It's because of the computer telling him, once directly in a pre-film script and by implication in the film itself.
Impossible! Question: Maybe they weren't all Runners. Maybe most of
them reached Life Renewal on Carousel.
The computer informs him that said possibility is negative. Logan then finds that hard to believe.
Question: None of them reached renewal?
Again, the computer asserts the number is zero.
But everyone believes that some....
Logan at this point is struggling with the idea that of that many people, surely some of them should have reached Renewal based on the belief people have that while most people don't make it to Renewal, surely some do. He's still talking about the 1,056 missing people, but then the computer chimes in:
UNACCOUNTED CITIZENS NUMBER 1056....
LOGAN-5 ASSIGNED TO LOCATE SANCTUARY AND REPORT....
Logan was only talking about the unaccounted citizens. The computer misinterpreted the statement as him talking about all citizens, and thus told him the number that have been Renewed, namely none. Logan immediately picks up on this and seeks clarification.
Question: You mean nobody's ever been renewed?
And then the computer answers:
The film as released had a different version of the conversation (which I just transcribed from watching it a few minutes ago):
THE OBJECT ANKH HAS BEEN IDENTIFIED WITH THE CODEWORD SANCTUARY. THE OBJECT AND THE WORD BOTH RELATE TO RUNNERS WHO HAVE NOT BEEN ACCOUNTED FOR.
HOLD. UNACCOUNTED RUNNERS ONE ZERO FIVE SIX. YOU MAY STATE YOUR QUESTION.
One thousand and fifty-six unaccounted for?
THE NUMBER IS CORRECT.
That's impossible. Question: maybe they weren't all runners? Maybe most of them reached Life Renewal on Carousel.
(computer stays silent)
Question: Nobody reached Renewal?
(computer stays silent)
But...everybody believes that some--
THE QUESTION HAS BEEN ANSWERED LOGAN-5.
You mean...nobody's ever been renewed?
THE QUESTION HAS BEEN ANSWERED.
(Logan realizes what the computer is implying).
In the case of the film, again, he's having difficulty with the idea that none of the 1,056 could have been Renewed, but the computer won't say anything. When he presses the computer with "Nobody reached Renewal?" he's still talking about the missing people, but the computer refuses to respond. In the next statement "But everyone believes that some--" he's still talking about the missing people, but the computer cuts him off and tells him the question has been answered.
It's at that point that Logan realizes the computer is intentionally avoiding the issue of Renewal, and that's what prompts the final question, which could still refer to the missing 1,056 or refer to the population in general. But the computer again refuses to actually answer the question or seek clarification on who Logan is talking about, and sounds peeved. It's then that Logan comes to the conclusion as to why the computer refuses to confirm or deny the possibility of some of the missing being Renewed.
(This is something that doesn't necessarily come across in the script as Michael York's expression is one of dawning comprehension and a bit of horror, making it clear that Logan-5 has realized no one has ever been renewed.)
So in both cases it's not a matter of the escaped Runners proving no one has been Renewed, it's the computer operating the place that informs Logan no one has ever been Renewed in a conversation regarding said Runners in the script, and in the film the computer actively refusing to talk about people who have been Renewed and Logan realizing what that implies.