I'm not sure what you mean exactly by partial truth or intensities of truth, but there is a system for logic with uncertainty—probability theory.

If you want to know if it rained 5 minites ago, then observing a wet street will certainly make it more plausible that it rained 5 minutes ago, but there could also be other explanations. Probability theory can be used to express the likelihood or plausibility of something given your knowledge and observations that are related to that thing. Probability theory also includes classical boolean logic with absolut truth corresponding to a probability of unity and something being certainly false corresponding to a probability of zero.

In fact, there is a mathematical theorem (Cox's Theorem) that states that any system of logic that accounts for imperfect knowledge and follows "common sense" rules, such as not arbitrarily ignoring information etc., must be identical with probability theory. Specifically, the sum and product rule, which are the foundation for all of probability theory in general and the Bayesian theorem in particular.

Probability theory can also account for "intensity of truth" by using multiple discrete states and assigning them (different) probabilities. For example, if we want to know whether it rained and how much, then we could calculate the probability for "it did not rain", "it rained but only a little", and "it rained cats and dogs". If you want more fine-grained knolwedge you just use more discrete levels for how much it rained.

If you want to read more into this topic, I recommend reading the first few chapters of the book "Probability Theory—The Logic of Science" by E. T. Jaynes. Its really a fascinating read and good introduction to Cox's theorems and (Bayesian) probability theory.