)Fx”, where Fx is “x failed the test”, I could mean “every single thing in existence (to include but not limited to the jar of M&Ms sitting on my desk) failed the test.” But that’s generally not what we mean by the phrase “everyone failed the test”. Instead what we mean is “of the people who took the test, everyone failed” so when we specify our universe of discourse, we specify that we’re only speaking about the people who took the test. I’m not actually familiar with any symbolized way of specifying the universe of discourse so I will usually either do it in plain text before beginning the proof, or I will just keep it in mind that I’m only talking about the relevant set of things.

Some (Existential Quantifier)

Bringing things down a notch from “all” sometimes we only want to say that a statement is true for some things. We may want to that “some people like football games”, but we’re not quite willing to go all in and claim that everyone does.

For that we would need the “Existential quantifier” “(∃x)”. For our purposes, you can read (∃x) as “There is at least one thing you can replace ‘x’ with such that…” So “Some people like football” could be symbolized (∃x)(Fx) (or, "there is at least one x such that x likes football").

Do note that most of the time that you’re using (∃x), the thing that follows will either be a single proposition (e.g. “There is at least one Skittle”, or “(∃x)(Sx)”) or multiple propositions separated by either “And” or “Or” (e.g. “There is at least one Purple Skittle” or “(∃x)(Sx ^ Px)” or roughly “there is at least one x such that x is a Skittle, and x is Purple”).

It’s comparatively uncommon to find an “(∃x)” followed by “If/Then”, since “If/Then” statements tend to imply that all things that have the first property also have the second. Having said that, it is possible to use DeMorgan’s, and Conditional Exchange to turn just about any “And” or “Or” statement into an “If/Then”. So it is possible to have an “(∃x)” followed by an “If/Then” statement. But if you are symbolizing a sentence and you find yourself writing out “(∃x)(Ax → Bx)” it might be good to double check and make sure that that’s what the original sentence actually says, and not “(∃x)(Ax ^ Bx)”.

*****
Ok stopping point. I think that’s enough for today. Next time we’ll come back and look over the opposites of “All” and “Some”. What they mean, and how they work. As usual, we'll have some practice below.

Practice:

To begin, let's look at some propositions and symbolized them.

A. If Sue lives in a glass house, then she should not throw stones.
B. Rocky is a master safe cracker
C. If Andrew does not get enough sleep, then he misses a lot, but he is very sharp if he is alert.
D. Amy and Kathy both like to climb trees, but both hate ice fishing.

Now lets try a couple with quantifiers:

E. Everyone who took the class passed.
F. Some people who took the class got an A.
G. At least one person passed despite not doing the homework. 


Answers:

(As always, the precise letters you use don't matter, as long as the structure is correct.

A. (Gs → ~Ss)
  1. Gx = x lives in a glass house
  2. Sx = x should throw stones
  3. s = Sue
B. Mr
  1. Mx = x is a master safe cracker
  2. r = Rocky
C. (~Sa → Ma)^(Aa → Ha)
  1. Sx = x gets enough sleep
  2. Mx = x misses a lot
  3. Ax = x is alert
  4. Hx = x is very sharp
  5. a = Andrew
D. (Ca ^ Ck) ^ (~Ia ^ ~Ik)
  1. Cx = x likes to climb trees
  2. Ix = x likes to go ice fishing
  3. a= Amy
  4. k = Kathy
E. (x)(Tx → Px)
  1. Tx = x took the class
  2. Px = x passed the class
F. (∃x)(Tx ^ Gx)
  1. Tx = x took the class
  2. Gx = x got an A
G. (∃x)(Px ^ ~Hx)
  1. Px = x passed the class
  2. Hx = x did the homework

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