How to find the probability of truth? A and B are independent witness in a case. The probablity that A
  speaks the truth is 'x' and that of B is 'y'.If A and B agree on a
  certain statement, how to find the probability that the statement is
  true ?
$ P(A)=x $ and $ P(B)=y $,

$A$ and $B$ are independent, so $P(A \cap B)=P(A)P(B)$

Therefore, the probability that both speak the truth will be $P(A \cap B)=P(A)P(B)=xy$.

And then, the probability that they agree on a certain statement is, they both speak the truth or they both tell lie, which would be:

$$xy + (1-x)(1-y) $$

As a result, the probability that the statement is true is:

$$ \frac{xy}{xy+(1-x)(1-y)} $$.
Say the fact is True or False (T/F), and independent statements A, B are binary.

We are given P(A=1|T)=x and P(B=1|T)=y.
By convexity P(A=0|T)=1-x and P(B=0|T)=1-y.

We can surmise P(A=0|F)=x and P(B=0|F)=y.
Also by convexity P(A=1|F)=1-x and P(B=1|F)=1-y.

We want P(T|A=1,B=1).

By Bayes rule P(T|A=1,B=1) = P(A=1,B=1|T)P(T)/P(A=1,B=1).

By independence P(T|A=1,B=1) = P(A=1|T)P(B=1|T)P(T)/(P(A=1)P(B=1)).

Plugging in, the numerator is xyP(T).

The denominator requires P(A=1) = P(A=1|T)P(T) + P(A=1|F)P(F) = xP(T) + (1-x)P(F), same for P(B=1).

So I don’t think we are identified without knowing the marginal probability of the truth P(T).

Of course, for a given x and y you can provide P(T|A=1,B=1) as a function of P(T) between 0 and 1.