The 8 statements occurring in this problem are, respectively: 1 1 2 2 3 3 4 4. These are the values at which the V switches (the statement dial switches, Unit 2) from V₁ to V₈ are set. The 7 connectives occurring in this problem are, respectively: and, if-then, and, if-then, and, if-then, if-then. These are the values at which the k switches (the connective dial switches, Unit 4) from k₁ to k₇ are set.
A grouping (one of several possible groupings) that specifies the antecedent and consequent of each connective is the following:
| 1 | and | 1 | if-then | 2 | and | 2 | if-then | 3 | and | 3 | if-then | 4 | if-then | 4 |
| | | | | | | | | | | | | |||||||||
| k₂ | k₄ | k₆ | ||||||||||||
| | | | | | | | | |||||||||||
| k₁ | k₅ | |||||||||||||
| | | | | |||||||||||||
| k₃ | ||||||||||||||
| | | | | |||||||||||||
| k₇ | ||||||||||||||
The grouping has here been expressed graphically with lines but may be expressed in the normal mathematical way with parentheses and brackets as follows:
{[ 1 and (1 if-then 2)] and [(2 if-then 3) and (3 if-then 4) ] } if-then 4.
So the values at which the antecedent and consequent dial switches are set are as shown in [Table 13].
Table 13
| Connective | Antecedent Switch | Set at | Consequent Switch | Set at |
|---|---|---|---|---|
| k₁ | A₁ | V | C₁ | k₂ |
| k₂ | A₂ | V | C₂ | V |
| k₃ | A₃ | k₁ | C₃ | k₅ |
| k₄ | A₄ | V | C₄ | V |
| k₅ | A₅ | k₄ | C₅ | k₆ |
| k₆ | A₆ | V | C₆ | V |
| k₇ | A₇ | k₃ | C₇ | V |
In any problem, statements that are different are numbered one after another 1, 2, 3, 4 ···. A statement that is repeated bears always the same number. In nearly all cases that are interesting, there will be repetitions of the statements. If any statement appeared with a “not” in it, we would turn up the denial switch for that statement (Unit 2).
The different connectives available on the machine are “and,” “or,” “if ··· then,” “if and only if.” If a “not” affected the compound statement produced by any connective, we would turn up the denial switch for that connective (Unit 8).