What Is Is?

And as we use the verb is; so the Latins use their verb est, and the Greeks their εστι through all its declinations.
Thomas Hobbes
Leviathan, ch. 46
Penguin Classics, 1982

The following is an excerpt from
T. Gowers (ed.), The Princeton Companion to Mathematics,
Princeton University Press, 2008, p. 9

Another word that famously has three quite distinct meanings is "is". The three meanings are illustrated in the following three sentences,

  1. 5 is the square root of 25.
  2. 5 is less than 10.
  3. 5 is a prime number.

In the first of these sentences, "is" could be replaced by "equals"; it says that two objects, 5 and the square root of 25, are in fact one and the same object, just as it does in the English sentence "London is the capital of the United Kingdom." In the second sentence, "is" plays a completely different role. The words "less than 10" an adjectival phrase, specifying a property that numbers may or may not have, and "is" in this sentence is like "is" in the English sentence "Grass is green." As for the third sentence, the word "is" there means "is an example of," as it does in the English sentence "Mercury is a planet."

These differences are reflected in the fact that the sentences cease to resemble each other when they are written in a more symbolic way. An obvious way to write (1) is 5 = 25. As for (2), it would usually be written 5 < 10, where the symbol "<" means "is less than." The third sentence would normally not be written symbolically because the concept of a prime number is not quite basic enough to have universally recognized symbols associated with it. However, it is sometimes useful to do so, and then one must invent a suitable symbol. One way to do it would be to adopt the convention that if n is a positive integer, then P(n) stands for the sentence "n is prime." Another way, which does not hide the word "is," is to use the language of sets.

...

... [the] way to do it is to define P to be the collection, or set, of all prime numbers. Then we can write (3) as "5 belongs to the set P." This notion of belonging to a set is sufficiently basic to deserve its own symbol, and the symbol used is "∈." So a fully symbolic way of writing the sentence is 5 ∈ P.

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