Conditional probabilties in the OR-gate $T=A\cdot B$ with zero-probabilities in $A$ and $B$? My earlier question became too long so succintly: 

What are $P(T|A)=P(T\cap A)/P(A)$ and $P(T|B)=P(T\cap B)/P(B)$ if $P(A)=0$ and $P(B)=0$?

I think they are undefined because of the division by zero. How can I specify the conditional probabilities now? Please, note that the basic events $A$ and $B$ depend on $T$ because $T$ consists of them, namely $T=A \cup B$.

I discussed with my teacher and he said that the conditional probability formula is a tautology. If you assume zero probability in some input of the OR, then you have different topology and hence you cannot use the formula like that -- you need reform the problem -- because OR port can be simplified with certain zero-probabilities. 
The key is to differentiate the different layers: logic, probability and real life. You can claim whatever you want in the logical layer before you think about probabilities -- even though you may get undefined in probabilistic layer. The real-world can be then anything between the logic and probability.
Example with interval probability

Suppose $T=A\cup B$. Now if $P(B)=0$, then $T=A$. Now if $P(B)\in [0,0.2]$, then $T=A$ if $P(B)=0$ and $T=A\cup B$ if $P(B)\not = 0$. So interval probabilities become pretty messy if you assume extreme probabilities such as $0$.

