Logic for Computer Scientists/Modal Logic/Axiomatics

Axiomatics
The simplest modal logics is called $$K$$ and is given by the following axioms: and the inference rules A $$K$$ derivation of $$X$$ from a set $$S$$ of formulae is a sequence of formulae, ending with $$X$$, each of it is an axiom of $$K$$, a member of $$S$$ or follows from earlier terms by application of an inference rule. A \defined{$$K$$ proof} of $$X$$ is a $$K$$ derivation of $$X$$ from $$\emptyset$$.
 * All classical tautologies (and substitutions thereof)
 * Modal Axioms: All formulae of the form $$\square (X \to Y) \to (\square X \to \square Y)$$
 * Modus Ponens Rule: Conclude $$Y$$ from $$X$$ and $$X \to Y$$
 * Necessitation Rule: Conclude $$\square X$$ from $$X$$

As an example take the $$K$$ proof of $$(\square P \land \square Q) \to \square (P \land Q)$$: $$\begin{matrix} & \mbox{Tautology: } & P \to (Q \to (P \land Q)) \\

&\mbox{Necessitation: }& \square (P \to (Q \to(P \land Q))) \\

&\mbox{Modal axiom:}& \square (P \to (Q \to(P \land Q))) \to (\square P \to \square (Q\to (P\land Q))) \\

&\mbox{Modus ponens: } & \square P \to \square (Q \to(P \land Q)) \\

&\mbox{Modal axiom:}& \square (Q \to (P \land Q)) \to (\square Q \to \square (P\land Q)) \\

& \mbox{Classical arg: } & \square P \to (\square Q \to \square (P \land Q)) \\

& \mbox{Classical arg: } & (\square P \land \square Q) \to \square (P \land Q) \\ \end{matrix}$$ There is a similar proof of the converse of this implication; hence it follows that in $$K$$ we have $$ \square (P \land Q) \leftrightarrow (\square P \land \square Q) $$

Note that distributivity over disjunction does not hold! (Why?)

Extensions of K
Starting from the modal logic $$K$$ one can add additional axioms, yielding different logics. We list the following basic axioms: $$K$$ :  $$\square (X \to Y) \to (\square X \to \square Y)$$ $$T$$ : $$\square A \to A$$ $$D$$ :  $$\square A \to \diamond A$$ $$4$$ :  $$\square A \to \square \square A$$ $$5$$ :  $$\diamond  A \to \square \diamond  A$$ $$B$$ :  $$ A \to \square \diamond  A$$ Traditionally, if one adds axioms $$A_1, \cdots, A_n$$ to the logic $$K$$ one calls the resulting logic $$KA_1 \cdots A_n$$. Sometimes, however the logic is so well known, that it is referred to under another name; e.g. $$K T 4$$, is called $$S4$$.

These logics can as well be characterised by certain classes of frames, because it is known that particular axioms correspond to particular restrictions on the reachability relation $$R$$ of the frame. If $$\langle W,R\rangle $$ is a frame, then a certain axiom will be valid on $$\langle W,R\rangle $$, if and only if   $$R$$ meets a certain restriction. Some restrictions are expressible by first-order logical formulae where the binary predicate $$R(x,y)$$ represents the reachability relation:

$$\begin{matrix} T: & \mbox{Reflexive}  &  \forall w \text{:} R(w,w) \\

D: & \mbox{Serial}    & \forall w \exists w': R(w,w') \\ 4: &\mbox{Transitive} & \forall s,t,u: (R(s,t)\land R(t,u)) \to R(s,u) \\ 5: & \mbox{Euclidean} & \forall s,t,u: (R(s,t)\land R(s,u)) \to R(t,u) \\ B: & \mbox{Symmetric} &\forall w, w': R(w,w') \to R(w',w) \end{matrix}$$