My point was that reflection is defined for the meta language, and quantifies over all the sentences in the object language. This is not something that can be added as an axiom in the object language at all—P (without the ’) has no meaning there.
So none of the sub-statements of reflection get assigned probability 1 - they just aren’t the kind of thing P works on. So I felt that calling this an schema distracted attention from what reflection is, which is a property of P, not axioms for the lower system.
The idea is not that this whole statement is an axiom schema. Instead, the idea is that the schema is the collection of the axioms
%20%3C%20b) for all rational numbers a,b and sentences phi such that %20%3C%20b). The full statement above is saying that each instance of this schema is assigned probability 1. (From what I remember of a previous conversation with Paul, I’m pretty confident that this is the intended interpretation.) The language in the paper should probably be clearer about this.
My point was that reflection is defined for the meta language, and quantifies over all the sentences in the object language. This is not something that can be added as an axiom in the object language at all—P (without the ’) has no meaning there.
So none of the sub-statements of reflection get assigned probability 1 - they just aren’t the kind of thing P works on. So I felt that calling this an schema distracted attention from what reflection is, which is a property of P, not axioms for the lower system.
But I may be misunderstanding the setup here...
Right, here’s what’s going on. The statement in the paper is,
%20%3C%20b)\implies\mathbb{P}(a%20%3C%20\mathbb{P}(\ulcorner\varphi\urcorner)%20%3C%20b)%20=%201.)The idea is not that this whole statement is an axiom schema. Instead, the idea is that the schema is the collection of the axioms
%20%3C%20b) for all rational numbers a,b and sentences phi such that %20%3C%20b). The full statement above is saying that each instance of this schema is assigned probability 1. (From what I remember of a previous conversation with Paul, I’m pretty confident that this is the intended interpretation.) The language in the paper should probably be clearer about this.That makes sense...