In the discussion of constants in ⟨02,01⟩, we noted that in Classical First-Order Logic, constants must name objects which exist. There can be no non-existent objects in our domain, and there can be no names for non-existent objects.

One funny result of this is that all fictional names—like Sherlock Holmes, Superman, Bilbo Baggins—are empty, and so effectively name the empty set (∅). But that seems odd!

In contrast, FREE LOGICS do not have this requirement: they are free from existence assumptions.

This introduces several new problems and puzzles, however. Here we look at just two.

First, we will want to have a way of saying that things exist. So we need a new unary predicate, Є(_), where "Є(santa)" just says "Santa exists". There is a very long standing debate about whether existence can be treated as a predicate in this way, and it's generally thought that Anselm's Ontological Proof of the existence of God requires it to be.

Second, the symmetry of complimentary predicates will be undermined. If we assume all the objects under discussion exist, then Sees(sara, kelly) should entail SeenBy(kelly, sara). But this does not extend to fictional entities: LivesIn(sherlock, london) is true, but IsLivedInBy(london, sherlock) is false. Likewise, Sees(sherlock, the prime minister) can be true, but SeenBy(the prime minister, sherlock) is false.

I owe this second point to the discussion in Terence Parsons's fascinating and lively book, Nonexistent Objects (New Haven: Yale UP, 1980). Link: https://philpapers.org/rec/PARNO

If you'd like to hear more about this topic, let me know in the comments or on Piazza, and I'll make some more optional videos.