I agree with Lee Mosher's characterization of the sentence. I'll offer up this suggested rewrite:
No amount of testing can show that a computer program produces the desired output for all input values, unless either every single input value is tested, or the correctness of the program is [theoretically] established.
I appreciate you wanting to translate this into propositional logic, but it doesn't distinguish between truth that is established theoretically vs. experimentally. What the author is saying is that you can't experimentally verify a program without testing all cases.