There literally is no evidence for or against the proposition. MEANING: The accuracy of this proposition is simply unknown.That is a strikingly wide range of interpretations. I am guessing that which interpretation is selected depends on the listener's opinion of the reliability and trustworthiness of the person making the statement.
The balance of the evidence which is available is against the proposition. MEANING: Based on what we know, the proposition is wrong.
All the evidence which is available is against the proposition. MEANING: We know the proposition to be wrong.
There is no evidence I can find for the proposition. MEANING: As best as I can tell, the proposition is wrong.
There is no evidence of which I am aware for the proposition. MEANING: I am guessing the proposition is wrong.
There is no evidence which I acknowledge for the proposition. MEANING: It is my belief that the proposition is wrong.
Interestingly, the degree to which there actually is or is not evidence on the proposition is independent of the listener's opinion of the speaker.
If I am a sophmore in high school and my biology teacher says "There is no evidence that living next to power lines causes cancer," I may choose to understand that there is no evidence at all and that the issue is open to debate. Alternatively, I may understand that the teacher is making an assessment of the balance of the evidence and is telling me that it is her opinion that the bulk of the evidence is against a link between power lines and cancer. My interpretation of the statement hinges on my trust and opinion of the teacher and has nothing to do with the actual volume, availability, and quality of evidence to support the statement.
A person who makes the statement, "There is no evidence . . . " may understand themselves to be resolving an open question and might fail to realize that their personal brand leaves it open as to what the listener actually concludes.
It is an interesting twist on how information gets processed within a community.