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 constants (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 constants 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. 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.

In MPL, 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 constant like this: "Fa". But, in MPL, 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 constant "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. MPL lets us do that with an open sentence such as "Fx". Here "x" is a variable, not a constant. A constant 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 constant, so avoids having to refer to a specific thing.

Now we know how to talk about the property of being red in MPL, without talking about a particular red thing. But we still need to see how, in MPL, 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 MPL, we can say that the property of being red is instantiated by combining the open sentence "Fx" with an existential quantifier:

∃xFx

Similarly, in MPL, we can translate "Everything is red" by combining an open sentence like "Fx" with a universal quantifier:

∀xFx

Exercises

Translate the following statements into MPL:

1. Something is red.
2. Nothing is red.
3. Not everything is red.
4. Something is not red.
5. Something is red and something is tasty.
6. Everything is both red and tasty.
7. If something is red, then something is tasty.
8. If something is red, then it is tasty.

Translate the following argument into MPL:

All hackers are nerds.
Mitch is a hacker.
So Mitch is a nerd. ∀x(Hx → Nx), Hm Nm'); ?>