Why aren't Kripke semantics "syntax in disguise"?

Question

The Wikipedia article on Kripke semantics suggests that they were considered a major breakthrough in part because algebraic semantics were seen as merely "syntax in disguise". But Kripke frames strike me as very algebraic in flavour, even if they don't technically fit anywhere obvious in the taxonomy of algebraic structures. So what is it specifically about Kripke semantics that transcends "syntax in disguise"?

Answer 1

Algebraic semantics give a good organising framework for models of a logic, but they don’t give examples of models, except syntax itself. Kripke models give an easy way to construct lots of concrete examples, with a wide range of properties.

Take for instance the modal logic S4: classical propositional logic plus the axioms □ϕ → ϕ , □ϕ → □□ϕ, and (ϕ→ψ) → (□ϕ→□ψ). You can define an “S4-algebra” as a Boolean algebra B with an interior operator, i.e. an operation □:B → B satisfying axioms □x≤x, □x≤□□x, and monotonicity. These give a sound and complete algebraic semantics for S4: sound since any such algebra admits an interpretation of the syntax, and complete since syntax modulo provability forms such an algebra, so anything that holds in all such algebras must be provable. And all this wasn’t specific to S4: if you think of some new axioms for a new modal logic, you can translate those similarly into Boolean algebra language, and get similarly sound and complete algebraic semantics.

So the algebraic semantics give a great organising framework for models of logics, and they do this uniformly off-the-shelf for a very wide range of kinds of logic. But for most applications (e.g. consistency or independence results), you still need a way to give examples of such models. The only automatic examples are things like the syntactic model (and e.g. finite products thereof). So this is the sense in which algebraic semantics can be called “syntax in disguise”.

What Kripke models give is a flexible way to build lots of different concrete examples of models of modal logics, in which lots of different properties can be made to hold or fail — so they are very powerful for proving independence/consistency results. (And they can either be set up from scratch, or within the algebraic-semantics framework — e.g. given a Kripke model of S4, its power-set forms a Boolean algebra with an interior operator, and the Kripke interpretation of syntax can be seen as arising from that.)

Answer 2

Answer

EDIT 2021-12-24

The initial presumption of my response that all model-theoretic models is wrong per the comments below. I'm letting the answer remain as the text is instructive to people such as myself with a marginal grasp of the concept.

Algebraic semantics isn't an algebraic structure as you point to in the taxonomy, they're a type of algebraic logic, which isn't an abstract algebra at all despite using the modifier 'algebraic'. The general difference between an algebra which is concerned with elements and operations generally, say group (G,*), and an algebraic logic is that algebraic logics study particular classes of algebras as models, as in model theory.

Kripke semantics in particular is more than "syntactic" because it grounds modal logics in a metalanguage to demonstrate soundness and completeness. Both ideas are very important in differentiating syntactic implication from semantic implication in mathematical logic. (See "A "paradox" of coherentism? (PhilSE)" for a better explanation of how the provability in syntax relies on the proof of consistency for grounding.) From WP:

Semantics is useful for investigating a logic (i.e. a derivation system) only if the semantic consequence relation reflects its syntactical counterpart, the syntactic consequence relation (derivability).4 It is vital to know which modal logics are sound and complete with respect to a class of Kripke frames, and to determine also which class that is.