[PL03] Quantifiers 1


§1. Something more

As we have seen, the sentence "John eats" is translated into PL by choosing a predicate letter to translate "eats" (for example "F"), choosing a name to translate "John" (for example "a") and putting them together: "Fa".

But what about statements like:

Someone eats.
Something is red.
Everyone owns a Porsche.

One might think that the answer is obvious. All of the above sentences can be translated in just the way a simple statement like "Archibald eats" is translated, for example, as "Fa".

But that answer is not acceptable. One should not translate words like "someone" and "everything" by names (as names like "Archibald" and "John" are translated). To see why, notice that "Archibald runs and Archibald does not run" is a contradiction which, quite rightly, can be translated by an inconsistent formula: "(Ra & ~Ra)". However, "Someone runs and someone does not run" is not a contradiction. But if we treated that sentence like "Archibald runs and Archibald does not run", its translation would be an inconsistent formula.

Similarly, "Either this car is mine or this car is not mine" has to be true. So, quite rightly, it can be translated by a tautology: "(Mc v ~Mc)". However, "Either everything is mine or everything is not mine" is no tautology.

For a related reason why one should not translate words like "someone" and "everything" by names consider the following valid argument:

John eats. So, someone eats. If the argument is translated thus:

Fa ⊧ Fb

The result is an invalid sequent, which is not what we want.

§2. Speaking of everything

An innovation, due to the logician Gottlob Frege, provides a way to translate sentences containing words like "something" and "everything". According to Frege, a sentence like "Something is red" is not like the sentence "John is red". "John is red" says that a certain individual, John, has the property of being red. The sentence "Something is red", according to Frege, is different. It says that the property of being red has a certain property: the property of being red has the property of being instantiated in the world.

We have already learned how to translate sentences like "Archibald is red" or "This ball is red". You just put together a predicate letter and a name like this: "Fa". But, in PL, how can we talk about the property of being red without talking about a particular red object? We can't use a formula like "Fa", because in this formula the name "a" refers to a specific thing, like this ball, or a specific person, like Archibald.

We need to talk about the property of being red without talking about a particular red object. Predicate Logic lets us do that with an open sentence such as "Fx". Here "x" is a variable, not a name. A name refers to a specific thing. A variable does not. We could think of "Fx" as translating the English sentence "It is red", where the word "it" is not referring to any particular thing. (Sometimes, logicians say that "Fx" translates the not quite English sentence "x is red".) Using variables avoids having to use a name, so avoids having to refer to a specific thing.

Now we know how to talk about the property of being red in PL, without talking about a particular red thing. But we still need to see how to translate the sentence "Something is red". Frege's insight was the "Something is red" says that the property of being red is instantiated.

In PL, we can say that the property of being red is instantiated by combining the open sentence "Fx" with an existential quantifier:

∃xFx

We can read this as saying "There is at least one thing x such that x is red". Or, more simply, "There is at least one thing that is red", in other words, "Something is red". Similarly, we can translate "Everything is red" by combining an open sentence like "Fx" with a universal quantifier:

∀xFx

We can read this as saying "Every x is such that x is red". Or, more simply, "Everything is red".

In predicate logic quantifiers are constructed by prefixing either the symbol "∀" or "∃" to a variable. Here are some examples :

∀x     ∃x     ∀y     ∃z

Any quantifier that starts with "∃" (such as "∃x") is an existential quantifier. "∃x" is translated as "there exists an x such that ..." We can combine a quantifier with a predicate to make a well-formed formula, as in :

∃xBx

To understand such a wff, we can rewrite it as a semi-formal statement :

There exists an x such that Bx.

Or alternatively,

There is at least one x such that Bx.

So what it says is that there is some object x, and x is B. In other words, it simply says that something is B. If the predicate "Bx" means x is a boy, then the wff can be translated as there is at least one boy, or a boy exists. Notice that "∃xBx", "∃yBy", "∃zBz", etc. all say the same thing. They are different wffs since they employ different variables. But they are logically equivalent nonetheless. Notice also that the truth of "∃xBx" is consistent with the claim that there is more than one B. It is just that this is not what "∃xBx" says. The latter wff says that there is one or more. It might be the case that there is just one, or it might be that there is more than one.

Any quantifier that starts with "∀" is a universal quantifer. "∀x" means "for all x, ... ". Again we can combine a universal quantifier with a predicate to form a wff, such as :

∀xBx

In semi-formal notation, this means the same as "for all x, Bx". What this says is that for any object x, x is B. If "Bx" means x is a boy, this wff would mean for every x, x is a boy. In other words, take any object whatsoever, it is a boy, which is just the same as saying that everything is a boy. Under such a translation, the wff is of course actually false.

Now consider the following formula, what do you think it means? (Suppose "Dy" means y is dirty, and "By" as before.)

∀y(By→Dy)

To work out what this wff is saying, we can rewrite the wff step-by-step as follows :

For all y, (By→Dy)
For all y, if y is a boy, then y is dirty.
Everything is such that if it is a boy, then it is dirty.

You might find the interpretation of wffs difficult at first, but if you try to understand them step-by-step, then it might become easier. We can see in the above example that the last sentence says the same as "Every boy is dirty", which is just what the wff means.

What do these wffs mean? Use the same translation scheme as before.

  1. ∀zDz
  2. ∀z(Dz→Dz)
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