## 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 $$R'_2$$.