I have read that Graph theory is useful to mathematical logic, but I don’t understand how. How exactly does a logic of graphs differ from standard predicate logic?
Here is an example from an article by Van Benthem:
2.2 Priority graphs and derived betterness ordering
Definition 1 (P-graphs). Let L(P) be a propositional language built on the set of atoms P. A P-graph is a tuple G = 〈 Φ, ≺ 〉 such that:
– Φ ⊂ L(P) with | Φ | < ω;
– ≺ is a strict order on Φ : property ψ is strictly better than ϕ ; also, for all propositions ϕ , ψ ∈ Φ: if ϕ ≺ ψ then ϕ logically implies ψ .
Intuitively, a P-graph is a finite graph of formulae from a propositional language, where each formula logically implies its immediate successor in the order. In what follows, we will often drop the index referring to the current language.
P-graphs where ≺ is a strict linear order are called P-sequences, they are referred to as sequences 〈ϕ 1 , . . . , ϕ n 〉 of propositional formulae and are denoted by letter S. Given a P-graph G = 〈 Φ, ≺ 〉 we denote by S G the set of maximal P-sequences (i.e., ≺-chains) included in the order ≺”
I know graph theory is studied in Math, but I didn’t know there was a relationship with logic and with deontic logic, in particular.
It seems to me from the article by Benthem, that was is being described is a finite directed graph where a directed edge denotes immediate successor in the described order. This is not surprising. The idea of graphs is a very general one and directly models one aspect of a situation, namely an edge models a particular relation between the objects (whatever they are) corresponding to the vertices of the graph. The graph itself is not logic, but only models one particular aspect of a particular kind of logic. Certain aspects of that kind of logic can be expressed in the language of graph theory. Many other aspects can not.