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The article deals with possibilies of the use of some means of the current logics for specifying terms and the concept aparatus used for verbal expressing of methods and processes used in the proofs process. The initial part of the article deals with the usage of elementary means of theoretical semantics to specify the basic terms, especially to substantially narrow their broad interpretation. Above all, attention is paid to narrowing the broad interpretation of the term of proof, the proofs process, and further terms which play an important role in the field of proofs in law. The next problematic area is devoted to the problems of „types of proofs“ and their specification in the current legal literature. A part of it is giving reasons for the fact that in real proofs, we do not encounter „clear“ types of proofs in the field of law which are generally stated in the literature (extensional and intensional types of proofs). The final part of the article deals with the problems of „consequence“ relations both logical and factual. It points to the fact that „logically“ decided statements – like logical consequent relations – predicate about the charecter and quality of our deduction (thinking) and not about the „real fact“. For judging the „merits of the case“, factually decided statements and identified factual consequent relations are determining (while preserving the logical correctness).