Is there a name for this phenomena with the English copula "to be"?
1a - "My day off is Saturday"
1b - "Saturday is my day off"2a - "John is a doctor"
2b - *"A doctor is John"
I think I've heard it before but can't remember. Also what are the conditions under which it can occur?
The difference you've identified is between predicational and specificational copular clauses (terms coined by Higgins, 1979, i think). In a predicational copular sentence, the subject denotes an individual and the complement (which may be either a Noun Phrase, or an Adjectival Phrase) denotes a property. The subject is predicated of the complement. Your examples 1b (i) and 2a (ii) are predicational copular clauses.
(i) [Saturday]
<e>is [my day off]<e,t>
The entity denoted by 'Saturday' satisfies the property of being the speaker's day off.(ii) [John]
<e>is [a doctor]<e,t>
The entity denoted by 'John' satisfies the property of being a doctor.
In specificational copular clauses, the relationship between subject and complement is reversed - the subject (which may only be a Noun Phrase, due presumably to the EPP), denotes a property, and the complement denotes an individual, which is predicated of the subject. Your examples 1a (iii) and 2b (iv) are specificational copular clauses:
(iii) [My day off]
<e,t>is [Saturday]<e>
The entity denoted by 'Saturday' satisfies the property of being the speaker's day off.(iv) #[a doctor]
<e,t>is [John]<e>
The entity denoted by 'John' satisfies the property of being a doctor.
Note that specificational copular clauses and their predicational counter-parts are truth-conditionally equivalent. The difference is generally considered to be driven by information-structural considerations, such as topic and focus (see Line Mikkelsen's dissertation on the topic). One common assumption is that predicational and specificational copular clauses have the same underlying D-structure, as follows (using X'-notation):
(a) [SpecIP ... [I' [+pres] [VP ... [V' is [predP John [pred' pred [NP a doctor]]]]]]]
The copular takes a small clause complement (a "predP", in Mikkelsen's terminology). The functional head pred takes a specifier of type <e> and a complement of <e,t>. In a predicational copular clause, the individual-denoting element in specPredP raises to the matrix subject position, and the complement remains in situ. In a specificational copular clause, the property-denoting element in compPredP raises to the matrix subject position, and the subject remains in-situ. See Mikkelsen's thesis for further discussion.
EDIT: @Daincihi points out that i don't explain why (iv) is bad in my answer. (v), below, illustrates that there isn't any general ban on indefinite specificational subjects:
(v) [A philosopher who seems to share Kiparskys’ intuitions on some factive predicates] is [Unger] (1972), who argues that ...
I mentioned that under Mikkelsen's account, the difference between a predicational and specificational copular clause concerns information structure. One of the restrictions on a specificational copular clause is that the subject must be a topic (i.e. it must be given/old information). A bare indefinite such as "a doctor" can't function as a topic for fairly obvious pragmatic reasons, whereas an indefinite modified by a relative clause can, hence the difference in acceptability between (v) and (iv).
N.b. I presuppose a very basic knowledge of type-theoretic semantics in my answer.
Some terminology, following @cerberus's suggestion:
e = the semantic type 'individual' (the type of, e.g. a proper name)
t = the semantic type 'truth-value' (the type of a sentence)
<e,t> = a function from individuals to truth-values (i.e. the type of a property, or predicate)
EPP = The Extended Projection Principle; The requirement that all English sentences have a subject.
