Is there a modal logic axiom corresponding to the condition that for every world w there is some world v such that vRw? (Reverse Seriality)

Expanding on the other answer, there is no such formula in the standard semantics.

Indeed, consider the following two models (here every world will be a $p$ world, where $p$ is the only atomic proposition):

• $\mathfrak{M}_1$ with exactly one world $w_1$ and the empty relation.

• $\mathfrak{M}_2$ with two worlds $x$ and $w_2$, such that $x$ sees $w_2$.

There is a bisimulation linking $w_1$ and $w_2$, so they satisfy the same modal formulas. But if there were a formula that expressed $\exists v . v R w$, then $w_2$ would satisfy it while $w_1$ wouldn't! Thus no such formula exists.

If you really want to express this formula, you need to pass to a richer language. As the other answerer has said, the standard way to do this is by adding symbols that express moving backwards along relations. Often in this context we write $\overset{\rightarrow}{\square}$ and $\overset{\rightarrow}{\lozenge}$ instead of $\square$ and $\lozenge$. We also add symbols $\overset{\leftarrow}{\square}$ and $\overset{\leftarrow}{\lozenge}$ to represent moving backwards along the relation. That is:

$$w \models \overset{\rightarrow}{\square} \varphi \iff \forall v . wRv \to v \models \varphi$$

$$w \models \overset{\leftarrow}{\square} \varphi \iff \forall v . vRw \to v \models \varphi$$

It turns out the following axioms (plus Necessisitation for $\overset{\rightarrow}{\square}$ and $\overset{\leftarrow}{\square}$ and Modus Ponens) are sound and complete wrt the class of all frames with the above semantics:

• All classical tautologies

• $\overset{\rightarrow}{\square}(\varphi \to \psi) \to \overset{\rightarrow}{\square} \varphi \to \overset{\rightarrow}{\square} \psi$

• $\overset{\leftarrow}{\square}(\varphi \to \psi) \to \overset{\leftarrow}{\square} \varphi \to \overset{\leftarrow}{\square} \psi$

• $\varphi \to \overset{\rightarrow}{\square} \overset{\leftarrow}{\lozenge} \varphi$

• $\varphi \to \overset{\leftarrow}{\square} \overset{\rightarrow}{\lozenge} \varphi$

In this system, "reverse seriality" can be expressed as $\overset{\leftarrow}{\square}\varphi \to \overset{\leftarrow}{\lozenge}\varphi$.

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I hope this helps ^_^