
| Pr B C L C"   | Pr B CL C"   | Pr B C    |            |    |    |
|---------------|--------------|-----------|------------|----|----|
| ))            | ))           |           |            |    |    |
| Pr L C"       | Pr L C"      | Pr L C"   |            |    |    |
| ))            | ))           | ))        |            |    |    |
| Pr BC L C"    | Pr BC L C "  | yv        | yv         |    |    |
| ))            | ))           |           |            |    |    |
| Pr L C"       | Pr L C"      | xu        | x          |    |    |
| ))            | ))           | Pr BCL C" | Pr BCL C " | y  | yv |
| ))            | ))           |           |            |    |    |
| Pr L C"       | Pr L C"      | x         | x          |    |    |
| ))            | ))           |           |            |    |    |
| Pr BC L C"    | Pr BCL C "   | yv        |            |    |    |
| ))            | ))           |           |            |    |    |
| Pr L C"       | Pr L C"      | x         |            |    |    |
| ))            | ))           |           |            |    |    |
| Pr C L C" ))  | y            |           |            |    |    |
| Pr L C" ))    | y x          | y         | y          |    |    |

b) Let u, , , and y be real numbers from (0; 1]. For al l probabilistic interpretations Pr with Pr B , the conditions u Pr B Pr BL G)), Pr L G)) Pr BL G)),

Pr B Pr BL H )), and y Pr L H)) Pr BL H )) are equivalent to:

| Pr B L G L H))   | Pr B L G L H))      | Pr B L G))    |                     |    |
|------------------|---------------------|---------------|---------------------|----|
| Pr B             | Pr B                | Pr B          |                     |    |
| Pr BL G L H))    | Pr B L G L H ))     | u v           | u                   |    |
| Pr B             | Pr B                | Pr BL G L H)) | Pr BL G L H )) Pr B | u  |
| Pr B             |                     |               |                     |    |
| Pr BL G L H))    | Pr BL G L H )) Pr B | u             |                     |    |
| Pr B             |                     |               |                     |    |
| Pr B L H))       | x                   |               |                     |    |
| Pr B             | y                   |               |                     |    |

Proof. The claims can be veried by straightforward arithmetic transformations based on the properties of probabilistic interpretations. 2

We are now ready to prove Theorems 4.7 and 4.8.

Proof of Theorem 4.7. The claims are proved by induction on the recursive denition of H The proof for C B B : : : Bk with k is done for k 2. It can easily be generalized to k 2. For C B we dene u Pr C B), Pr B C), H C; C"), and y H C; C"). Note that  entails ; y and + 1. For C B B we dene G B H B u H  B ; G), H  B ; G), H  B ; H), and y H  B ; H). Note that  entails u ; v ; x ; y and u + 1. a) All models Pr Mo B ; C with Pr L C)) satisfy the indicated condition. In the sequel, let Pr Mo B ; C with Pr L C)) and thus also Pr B 0. Basis: Let C B Since C L C), we get:  Pr L C)) Pr L C)) Pr CL C)) Pr BL C)) : Induction: Let C B For all models Pr Mo C; C"), we get Pr C Pr CL C")) by Theorem 4.3 a), and y Pr L C")) Pr CL C")) by the induction hypothesis. Thus,

Theorem 4.3 a) and by the induction hypothesis, respectively:

Pr satises the same conditions. Since L C"L C and by Lemmata A.1 and C.6 a):

$y$$\frac{y}{x}+\frac{y}{x}$$Pr$$CL$$C^{\dagger}$$Pr$$L$$C^{\dagger}$$Pr$$CL$$C$$Pr$$L$$C$$G$$G$$H$$Pr$$Mo$$B,G$$Pr$$Mo$$B,H$
u Pr B Pr BL G)); Pr B Pr BL H)) Pr L G)) Pr BL G)); y Pr L H)) Pr BL H)) :
235