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The book combines a research report on semantical dependence relations with an elementary introduction to probability semantics. Probability semantics is that branch of probability theories which assigns probability values to formulae of logical languages. The theoretical framework of the first part of the book is a version of standard probability theory, that of the second part a version of one of Popper's probability theories. The method in both cases is strictly axiomatic. Since in such an approach it is the theorems derived from the axioms that tell us what there is to tell, this book consists mostly of theorems and proofs, although, of course, the main theorems obtained are discussed, and their relevance for philosophy of science, especially for Popper's anti-inductivism, is argued. The book may be studied in logic courses as well as in seminars on philosophy of science; it can also serve as a systematic reference book to more than 1000 theorems in probability semantics.