Soft Truth Values in the PSL model This might sound like a trivial question. But since am starting out with my research in an area that is entirely new to me, I would really appreciate it if someone could kindly elucidate what Soft truth values mean.

I came across this in the context of PSL(Probabilistic Soft Logic).
This paper stated that;


  PSL uses first order logic rules as a template language for graphical models over random variables with soft truth values from the interval [0,1].


Any clarifications in this regard will be much appreciated.
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.