There are two main sources of the idea that truth has degrees.
One is objective idealism, as explained under coherence theory of truth.
The other arises because many predicates are essentially vague. When does a heap of sand become large? It seems plausible that a 100-grain heap is not large, and that a heap which is not large can never become large by acquiring just one extra grain.
It seems to follow that a small heap which has grains added one at a time can never become large. (This is one version of the paradox known as that of the heap, the sorites, or the bald man; being invented in ancient Greece and only revived quite recently.)
The paradox (like many) is harder to answer than appears at first, and one answer takes the form of letting truth have degrees.
R M Sainsbury, Paradoxes (1988), ch. 2
In classical logic, propositions are typically unambiguously considered as being true or false. For instance, the proposition one is both equal and not equal to itself is regarded as simply false, being contrary to the Law of Noncontradiction; while the proposition one is equal to one is regarded as simply true, by the Law of Identity. However, some mathematicians, computer scientists, and philosophers have been attracted to the idea that a proposition might be more or less true, rather than wholly true or wholly false. Consider My coffee is hot.
In mathematics, this idea can be developed in terms of fuzzy logic. In computer science, it has found application in artificial intelligence. In philosophy, the idea has proved particularly appealing in the case of vagueness. Degrees of truth is an important concept in law.
The term is an older concepts than conditional probability. Instead of determine the objective probability only a subjective assessment is defined. Especially for newbies in the field, the chance for confusion is high. They will confound the concept of probability with degree of truth for sure. To overcome the misconception, it make sense to see probability theory as the preferred paradigm to handle uncertainty.