Figure 2 description

A diagram consisting of eight circles labelled A through H. These are laid out in a five columns by three rows array. A is at in the first column from the left and first row from the top; B is in the same column but in the bottom row; both have arrows pointing to C which is in the second column, middle row; A's arrow is labelled \(C \dequal A \land B\). D is in the third column top row and E in the same column bottom row; C has arrows going to both with the arrow to D labelled \(D \dequal \neg C\) and the arrow to E labelled \(E \dequal C\) . F is in the fourth column top row and G in the same column bottom row; an arrow goes fro D to F labelled \(F \dequal \neg D\); an arrow goes from E to G labelled \(G \dequal E\). H is in the fifth column middle row; arrows go to it from F and G; the arrow from F to H is labelled \(H \dequal F \lor G\).

Figure 4 description

The diagram is a circle outlined in red and split into four parts by a horizontal line and a vertical line. The horizontal line is labelled ‘Antecedent’ with a \(\phi\) above and a \(\neg \phi\) below. The vertical line is labelled ‘Consequent’ with a \(\psi\) to the left and a \(\neg\psi\) to the right. The center of the circle is labelled \(w_0\). The upper half of the circle is tinted blue and the upper right quadrant is shaded with a pattern. This upper half is also labelled ‘Constant Domain’. The whole diagram is labeled ‘Accessible Worlds \(R(w_0)\). Above the diagram is a legend: “Strict Analysis of \(\phi > \psi\): All φ-worlds accessible from \(w_0\) (indicated in blue in the diagram) are ψ-worlds. I.e., shaded regions must be empty”

Figure 5 description

The diagram is three concentric circles outlined in red (so forming two rings, outer and middle, and an innermost circle) and split into four parts by a horizontal line and a vertical line. The horizontal line is labelled ‘Antecedent’ with a \(\phi\) above and a \(\neg \phi\) below. The vertical line is labelled ‘Consequent’ with a \(\psi\) to the left and a \(\neg\psi\) to the right. The center of the circles is labelled \(w_0\). The upper half of the circles is tinted blue with the tint with the lightest tint in the outer ring, slightly darker in the middle ring and darkest in the innermost circle; this half is labelled ‘Variable Domain’. The upper right quadrant of the innermost circle is shaded with a pattern. In the middle ring in the upper right quadrant is a dot labelled \(w_1\). The whole diagram is labelled ‘System of Spheres of Similarity \(\mathcal{R}(w_0)\)’. Above the diagram is a legend: “ Similarity Analysis of} \(\phi > \psi\) All \(\phi\)-worlds most similar to \(w_0\) (indicated in blue in the diagram) are \(\psi\)-worlds. I.e. shaded region must be empty.

Figure 6 description

The diagram is a circle outlined in red and split into four parts by a horizontal line and a vertical line. The horizontal line is labelled ‘Antecedent’ with a \(\phi_1\) above and a \(\neg \phi_1\) below. The vertical line is labelled ‘Consequent’ with a \(\psi\) to the left and a \(\neg\psi\) to the right. The center of the circle is labelled \(w_0\). The whole diagram is labeled ‘Accessible Worlds \(R(w_0)\). The upper half of the circle is tinted blue and the upper right quadrant is shaded with a pattern. So far very similar to Figure 4; the major difference is a yellow circle outlined in black above the center of the first circle and bisected by the vertical line. Most of the yellow circle is inside the first circle, but, a small portion is outside. The yellow circle is labelled \({\llbracket}\phi_2{\rrbracket}^R_v\)

Figure 7 description

This figure has two similar diagrams, a and b. Each diagram has three overlapping circles (left, center, right); however, the overlap between left and right is completely within the center circle. In the non-overlapping section of the right circle is a dot labelled w.

For diagram a the left circle is tinted yellow and labelled \({\llbracket}\phi{\rrbracket}^R_v\); the center circle is gray and labelled \({\llbracket}\psi{\rrbracket}^R_v\); the right is blue and labelled \(R(w)\). The whole is labelled \(\llbracket\phi\strictif\psi\rrbracket^R_v\).

For diagram b the left circle is tinted red and labelled \({\llbracket}\neg\psi{\rrbracket}^R_v\); the center circle is gray and labelled \({\llbracket}\neg\phi{\rrbracket}^R_v\); the right is purple and labelled \(R(w)\). The whole is labelled \(\llbracket\neg\psi\strictif\neg\phi\rrbracket^R_v\).

Figure 8 description

The diagram is made up of four circles (yellow, orange, blue, gray). The center of the orange circle is a point labelled \(w\); the circle as a whole is labelled \({\llbracket}\phi_1{\rrbracket}^R_v\). The yellow circle is to the left and overlaps the orange circle (the overlap includes the \(w\) point); the yellow circle is labelled \({\llbracket}\phi_2{\rrbracket}^R_v\). The blue circle is to the right and overlaps the orange circle (the overlap again includes the \(w\) point); the blue circle is labelled \(R(w)\). The orange circle is completely within the union of the blue and yellow circles. The gray circle is to the lower right of the orange circle; it includes all of the yellow/blue overlap and orange/blue overlap but not all of the orange/yellow overlap; it is labelled \({\llbracket}\psi{\rrbracket}^R_v\).