cp's OEIS Frontend

n-place formulas in first-order logic like Ax Ey P(x, y) or Ex P(x, x) can be ordered by implication. This Hasse diagram has A300696(n) vertices and a(n) edges.

The corresponding sequence for convex concertina n-cubes is A300693.

The characters of zeroth-order logic include the tilde (~), ampersand (&), wedge (∨), horseshoe (⊃), triple bar (≡), left and right parentheses, and variables (upper-case letters with or without subscripts.) However, since the set of upper-case letters with or without subscripts is infinitely large, it is then, for the sentences of zeroth-order logic containing k variables, restricted to the set {A1, ..., Ak}, with an additional restriction as follows: a sentence may only contain Ai iff it contains every Aj for j=1..i-1 (this gives a total of A000670(k-1) legal permutations for a sentence containing k variables.)

The rules for a well-formed formula (wff) of zeroth-order logic are defined recursively as follows (see M. Bergmann et al.):

1. Every variable is a wff.

2. If P is a wff, then so is ~P.

3. If P and Q are wffs, then so is (PxQ), where 'x' is any binary logical operator.

It is also customary to remove the outermost parentheses of a sentence.