#### Supplement to Philosophical Aspects of Multi-Modal Logic

## Long descriptions for some figures in Philosophical Aspects of Multi-Modal Logic

### Figure 1 description

A diagram consisting of three labeled small, light orange circles with
darker orange rims in a roughly triangular layout enclosed by a large
rectangle with the whole labeled *M*. The first circle is labeled
\(w_1\) and encircles the letter *p*; this circle has two arrowed
lines, one dashed and one solid, each of which loops back to it. The
second circle is labeled \(w_2\) and also contains the letter
*p*. It also has two arrowed lines, one dashed and one solid,
looping back to it. In addition it has a dashed arrowed line coming to
it from the third circle and a solid line with arrowheads at both ends
between it and the third circle. The third circle is labeled \(w_3\)
and does not contain any letters. Besided the lines connecting it to
the second circle, it has an arrowed solid line looping back to
itself.

### Figure 2 description

Like figure 1 above, this has three small orange circles but spaced in
a line. The first two circles on the left are enclosed by a large
rectangle labeled *M* and the last circle is enclosed in its own
rectangle labeled \(M_{(p \land \lnot Kp)!}\). The two rectangles are
connected by an arrowed double line pointing to the right rectangle
with the line labeled \((p \land \lnot Kp)!\). The first circle in the
first rectangle has a double line dark orange rim, is labeled *w*
and contains the letter *p*. The second circle in the first
rectangle has a single line dark orange rim, is labeled *u*, and
contains nothing. A solid line connects these two circles. The circle
in the second rectangle has a double line dark orange rim, is labeled
*w*, and contains the letter *p*.

### Figure 3 description

Two large rectangles each containing several orange circles. The first
rectangle is labeled *M* and contains two circles. The first has
a double line dark orange rim and is labeled *w*. The second has
a single line dark orange rim, is labeled *u*, and contains the
letter *p*. Two solid lines connect the two circles; one labeled
\(R_1\) and the other \(R_2\).

The second rectangle contains 5 orangle circles on a 3x3 matrix. The
first column contains, in the second row, the first circle, which has a double line dark
orange rim. The other two columns each have two
circles, one in the first row and one in the third row, with all of them having only a single line dark orange rim. The first circle is labeled
\(w'\). The circles in the second column each contain the letter
*p* and are labeled \(u'_1\) (first row) and \(u'_2\) (third
row). The circles in the third column contain nothing and are labeled
\(v'_1\) (first row) and \(v'_2\) (third row). The \(w'\) circle has a
solid line labeled \(R'_1\) to the \(u'_1\) circle which in turn has
two solid lines to the \(v'_1\) circle labeled \(R'_1\) and \(R'_2\).
The \(w'\) circle also has a solid line labeled \(R'_2\) to the
\(u'_2\) circle which in turn has two solid lines to the \(v'_2\)
circle labeled \(R'_1\) and \(R'_2\).