What does occurance, expansion and renaming mean for abstract logic?

Question

According to this article an abstract logic

is a formal system consisting of a class of sentences and a satisfaction relation with specific properties related to occurrence, expansion, isomorphism, renaming and quantification.

They include propositional, first-order & infinitary logic. The

Now, what does occurance, expansion and renaming mean? (I expect isomorphism & quantification are the usual ones).

I'm interested in them, because first-order logic is charactereised as the unqiue abstract logic which is compact and has the smallest possible lowenheim-skolem number.

Answer 1

The article you mentioned doesn't aim to give a complete definition of abstract logics, but since we're going to talk about occurrence, expansion and renaming, it might be useful if we look at all the properties a system has to meet in order to be called an abstract logic. Here's a definition.

Abstract logics are pairs (σ, Sat_σ) where σ is a set of sentences and Sat_σ is a satisfaction relation meeting the following conditions: (1) occurrence, (2) expansion, (3) isomorphism, (4) renaming, (5) closure, (6) quantifier, and (7) relativization. (Chang & Keisler, p. 128)

Here's how they define each of those properties. I apologize for the ugly notation; with no TeX option, I've tried to do all I can with alternative formulations of relations and HTML characters. Let each sentence s ε σ ('ε' stands for the set membership relation) have an associated finite language L(s) called the set of symbols occurring in s. According to C&K, system (σ, Sat_σ) is an abstract logic iff it meets the following conditions:

1. Occurrence: if s ε σ and M models L, then (M, s) ε Sat_σ is defined iff L includes L(s).

2. Expansion: if (M, s) ε Sat_σ and M' expands M to a larger language, then (M', s) ε Sat_σ.

3. Isomorphism: if (M, s) ε Sat_σ then (N, s) ε Sat_σ for any N isomorphic to M.

Let ρ be a bijection from a language L to a language ρ(L) that preserves the number of places of all symbols, and for each model M for L let ρ(M) be the model for ρ(L).

4. Renaming: for each s ε σ with L(s) included in L, there is a sentence ρ(s) ε σ with L(ρ(s)) = ρ(L(s)) s.t. for each model M for L, (M, s) ε Sat_σ iff [ρ(M), ρ(s)] ε Sat_σ.

5. Closure: (i) σ contains all atomic sentences closed under the first-order connectives, (ii) Sat_σ satisfies the usual rules for satisfaction of atomic sentences and first-order connectives, and (iii) L(s) is well behaved.

6. Quantifier: same as (5), but with "quantifiers" substituted for "first-order connectives".

The last, relativization condition is a bit more involved and hard to express without TeX, but here's a try.

7. Relativization: for each sentence s ε σ and relation R(x, ã) with R, ã ε L(s), there is a new sentence s|R(x, ã) read as "s relativized to R(x,ã)" with language L(s) ∪ {R} ∪ a (for each a ε ã) s.t. whenever N is a submodel of a model M for L(s) with the universe {b ε M: R(b, ã)}, we have: [ (M, R, ã), s|R(x, ã) ] ε Sat_σ if and only if (N, s) ε Sat_σ.