Leon Gumański

1.

True		False
u  w u w		u  w
1	2	1 u	2 w

It should be immediately obvious that if u or w is not atomic, succeeding steps lead ultimately only to closed tableaux with atomic expressions at the end of the stage and for each of these tableaux there will be such an atomic expression which appears in both columns of the tableau (a precise proof of it is by induction on the number of truth–functors).

REMARK 2.4. For all that reason we shall hereafter take into consideration k–stage tableau–constructions of t with terminated stages exclusively. Definition 2.2 has already been formulated in accordance with this intention.

CONVENTION 2.1. From now on we shall use the letter “n” to denote the number of subtableaux which are present at stage 1 of any k–stage tableau–con­stru­ction of t.

Now, there is an important point which must not be overlooked.

REMARK 2.5. Since every tableau of a k–stage tableau–construction of t ter­minates in atomic expression, in the light of previous Remark 2.3 and 2.4 it should be obvious to the reader that it is enough to inspect only the atomic expressions in order to determine whether all the tableaux of the construction are closed.

In view of this it will be convenient to characterize any k–stage tableau–con­struction of t in the shape of a tree, taking into account only the atomic expre­ssions and supposing that all subtableaux have been numbered in succession in conformity with Assumption 2.2.

ASSUMPTION 2.3. Every atomic expression occurring in the left (True) side of a subtableau (tableau) is preceded by the sign “+”.

Graph G

Stage	1	2	k–1	k
t

where p=n+n²+n³+…+n^k–2, q=p+n^k–1, r= q+n^k (hence r is the number of sub­ta­bleaux whereas n^k is the number of tableaux in this construction). The set is defined as follows (1jn, 1er):

DEFINITION 2.6. is the ordered set of all atomic expressions occurring in the e–th subtableau which belongs to stage 1 if en (e=j) and if e>n, then is on the j–th branch of the anterior subtableau. Atomic expressions v, +v are separate elements of

ASSUMPTION 2.4. For every jn, dr, er: the elements of are ordered according to the same principle as the elements of

CONVENTION 2.2. From now on we shall use the abbreviation “in analogous places in the sets X, Y” for “in the place g in the e–th expression of X and in the place g in the e–th expression o Y”.

In the light of assumptions 2.1, 2.2, 2.3, 2.4 the following observation is obvious:

OBSERVATION 2.2. The sets (e ≠ d) have the same structure: they differ at most in the shape of y–variables in analogous places and if a y–variable has been substituted by rule F(a) (or F(Ea)) in a certain place g in then a y–variable has also been substituted by the same rule in the place g in

REMARK 2.6. We can characterize any tableau T of the construction as a sequence of sets of atomic expressions like this: