OFFSET
1,3
COMMENTS
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.
REFERENCES
Merrie Bergmann, James Moor, and Jack Nelson. The logic book. Vol. 2. New York: McGraw-Hill, 1990, p. 54.
LINKS
Sean A. Irvine, Java program (github).
EXAMPLE
a(4) = 25, since the number of sentences of zeroth-order logic containing four characters are as follows: ~~~A, ~AxA, Ax~A, ~AxB, Bx~A, ~BxA, and Ax~B, where 'x' is any of the four binary logical operators.
CROSSREFS
KEYWORD
nonn
AUTHOR
Christoph B. Kassir, Jun 01 2021
EXTENSIONS
More terms from Sean A. Irvine, Jul 24 2021
STATUS
approved