Formal Logic 101 – Part 17: Predicate Logic Symbolization Part 2 (Not All and None)

Last time we covered “All” and “Some”. But what if I want to deny that something exists entirely? Some guy is telling me about the person he read about in the tabloids who was born with working gills and lives entirely underwater, and I get fed up and say “No human has gills!” Now how do we symbolize that?

Let’s say Hx is “x is a human” and Gx is “x has gills.”

From what we’ve covered so far, there are two major ways that people try to symbolize that:

1) ~(x)(Hx→Gx)
2) ~(∃x)(Hx ^ Gx)

(and if you don’t fall into one of these camps, don’t worry, I think I know what you did, and you’re on the right track.)

Let’s start with option 1 and translate it back to English. If we start with just (x)(Hx→Gx), it comes to “For whatever you fill in “x” with, if that thing is human, then it has gills.” Or roughly “All humans have gills". Sticking the “Not” in front of it, thus changes the statement to “Not all humans have gills.”

While “Not all humans have gills” leaves open the possibility that there are no humans with gills (After all, if there are 7 billion humans and none of us have gills, then it is true that not all of us have gills), it similarly leaves open the possibility that all humans but one have gills. So ~(x) or “Not all” is much more non-committal than “None”.

Turning our attention to option 2, note that (∃x)(Hx ^ Gx) translates to “There is at least one thing that is both a human and has gills”. So, sticking the “Not” back in front of it changes the statement to “There is not even one thing that is both a human and has gills.” “Not even one”, that is the kind of commitment we’re looking for when we use the word “None”.

So to recap, ~(x) translates to “Not all”, as in “there’s at least one thing that doesn’t meet these criteria”, while ~(∃x) translates to “None” as in “Not even one, no matter how rare”.

If you find yourself asking “Well why don’t we just use (x)(Hx → ~Gx) give yourself a pat on the back and a gold star, that is logically equivalent to ~(∃x)(Hx ^ Gx), and we’re going to prove it!

Consider the phrase “Not everything is easy”, which we can symbolize ~(x)(Ex) (or "It’s not the case that all things are easy”). Let’s just say there are 100 things you could do with your day. If all 100 of them are easy, then everything actually is easy. So, in order for the phrase “Not everything is easy” to be true, then at MOST 99 out of those 100 things can be easy, and at LEAST one thing must not be easy.

So from “Not everything is easy” ~(x)(Ex), we can conclude that at least one thing is not easy (∃x)(~Ex).

Similarly, if we start from “Everything has a purpose” (x)(Px) we can conclude that “not even one thing does not have a purpose” ~(∃x)~Px.

These previous examples illustrate our newest rule “Quantifier Negation” (Q.N.). Quantifier Negation takes 4 forms:
1) ~(∃x)Ax = (x)~Ax
2) ~(x)Ax = (∃x)~Ax
3) ~(∃x)~Ax = (x)Ax
4) ~(x)~Ax = (∃x)Ax

Using Q.N. we can go ahead and prove that (x)(Hx → ~Gx) is logically equivalent to ~(∃x)(Hx ^ Gx) as such:
1. (x)(Hx → ~Gx)            P – Prove ~(∃x)(Hx ^ Gx)
2. (x)(~Hx v ~Gx)           1 C.E.
3. (x)~(Hx ^ Gx)              2 DeM
4. ~(∃x)(Hx ^ Gx)            3 Q.N.


Alright, so we’ve covered one of our final two replacement rules. Next time we should be able to get through two of our final four inference rules, and then we’ll be in the home stretch as far as new rules go. In the meantime, there are some practice symbolizations below.

Practice

Symbolize the following:

A. There's no such thing as a free lunch.
B. There are no unicorns.
C. Not everything is enjoyable.
D. Not everything can be understood.

Now a couple more complex ones:

E. Not every politician who is a millionaire abuses their power.
F. If John and Nancy both take bribes, then not all politicians are honest.



Answers

A. ~(∃x)Lx
        Lx = x is a free lunch.

B. ~(∃x)Ux
        Ux = x is a unicorn.

C. ~(x)Ex
        Ex = x is Enjoyable.

D. ~(∃x)Lx
        Lx = x is a free lunch.

E. ~(x)[(Px^MX) → Ax]
        Px = x is a Politician.
        Mx = x is a Millionaire
        Ax = x Abuses their power

F. (Bj^Bn) → ~(x)(Px→Hx)
        Bx = x takes Bribes
        Px = x is a politician
        Hx = x is honest
        j = John
        n = Nancy


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