21.4 Atomic Sentences
You already know how to write atomic sentences in BOOL and PROP. You just write a capital letter.
We can still do that in FOL. So sometimes you'll still see us write P as an atomic sentence or PvQ as a complex sentence in FOL.
But in FOL we can also write atomic sentences using names and predicates. Here's the formula: take any predicate and insert the right number of names, and presto! You get an atomic sentence.
For example, last section you wrote: Guilty(pia) and G(r). Both of those are atomic sentences.
Think about it this way: names and predicates are like the subatomic parts (the protons and electrons), which we use to build the atoms of the language.
We said that an atomic sentence is just a predicate with the correct number of names, so you might be wondering: What is the correct number of names?
That just depends on the arity of the predicate. For a binary predicate, you need two names separated by a comma. For a 10-ary predicate, you need 10 names, separated by commas.
Of course, the names don't have to be different. "Pia likes Pia" is a perfectly grammatical sentence of English.
Now that we have atomic sentences, we can build complex sentences using the connectives.
Using connectives is not the only way we can make complex sentences. The other way is with quantifiers, which we'll learn about next.