Figure 2 description

A square with the four corners labeled in clockwise order as Necessary, Impossible, Non-Necessary, and Possible. The legend describes four types of lines:

• Implication [arrowed line],
• Contraries [dotted blue line],
• Subcontraries [green line],
• Contradictories [dashed purple lines].

Implication lines connect Necessary to Possible and Impossible to Non-Necessary. A Contraries line connects Necessary and Impossible. A Subcontraries line connects Possible and Non-Necessary. Contradictories lines connect Necessary and Non-Necessary and also Impossible and Possible.

Figure 4 description

A square with the four corners labeled in clockwise order as Obligatory, Impermissible, Omissible, and Permissible. Assuming the legend from figure 2 Implication lines connect Obligatory to Permissible and Impermissible to Omissible. A Contraries line connects Obligatory and Impermissible. A Subcontraries line connects Permissible and Omissible. Contradictories lines connect Obligatory and Omissible and also Impermissible and Permissible.

Figure 5 description

A hexagon with the corners labeled clockwise as

1. \(\OB p\)
2. \(\IM p\)
3. \(\NO p\)
4. \(\OM p\)
5. \(\PE p\)
6. \(\OP p\)

Using the same legend as figure 2, the following lines connect each corner to all the others

• Implication [arrowed line],
  • \(\OB p\) (corner 1) to \(\NO p\) (corner 3)
  • \(\OB p\) (corner 1) to \(\PE p\) (corner 5)
  • \(\IM p\) (corner 2) to \(\NO p\) (corner 3)
  • \(\IM p\) (corner 2) to \(\OM p\) (corner 4)
  • \(\OP p\) (corner 6) to \(\OM p\) (corner 4)
  • \(\OP p\) (corner 6) to \(\PE p\) (corner 5)
• Contraries [dotted blue line],
  • \(\OB p\) and \(\IM p\) (corners 1 and 2)
  • \(\OB p\) and \(\OP p\) (corners 1 and 6)
  • \(\IM p\) and \(\OP p\) (corners 2 and 6)
• Subcontraries [green line],
  • \(\NO p\) and \(\OM p\) (corners 3 and 4)
  • \(\NO p\) and \(\PE p\) (corners 3 and 5)
  • \(\PE p\) and \(\OM p\) (corners 4 and 5)
• Contradictories [dashed purple lines].
  • \(\OB p\) and \(\OM p\) (corners 1 and 4)
  • \(\IM p\) and \(\PE p\) (corners 2 and 5)
  • \(\NO p\) and \(\OP p\) (corners 3 and 6)

Figure 6 description

A square with the four corners labeled in clockwise order as

1. All \(x:p\)
2. No \(x:p\)
3. Some \(x:p\)
4. Some \(x: \neg p\)

Assuming the legend from figure 2 Implication lines connect corner 1 to corner 4 and corner 2 to corner 3. A Contraries line connects corners 1 and 2. A Subcontraries line connects corners 3 and 4. Contradictories lines connect corners 1 and 3 and also corners 2 and 4.

Figure 7 description

Three boxes containg respectively the phrases:

1. All \(x:p\)
2. Some \(x: p\) & Some \(x: \neg p\)
3. No \(x: p\)

The first and second are braced as “Some \(x: p\)”. The second and third are braced as “Some \(x: \neg p\)”.

Figure 8 description

A diagram of six boxes in a row. All boxes have a blue dot at the bottom and \(A^i\) below.

1. first box is labeled “\(\OB p\):” and contains “All \(p\)”
2. second box is labeled “\(\PE p\):” and contains “Some \(p\)”
3. third box is labeled “\(\IM p\):” and contains “No \(p\)”
4. fourth box is labeled “\(\OM p\):” and contains “Some \(\neg p\)”
5. fifth box is labeled “\(\OP p\):” and contains “Some \(p\) and Some \(\neg p\)”
6. sixth box is labeled “\(\NO p\):” and contains “All \(p\) or No \(p\)”

Figure 10 description

A vertical bar with three vertical dots below it. The top of the bar is labelled \(\OB p:\) and an arrow pointing to the top of the bar is labelled “all \(p\)-worlds here”. A brace encloses both the bar and the dots and is labelled “the \(i\)-ranked worlds (the higher the level, the better the worlds within it, relative to \(i\))”.

Figure 11 description

A diagram of two boxes in a row. All boxes have \(R^i\) below.

1. The first box has “NEC\(p\);” above and “All \(p\)” inside.
2. The second box has “POS\(p\);” above and “Some \(p\)” inside.

Figure 12 description

A diagram of one box which has “\(d\): DEM” above and a blue dot labelled \(j\) inside.

Figure 13 description

Two overlapping blue boxes. In the intersection of the two boxes is a blue dot. Above both is “POS\(d\): DEM”; below both is “\(R^i\)”.

Figure 14 description

Six pairs of two overlapping blue boxes. Below each pair is “\(R^i\)”. Each pair also has a blue dot in the intersection of the two boxes.

1. first pair is labelled “\(\OB p:\) DEM” and has the blue dot labelled “All \(p\)”.
2. second pair is labelled “\(\PE p:\) DEM” and has the blue dot labelled “Some \(p\)”.
3. third pair is labelled “\(\IM p:\) DEM” and has the blue dot labelled “No \(p\)”.
4. fourth pair is labelled “\(\OM p:\) DEM” and has the blue dot labelled “Some \(\neg p\)”.
5. fifth pair is labelled “\(\OP p:\) DEM” and has the blue dot labelled “Some \(p\) and Some \(\neg p\)”.
6. sixth pair is labelled “\(\NO p:\) DEM” and has the blue dot labelled “All \(p\) or No \(p\)”.

Figure D.1 description

A graph of three dots: \(i, j, k\).

1. first dot is labelled \(i\). There is the comment “So \(\neg\OB(\OB p \rightarrow p)\)”.
2. second dot is labelled \(j\) and has \(\neg p\) above it. There is the comment “So \(\OB p\) and \(\neg(\OB p \rightarrow p)\)”.
3. third dot is labelled \(k\) and has \(p\) above it. There is no comment.

An arrow points from dot \(i\) to dot \(j\) and another from dot \(j\) to dot \(k\). An arrow also points from dot \(k\) back to itself.

Figure D.2 description

A graph of four dots: \(i, j, k, l\). Similar to figure D.1.

1. first dot is labelled \(i\). There is the comment “So \(\OB \OB p \rightarrow \OB p\) and \(\neg\OB(\OB p \rightarrow p)\)”.
2. second dot is labelled \(j\) and has \(\neg p\) above it. There is the comment “So \(\neg(\OB p \rightarrow p)\)”.
3. third dot is labelled \(k\) and has \(p\) above it. There is no comment.
4. fourth dot is labelled \(l\).

An arrow points from dot \(i\) to dot \(j\) and another from dot \(j\) to dot \(k\). An arrow also points from dot \(k\) back to itself. In addition an arrow points from dot \(i\) to dot \(l\) and another from dot \(l\) to dot \(j\).