An organization that determines its beliefs by voting can have inconsistent beliefs even if each voter has consistent beliefs and votes honestly.
For example, imagine that "(A AND B) logically implies C", and every voter agrees with that statement. Imagine that 1/3 of the voters believe (A, not-B, not-C), 1/3 believes (not-A, B, not-C), and 1/3 believes (A, B, C). If you hold a vote, 2/3s of voters believe A, 2/3s believe B, and 1/3 believes C; so the organization as a whole will hold the contradictory set of beliefs (A, B, not-C).
For example, imagine that "(A AND B) logically implies C", and every voter agrees with that statement. Imagine that 1/3 of the voters believe (A, not-B, not-C), 1/3 believes (not-A, B, not-C), and 1/3 believes (A, B, C). If you hold a vote, 2/3s of voters believe A, 2/3s believe B, and 1/3 believes C; so the organization as a whole will hold the contradictory set of beliefs (A, B, not-C).