OFFSET
1,5
COMMENTS
a(n) is the number of tautologies and contradictions that are n symbols long in propositional calculus with the connectives not (~), and (*), or (+), implies (->) and if and only if (<->).
When measuring the length of a formula all brackets must be included. The connectives -> and <-> are counted as one symbol each (but writing them as such requires non-ASCII characters).
Formally, the language used for this sequence contains the symbols a-z and A-Z (the variables), ~, *, +, ->, <->, ( and ).
The formulas are defined by the following rules:
- every variable is a formula;
- if A is a formula, then ~A is a formula;
- if A and B are formulas, then (A*B), (A+B), (A->B) and (A<->B) are all formulas.
A formula is a tautology if it is true for any assignment of truth values to the variables.
A formula is a contradiction if it is false for any assignment of truth values to the variables.
This sequence is increasing, as adding a ~ to the start of a tautology or contradiction gives a contradiction or tautology one symbol longer.
LINKS
M. Scroggs, Logical Contradictions
M. Scroggs, List of tautologies
M. Scroggs, List of contradictions
EXAMPLE
The contradictions of length 6 are ~(a<->a), ~(a->a), (~a*a), (~a<->a), (a*~a) and (a<->~a): 6 formulas, and the tautologies of length 6 are (~a+a) and (a+~a): 2 formulas. So a(6) = 6+2 = 8.
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Matthew Scroggs, Oct 08 2016
STATUS
approved