If free logic governs quantified modal logic, then (a=a) fails in worlds where Adams does not exist, breaking the derivation at P5 without any contradiction in the remaining premises.
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Worlds (possible worlds)(as the objects treated by fictionalism)
Hypothetical complete scenarios or ways reality could have been; philosophers use this concept to think through what's possible versus necessary.
a=a(as used in formal logic notation)
A symbol representing the logical rule that something is always equal to itself (a thing is identical to itself), which is considered a basic truth in most logical systems.
free logic(Contrasted with standard first-order predicate logic)
A logical system in which the existential generalization '∃x φ(x)' cannot in general be derived from 'φ(t)' for a singular term 't', meaning singular terms do not automatically carry existential import
quantified modal logic(Introduced as the result of moving past Quine's criticisms of combining quantification and modality.)
The logic obtained by combining pure quantificational logic with propositional modal logic.