Welcome to Philosophy SE. I'll just fill in some of the flesh on the bones of Conifold's succinct argument.
We should not agree that believing the contradictory or the contrary of a proposition entails not believing that proposition.
In other words, believing a contradiction is different from being a contradiction; the former is a process, whereas the latter is a state. To wit:
P1 [B(p)] I believe Socrates is in the kitchen.
P2 [~B(p)] I don't believe Socrates is in the kitchen.
P3 [B(~p)] I believe Socrates is not in the kitchen.
Presuming the Law of the Excluded Middle (no claiming he's standing in the doorway!), p and ~p are contradictions. Socrates state simply can't be in two rooms at once. Hence, believing p and ~p is believing a contradiction. On the other hand, it is not a logical contradiction to actually believe in a contradiction in the technical sense. Rather, it's a fallacy in the technical, because it violates principles of good argumentation. (One shouldn't generally admit contradictory notions to build solid arguments, although we often do when in a more sophisticated, inductive sense. See dialetheism at SEP.)
Contradictions are properties of propositions, and fallacies are properties of people who argue by means of propositions. The first might be considered atomic propositions because they speak to how language corresponds to the state of affairs in the real world, and the latter might be seen as attitudinal propositions because they describe language with describes the coherence of reasoning. This dichotomy arises from a more fundamental dichotomy that is built into our language with the Cartesian duality which presumes there is a crisp distinction between the objective and subjective.
Is this a deepity? Yes! Why? Cognitive dissonance! To confuse the difference between the technical meaning of contradiction, and the lay meaning of contradiction is an act of linguistic equivocation and is a basis for informal fallacy. So that odd sensation you get thinking about it is your intuition telling you you're making an error in reasoning.
From Google: Contradiction
A combination of statements, ideas, or features of a situation that are opposed to one another.
"the proposed new system suffers from a set of internal contradictions"
a person, thing, or situation in which inconsistent elements are present.
the statement of a position opposite to one already made.
- Is a technical contradiction, where as 2. is technically an act of fallacy. In ordinary language, we use the same word for two closely related (but technically distinct) concepts. The joys of definition!
