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A277276 Number of tautologies and contradictions in propositional calculus of length n. 2
0, 0, 0, 0, 2, 8, 14, 26, 63, 215, 527 (list; graph; refs; listen; history; text; internal format)



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.


Table of n, a(n) for n=1..11.

M. Scroggs, Logical Contradictions

M. Scroggs, List of tautologies

M. Scroggs, List of contradictions


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.


Equals A256120 plus A277275

Sequence in context: A161156 A125902 A295055 * A324785 A173974 A056677

Adjacent sequences:  A277273 A277274 A277275 * A277277 A277278 A277279




Matthew Scroggs, Oct 08 2016



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Last modified April 21 11:19 EDT 2021. Contains 343150 sequences. (Running on oeis4.)