[reposted July 19, 2008]
Every son has a father.
Notice that "every" = (x)
How do we say that x "is a son," given the vocabulary of Mx = x is male; Px = x is a person; Pxy = x is a parent of y? To begin with, x will be a male person; but that won't be sufficient: recall that Adam and Eve were neither son nor daughter. x will have to have a (= "some or other") parent as well. So "(Mx · Px) · (∃y)Pyx" will say that x is a son. Now to say that x has a (= some or other) father, we will have to say that x has a parent who is a male person: (∃z)[(Mz · Pz) · Pzx]. So to put this together:
(x){[(Mx · Px) · (∃y)(Py · Pyx] ⊃ (∃z)[(Mz · Pz) · Pzx)] }
No one buys from a single store everything that it sells.
We are given the vocabulary: Px = x is a person; Sx = x is a store; Bxyz = x buys y from z.
We begin with the "Nobody frame": "(x)Px ⊃ ~ …"
Next we wish to symbolize what remains: "x buys from a single (= some specific) store everything (=all) which it sells." Because the reference is to a particular store, the existential prefix—and the predicate belonging to it (i.e., "is a store")—need to be written before the universal prefix. We will begin:
(∃y)[Sy · "x buys from y everything which y sells"].
Notice that "everything which y sells" will require a universal prefix: so, we want to represent "for any z, if y sells z then x buys z from y." We will easily symbolize the last clause as "Bxzy." But how do we say "y sells z"? Well, if y sells z, y must sell it to some purchaser or other; so we need an additional existential prefix: "(∃u)Buzy" ("some u buys z from y").
So now we combine the whole shebang: (x)[Px ⊃ ~(∃y){Sy . (z)[(∃u)Buzy ⊃ Bxzy]}].
Consider an English sentence which sounds somewhat different than the one we just examined:
No one buys from any store everything which it sells.
We are given the vocabulary: Px = x is a person; Sx = x is a store; Bxyz = x buys y from z.
We should begin by symbolizing the "no one" frame: "(x)[Px ⊃ ~ …"
Now, we will go back to the remaining portion of the English sentence, the part expressing what is being denied, using "x" for "one": "x buys from any store everthing which it sells." We find that this is now ambiguous because of the "any": is it saying "x buys from some particular store everything which it sells" OR "x buys from every store everything which it sells"? Clearly the first of these is what the sentence is saying; so the "any" will be symbolized with an existential quantifier. But since the sentence goes on to talk of "everything," which will require a universal quantifier, we have a further question: does the universal prefix or the existential prefix go first? To answer this question, we must return to the "some" and ask ourselves whether it means "some store or other" or "some particular store." It seems not to make good sense to say "x buys from some store or other everything which it sells." We talk of "some or other" when we say that, for example, "x bought everything in his house from some store or other," implying that while he bought the furniture from one store (Ralph's Furniture), he might have bought the refrigerator from a different store (Elmer's Appliances); but if we, here, want to talk about everything which it sells, we had better be speaking about a single store. So the second choice—some particular store—is required; as a result, the existential quantifier will precede the universal one. So we will now symbolize "x buys from some particular store everything that it sells."
"(∃y)[Sy . (z)(if y sells z then x buys z)]." But in order to say "y sells z" using the vocabulary, "Bxyz," we must talk about the person (or the "thing"—the purchaser might be a company) to whom y sells z—that is, that which buys z. We will have to refer to such an individual with an existential quantifier.So, we will symbolize "y sells z" as "(∃u)(Buzy)"—"some u buys z from y." When we put the whole thing together, we get this:
(x)[Px ⊃ ~(∃y){Sy . (z)[(∃u)Buzy ⊃ Bxzy]}]
Let's consider a puzzling answer which Copi gives to the symbolization of *35 in Part II on page 129:
(x){Cx ⊃ (y)[(∃u)(∃v)Dyuv ⊃ (∃z)(∃w)(Dyzw . ~Dyzx)]}. We are given the vocabulary: Cx = x is a charity; Px = x is a person; Dxyz = x donates y to z.We can make this formula more tractable by moving the tilde to the front of the formula so that we have a "Nobody verbs" frame; but we will do it in stages. We will begin with the part in blue:
(x){Cx ⊃ (y)[(∃u)(∃v)Dyuv ⊃ (∃z)(∃w)(Dyzw . ~Dyzx) ]}.Moving the tilde to the beginning of this clause will change the dot to a horseshoe and change each of the existential prefixes to universal prefixes:
(x){Cx ⊃ (y)[(∃u)(∃v)Dyuv ⊃ ~(z)(w)(Dyzw ⊃ Dyzx)]}.But now notice the segment in red: its major operator is a universal prefix, its second rank operator is a horseshoe, and the universal prefix does not bind any variable in the consequent; that means that we may rewrite the red segment using Equivalence E.
(x){Cx ⊃ (y)[(∃u)(∃v)Dyuv ⊃ ~(z)[(∃w)Dyzw ⊃ Dyzx)] ]}.
As we continue pulling the tilde out to the beginning, the horseshoe (presently just to the left of the blue tilde) will change to a dot; but notice that the two existential prefixes in purple will not change, since they are part of the antecedent; but the universal prefix, "(y)," will change to an existential prefix. So we will get the following:
(x){Cx ⊃ ~(∃y)[(∃u)(∃v)Dyuv . (z)[(∃w)Dyzw ⊃ Dyzx)] ]}.Now, we have a schema which fits the "Nobody verbs" frame; so we will begin the translation. "No charity ..."; now we must look at the very last clause, "Dyzx," to get our verb, and because "y" is doing the donating, "x" will be passively "receiving" what "y" donates. So we get a bit furher: "No charity receives from ..." Now we must notice the existential "y" which is the major operator after the tilde: "from some particular, or single, ..." But what sort of thing is this "y"? It isn't said to be a person or money; instead what we know about "y" is that "y" donates some "u" to some "v": evidently, we can say that "y" is a "donor". So, thus far, we have: "No charity receives from any single donor ..." Here we need to look at the next prefix, which is a universal: we should read it as "every;" but what sort of thing is this "z"? The last clause goes on to say, "if y (the donor) donates z to anything, y donates z to x;" so evidently, "z" refers to donations. And now we can put it all together: "No charity receives from any single donor every donation which that donor donates." And that is recognizably equivalent to the original English sentence on page 129,
"No charity receives all of his donations from any single donor," or what would be more graceful English, "No charity receives from any single donor everything which he donates."
Remember to study Symbolizing in Predicate Logic