Liar, Liar, Pants on Fire!

Suppose Person A is speaking the truth. Then ultimately , Persons B,C,D speak lie which contradicts the statement of Person A. Hence , Person A speaks lies.

So if Person A speaks lies , Person B must speak truth since their statements are contradictory.So Person C can speak truth or lie.

If person C speaks truth , then person D also speaks truth which contradicts statement of person B because then there will be more truth tellers than liars.

So , person C speaks lie , which means person D also lies and person E also lies because it will make the statement of person $C$ true.

Hence , Person B only definitely speaks the truth.

(The phrases in bold say the conclusion)

If C: " $\text{Only two of us speak the truth.}$ " is $\color{#3D99F6}{True}$ , then:

• D: " $\text{Only three of us speak lie.}$ " is also $\color{#3D99F6}{True}$ and
• B: " $\text{More of us speak lie than truth.}$ " is also $\color{#3D99F6}{True}$

then, there are at least three persons speak truth which contradicts with C: " $\text{Only}$ $\text{two of us speak the truth.}$ " Therefore, C: " $\text{Only two of us speak the truth.}$ " must be $\color{#D61F06}{False}$ .

Since C: " $\text{Only two of us speak the truth.}$ " is $\color{#D61F06}{False}$ , then:

• D: " $\text{Only three of us speak lie.}$ " is also $\color{#D61F06}{False}$ and there are at least two liars.
• A: " $\text{More of us speak truth than lie.}$ " must be $\color{#D61F06}{False}$ because if it is true than B's and E's statements must be true, but B's contradicts with A's. So there is at least three true statements, then
• B: " $\text{More of us speak lie than truth.}$ " must be $\color{#3D99F6}{True}$
• E: " $\text{I always speak truth.}$ " can be $\color{#3D99F6}{True}$ or $\color{#D61F06}{False}$ .

Person $\boxed{B}$ definitely speaks the truth.

(1) A and B say opposite sentences so they are of different types.

(2) C and D say the same sentence logically speaking so they share the same type.

Let's suppose C and D are truthtellers, since there are already two honest people, A, B and E must be liars wich is a contradiction since in (1) we say that they are of different types. So, by (2) we get that C and D are both Liars.

E can't be honest, because if so, since either A or B is a truthteller, there will be exactly two honest people and the sentence of C (and D) would be right, so E is a liar.

The last step is to determine who is the liar and who is not from A and B. Obviously there are 3 liars so far so B is right and A is a liar too, therefore, the only one who is honest is B

A & B are saying 2 opposite things, so one must be truthful while the other is lying. On the other hand, C & D are saying complementary things, so the two must always be together in truths or in lies, but we know that they can't be the only 2 truthful people since we already deduced that either A or B must be one, too, thus C & D must both be lying.

By now, with 2 identified liars and 1 yet to be identified liar between A & B, we know that there are more liars than truth teller(s), so we have a truthful B and a liar A.

Finally, since D's statement is false, then the liars need another member to keep their lies intact, and we have E to be our last liar for the day.

Conclusion : B is the only truthful person in the group.

Considering the statements made by C and D, we find that both have the same meaning. Since there are 5 people, if 3 are lying for example, then $5-3=2$ people would be speaking the truth. This means that C and D have the same truth value -either both are true or both are false. Suppose both are true, then we already have two true statements. But if we consider statements made by A and B, they have opposing meanings so they have opposite truth values - one true and one false. This would give us at least three true statements which contradicts our assumption that C and D are telling the truth. So both C and D must be lying.

Now we have at most $5-2=3$ true statements since we already have identified 2 liars. Also considering the fact that either A or B is telling a lie, this leaves us with at most 2 true statements. But for the statement that 2 are telling the truth to be false, there cannot be exactly 2 true statements - there can be at most one. Since we also know that there is exactly one truth teller between A and B, there cannot be zero truth tellers either. So we have identified the number of truth tellers to be exactly one.

With that, statement B's statement must be true as there are 4 liars and 1 truth teller (which is B himself).