A mathematical proof is a finite sequence of steps, each following by valid deduction from axioms or earlier results. If the proof is correct, the conclusion is certain relative to the axioms: no new data can overturn it. Deductive validity is binary. An argument is either valid (the conclusion must be true if the premises are) or not. Probability theory, by contrast, deals with degrees: evidence can make a hypothesis more or less probable without settling it.
Policy debates almost never offer deductive proof. Claims about the effects of a policy are empirical and uncertain; claims about what ought to be done also depend on values. The relevant question is therefore the burden of proof: what must a side show, and to what standard, for the judge to accept its claim? In many formats the Proposition bears the burden of showing the motion is, on balance, desirable; each side bears the burden of supporting the claims it introduces. Legal systems use explicit standards, such as “balance of probabilities” in civil cases and “beyond reasonable doubt” in criminal cases in common-law jurisdictions, and they reflect a judgment about the relative costs of different errors.
In the televised-trials motion, the Opposition might claim that cameras make witnesses less willing to testify. This is an empirical claim; it cannot be proved deductively. Evidence (surveys of witnesses, comparisons of courts with and without cameras) can make it more or less likely. Whether it is likely enough depends on the standard: a judge might accept a plausible mechanism with weak evidence as enough to create doubt about the Proposition's case, while a stronger claim (“cameras will cause wrongful convictions”) requires more support. In the interactive, move the strength of evidence and see which standards are met.
Empirical claim: “Cameras make some witnesses less willing to testify.”
Deductive claim: “The sum of two even numbers is even.”
A proof does not come in degrees: either a valid proof exists or the claim is unproven. Once proved, no new data can overturn it.
Numerical thresholds for legal standards (for example “51%” or “90%”) are teaching glosses; courts generally do not define these standards numerically, and debate judging is not literally probabilistic. Treating a debate as a proof also misleads in the other direction: a case can be valuable and persuasive without being certain. The mathematical lesson is to be clear about what kind of support a claim has, not to demand certainty.