I'm currently working on the section that motivates Free Logic. I wrote up what I think are the 'standard' reasons in favor of Free Logic, and then added the following one, which I'm not 100% sure about yet. I don't think I've seen it anywhere else before, but please correct me if someone else has made this argument already!
Suppose s is a string of characters with all the grammatical or syntactic markers of a name of an individual. We explicitly leave it open whether s refers to exactly one individual, i.e. we leave it open whether s is like 'Angela Merkel', or is instead like 'Zeus'. For the classical logician, 'Zeus' cannot be a name. So the classical logician holds that some strings of characters with the form s=s are not true (e.g., ‘Zeus = Zeus’), while other strings with the same form are true (e.g., ‘Angela Merkel = Angela Merkel’). However, this fact about classical logic conflicts with the widely-accepted principle that a logical truth is true in virtue of its logical form. That is:
(FORMAL) If a string of characters is a logical truth, then every string with the same grammatical or syntactic form is also true.Since ‘Angela Merkel = Angela Merkel’ has the same grammatical or syntactic form as ‘Zeus = Zeus’, and classical logic classifies the first but not the second as a logical truth, classical logic violates this widely-held principle (FORMAL). [EDIT/ UPDATE (June 1 2021): This argument might be improved by replacing every instance of 'logical truth' with 'theorem', and 'logical form' with 'syntactic form'; see the 7th comment in the comment thread to this post]
'Positive' free logicians can avoid this problem: they make every instance of s=s, including `Zeus = Zeus', a logical truth (again, where s has all the grammatical features of a name). And 'negative' and 'neutral' free logics make no instances of s=s logical truths. (However, as Nolt points out in the Free Logic SEP entry, "[I]n negative or neutral free logic [it] is not the case [that] ... any substitution instance of a valid formula ... is itself a valid formula"; see the reference there for explanation.)