What is the real logical meaning of deny?

Solution 1:

Logically, deny means SAY NOT;
i.e, if x denies P, then x says that P is false.
- DENY (x, P) means SAY (x, (NOT P))

Deny is transitive and requires an object, at least a pronoun it.
The object must be a noun phrase, and the noun phrase may be any kind of complement clause except an infinitive complement. Most noun objects are reduced complement clauses of one kind or another:

• He denied ((his) having) any involvement.
  (noun phrase object comes from reduced gerund clause with A-Equi.
  NB: deny, as a negative, licenses Negative Polarity Items like any)
• He denied (his) leaving the room.
  (gerund clause with A-Equi; i.e, he is the subject of both deny and leaving)
• He denied that they had told him about it.
  (tensed that-clause)
• He denied what she had said.
  (tensed embedded question clause)
• *He denied (for Mary) to leave.
  (ungrammatical for infinitive complements, with or without subjects)

There are also idioms. To deny someone is 'to refuse a request from a person'; it may also mean 'to refuse to recognize a person', or 'to say one does not recognize a person'. It may also mean something more proactive, like 'to exclude a person from communion, family, or friendship'.

Solution 2:

deny verb (denies, denied, denying) ¹
1. to declare something not to be true.
2. to refuse to give or allow to someone.
3. to refuse to acknowledge a connection with someone or something.
deny oneself something to do without (things that one desires or needs).

denial noun ²
1. an act of denying or declaring something not to be true.
2. an act of refusing something to someone.
3. an act of refusing to acknowledge connections with somebody or something.
in denial psychol suffering from a condition which involves the suppression, usually at a subconscious level, of an emotion, truth, etc that is particularly painful or difficult to come to terms with • After the death of his mother, he was in denial for months.

In most uses, the words deny and denial have no direct connection with whether the matter being denied is true or false.

Both your examples involve denial.

• In your first example, you are denying you broke the mug, because you are afraid of admitting the truth. You are telling a lie.
• In your second example, you are denying you broke the mug, because that is the truth: you did not break it. You are telling the truth.

But deny is the correct word in both cases.

However, there are cases where using the word deny or denial implicitly means that it is not the truth. For example, the second definition above contains the sentence:

After the death of his mother, he was in denial for months.

That means that he refused to accept / acknowledge that his mother was dead. In that particular case, obviously he was denying the truth.

To repeat, in most cases the words deny and denial have no implication of whether the matter being denied is true or false.