## Franklin's Notes

### Model Theory for Modal Logic

Modal logic can be described as a variation of sentential logic that attempts to formalize the metaphysical notion of "possibility" or of a "possible world". In addition to the usual logical symbols $\land,\lor,\neg,\leftarrow,\leftrightarrow,\top,\bot$, we have two additional symbols $\square,\diamondsuit$ which respectively (informally) mean "necessarily" and "possibly". The language of modal logic also consists of (countably) infinitely many atomic sentence symbols $\mathbb P_0, \mathbb P_1, \mathbb P_2,\cdots$. I'm using Modal Logic by Brain Chellas as my reference book on this topic.

A model consists of a pair $\langle W,P\rangle$ in which $W$ is a set of "possible worlds", and $P=(P_0,P_1,P_2,\cdots)$ is an infinite sequence of subsets of $W$. The set $P_n$ consists of all possible worlds in $W$ at which $\mathbb P_n$ is true. Here are some basic observations:

• The $P_n$ can be empty. If $P_n=\varnothing$ for some $n$, then $\mathbb P_n$ is false at every possible world of $W$.

• The $P_n$ can also be equal to the entire set $W$. If $P_n=W$ for some $n$, then $\mathbb P_n$ is true at every possible world.

• There may exist $w\in W$ which is not contained in any of the $P_n$. This would correspond to a world at which every $\mathbb P_n$ is false.

Truth is notated/defined at worlds $w\in W$ in models $\mathfrak M=\langle W,P\rangle$ as follows. To express that a sentence $\mathrm A$ is true at some world $w\in W$ in a model $\mathfrak M$, we write $\vDash_w^\mathfrak M \mathrm A$, or just $\vDash_w \mathrm A$ when the model $\mathfrak M$ is understood from context. Truth is defined recursively on sentences as follows:

1. For all $n$: $\vDash_w^\mathfrak M \mathbb P_n$ iff $w\in P_n$
2. $\vDash_w^\mathfrak M \top$
3. $\vDash_w^\mathfrak M \neg\mathrm A$ iff not $\vDash_w^\mathfrak M \mathrm A$
4. $\vDash_w^\mathfrak M \mathrm A \land\mathrm B$ iff both $\vDash_w^\mathfrak M \mathrm A$ and $\vDash_w^\mathfrak M \mathrm B$
5. $\vDash_w^\mathfrak M \mathrm A \lor\mathrm B$ iff either $\vDash_w^\mathfrak M \mathrm A$ or $\vDash_w^\mathfrak M \mathrm B$
6. $\vDash_w^\mathfrak M \square\mathrm A$ iff for every $v\in W$, $\vDash_v^\mathfrak M \mathrm A$
7. $\vDash_w^\mathfrak M \diamondsuit\mathrm A$ iff for some $v\in W$, $\vDash_v^\mathfrak M \mathrm A$

Chellas includes some additional definitions for sentence of the form $\vDash_w^\mathfrak M \bot$ and $\vDash_w^\mathfrak M \mathrm A \rightarrow\mathrm B$ among others, but these are redundant because $\bot$ and $\rightarrow$ (and $\leftrightarrow$) can be expressed in terms of $\neg,\land,\lor$ and $\top$. (Technically, the above listing is redundant as well, since $\lor$ can be expressed in terms of $\neg$ and $\land$.) If some sentence $\mathrm A$ is true at every possible world of every model, then we say that $\mathrm A$ is valid and write $\vDash\mathrm A$.