The question is quite broad, so my answer will also probably be broad.
Traditional logic is Boolean, in the sense that every sentence evaluates to either true or false.
However, there are many contexts in which satisfaction is not Boolean. Examples of this include probabilistic reasoning (captured by PSL that you mentioned), fuzzy logic, quantitative reasoning, knowledge, belief, and trust representations, and many more.
In many such contexts, "truth" is defined as e.g., a value in $[0,1]$, and this is sometimes referred to as "soft truth values". The interpretation of such values can differ according to the usage. For example, a sentence with truth value $2/3$ can mean "the sentence is true with probability $2/3$" (i.e. it is true in $2/3$ of the models), or it could mean "the belief level I have in this sentence is $2/3$" (for some well defined concept of belief).
Or it could mean something like $2/3$ of the agents in this multiple agent system think the sentence is true (which is used to represent e.g. voting scenarios).
For some self-propaganda, I have some work on the topic of quantitative reasoning in temporal logic which you can use as a starting reference.