In math and computer science we have a fairly strict meaning of proof. Something is proven if you can build a chain of inferences that start at some given set of axioms and get to the theorem you want to prove. So when someone outside these sciences claims that something has been proven, I'm often reluctant to credit the claim. For instance, in natural sciences (physics, chemistry, etc.) there seems to be a conception that as long as theory fits the observed data, it can be considered valid, but you can not generally prove anything in physics (in this very strict mathematical sense), can you? In medicine, and social sciences is even worse: I often hear that it has been proven that something is good/bad for health, and what they really mean is that correlation is very high, but, is that a proof? I'm not trying to judge anyone, on the contrary, I would like to know what is the expected level of formality/confidence in other sciences, such that scientific claims are considered "proven" or valid by the community.
Logical proof is pretty much the same as in Math or Computer Science, just instead a formal representation in non-formal science natural language is used.
Application area of logic is much broader than verification in mathematical/formal methods sense. Herodot, Socrates, Platon, Aristotle gave the basic ideas of logic which are used, e.g., in philosophy. The classic example of informal representation of logic: "All men are mortal" and "Socrates is a man" imply together that "Socrates is mortal".
However, sometimes "proven" also means "proven by evidence", "proven with some probability", etc.
You are correct to discredit claims of proof in the inexact sciences. There certainly is a widespread misconception that as long as theory fits the observed data, then the theory is valid. That approach to validity is misguided and unscientific. The engineering approach to developing theories and models is based on a recognition that "all models are wrong" - i.e., all models are inexact. The objective is to assess HOW inexact, in order to make an educated assessment of the appropriateness or suitability of that model for a specific application. The validation process - usually defined as assessing "... the degree of accuracy..." - becomes corrupted when assumptions about validity drive conclusions like "proof."
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