Formal Logic 101 – Part 20: Relational Predicate Logic (Part 1)

Multi-place sentences

We have previously discussed how certain propositions (such as “Tolstoy was a Russian. All Russians write sad stories. Therefore, Tolstoy writes sad stories.”) simply cannot be usefully symbolized with propositional logic. So we introduced predicate logic. Well boy howdy, guess what: Some sentence can’t be usefully symbolized with the predicate logic we’ve learned so far either!

Consider: Some people don’t like any dogs. Puppies are dogs. Therefore some people don’t like puppies. If we try to symbolize this using what we have so far, we end up with something like this:
Px (x is a person), Lx (x likes dogs), Ux (x is a puppy), Dx (x likes dogs), Kx (x likes puppies)

1. (∃x)(Px ^ ~ Lx)            P
2. (x)(Ux→Dx)                 P – prove (∃x)(Px ^ ~ Kx)

If you look at this, it really leaves us with nowhere to go. The entire second premise is useless, “Kx” is in the conclusion but nowhere in the premises. This is an obviously valid argument, yet it seems unprovable.

This is where Relational Predicate Logic comes into play. Relational predicate logic tells you about the relationship that two things have to one another. For instance I might want to say “The penguin is on top of the box”, which describes the relationship between the penguin and the box (one being on top of the other). This could be symbolized as follows:

Txy (x is on Top of y)
p (Penguin)
b (Box)
Giving us a final symbolization of “Tpb”

We can now revisit our argument about dogs. We can now take “Some people don’t like dogs” and symbolize “x likes dogs” as Lxd. The whole sentence could thus become (∃x)(Px ^ ~ Lxd) (“There is something that is a person, and doesn’t like dogs”. Our full argument would then be:

1. (∃x)[Px ^ (y)(Dy → ~Lxy)]           P
2. (x)(Ux→Dx)                                  P – Prove (∃x)[Px ^ (y)(Uy → ~Lxy)]

Note that we could not say “d = dogs” and symbolize line 1 as (∃x)(Px^ ~Lxd). “d” in this context would be denoting a specific individual. It would be noting one singular creature named “Dogs”, rather than noting the broad category of dogs. From here, our path forward should be relatively simple:

3. Pa ^ (y)(Dy → ~Lay)                    1 EI
4. (y)(Dy → ~ Lay)                            3 simp
5. Pa                                                   3 simp
Taking a look at what we need to prove again, we have Pa all by itself. If we can get (y)(Uy → ~Lay), then we’re one EG away from the answer. Since we now need to prove a universal statement, it will probably be best to try a Flagged Subproof.

6. b                                                      F.S. (U.G.)
7. Ub → Db                                        2 UI
8. Db → ~Lab                                    4 UI
9. Ub → ~Lab                                    7,8 H.S.
10. (y)(Uy → ~Lay)                           9 U.G.
11. Pa ^ (y)(Uy → ~Lay)                   5,10 conj
12. (∃x)[Px ^ (y)(Uy → ~Lxy)]         11 UG.

Do note that we can use this sort of symbolization for any kind of word that describes a relationship. For instance:

Mary is taller than Susan (Txy=”x is taller than y”, m=Mary, s=Susan, Full symbolization: Tms)
John loves tacos (Lxy=”x loves y”, j=John, t=tacos, Full symbolization: Ljt)
Bill is Ted’s friend (Fxy=”x is a friend of y”, b=Bill, t=Ted, Full symbolization: Fbt
We can also use this format for sentences that relate three or more ideas, such as:
Nebraska is between Texas and Canada. (Bxyz=”x is between y and z”, full symbolization: Bntc)
Burr introduced Hamilton to Laurens (Ixyz=”x introduced y to z”, full symbolization: Ibhl)
Chicago is further from Dallas than Milwaukee is from St. Paul (Fxyzw=”x is further from y than z is from w, full symbolization: Fcdms)
Elizabeth Taylor gave George the Bulgarian Emerald in exchange for the Hope Diamond. (Gxyzw= x gave y to z in exchange for w, full symbolization: Gebgd)

If we wanted to we could keep going into five-place, six-place, or even seven-place sentences, but frankly, I’m just not willing to put that kind of effort into coming up with obnoxiously convoluted sentences. Regardless, at the end of the day, all of our rules apply to these more complex symbolizations. There are no new rules that we need for handling these symbolizations, we just plug them in to the same old rules we already have.

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