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Stream: theory: mathematics

Topic: Logical operations on classes of models (as a precursor t...

Julius Hamilton (Sep 23 2024 at 17:14):

Curious about this post:

Fix a signature $\Sigma$ and let $\mathsf{Struc}_\Sigma$ be the class of $\Sigma$-structures. We can think of first-order logic, FOL, in a “syntax-free” way by considering the hyperclass (class of classes) it yields:

$\mathsf{FOL}_{\mathsf{semantics}} := \{Mod(\phi):\phi \in \mathsf{FOL} \}$.

To paraphrase: instead of working with the language $\mathsf{FOL}$ in terms of endomorphisms (term- and formula-formation rules) on a set of symbols, we consider $Mod(\phi)$, the class of models for a formula $\phi$, for all well-formed formulae $\phi \in \mathsf{FOL}$. This yields the hyperclass $\mathsf{FOL}_{\mathsf{semantics}}$ containing a class of models for each formula in $\mathsf{FOL}$.

This hyperclass has structure: we can relate relations on FOL formulae to relations on classes-of-models, as in:

The fact that first-order logic has finite conjunctions is reflected in the fact that whenever $\mathbb{A}, \mathbb{B} \in \mathsf{FOL}_{\mathsf{semantics}}$ we also have $\mathbb{A} \cap \mathbb{B} \in \mathsf{FOL}_{\mathsf{semantics}}$. This is because we always have $Mod(\phi) \cap Mod(\psi) = Mod(\phi \wedge \psi)$.

Similarly, the fact that first-order logic has negations is reflected in the fact that whenever $C \in \mathsf{FOL}_{\mathsf{semantics}}$ we also have $\mathsf{Struc}_\Sigma \setminus C \in \mathsf{FOL}_{\mathsf{semantics}}$. This is because we always have $\mathsf{Struc}_\Sigma \setminus Mod(\phi) = Mod(\neg \phi)$.

Note that we're not intersecting structures here!

I think I get it. You map the symbols in a signature to sets in a domain. The map is a model if it respects the axioms (it sounds like a functor?). When we take the intersection of two classes-of-models, this is the class of models where both formulae hold true. When we take the complement of $Mod(\phi)$ in $\mathsf{Struct}_\Sigma$, this is the class of models where $\phi$ is not true. Etc. (Very nice.)

I’ll hopefully write my understanding of the next part of the post, in working up to defining “regular logic”. Any comments are welcome.