Exercise 8, Appendix 0

Let $R_1$, $R_2$, ..., $R_n$ be the logically irreducible relations in a theory $\mathscr{T}$ (Exercise 6). Assign to each of the relations $R_i$ one of the signs $0, 1, 2$. To each relation in a logical construction with base $R_1$, $R_2$, ..., $R_n$ we assign one of the signs 0, 1, 2 in accordance with the following symbolical rules (Exercise 7) :

(a) If $R$ is logically constructed from $R_1$, $R_2$, ..., $R_n$, the sign carried by $R$ is independent of the logical construction, with base $R_1$, $R_2$, ..., $R_n$, in which $R$ appears.

(b) Suppose that the only axioms of $\mathscr{T}$ are those provided by the schemes S2, S3, S4. Let $R$ be a theorem in $\mathscr{T}$. Show that however the signs 0, 1, 2 are assigned to the logical components of $R$, the corresponding sign carried by $R$ is 0. On the other hand, if $S$ is logically irreducible and has sign 2, then the sign carried by $(S$ or $S) \Rightarrow S$ is 2. Deduce that $\mathscr{T}$ is not equivalent to the theory which has the same signs as $\mathscr{T}$ and whose axioms are provided by the schemes S1, S2, S3, S4.

(c) Establish an analogous result for theories based only on the schemes S1, S3, S4, or on the schemes S1, S2, S4. (Use respectively the following rules :

(d) Establish an analogous result for a theory based only on the schemes S1, S2, S3. (Use four signs 0, 1, 2, 3 and the following rules :