There are many theories pertaining to what precisely is a truth as well as what it is that makes a truth true, but none of that is what will be discussed here. Similarly, whether truths are propositions, facts, statements, or sentences is quite irrelevant to what is of interest here – although what gets discussed here pertains to any expression which is regarded as a truth.
Whatever form a truth takes, it is always an expression of one sort or another. And, in order for any sort of expression to be a truth, it must refer to something, whether explicitly or not (were anyone to posit a truth which could not refer to something, then the matter of interest here would pertain only to expressions which are truths and which refer).
A characteristic of this referencing is that it always claims that whatever is referenced is what can be called a determinate matter, state, condition, or context – which is to say that what is being referenced is described as being set or settled or definite.
Qualifiers such as likely, possibly, and probably – in and of themselves – essentially deny that what is being referenced is a determinate or settled matter.
For example, the statement “Bob wears a hat on Wednesday”, expressly claims a determinateness which is absent from statements such as “Bob might wear a hat on Wednesday” or “It is possible that Bob wears a hat on Wednesday” or “Bob probably wears a hat on Wednesday.”
In order for a matter to be possibly true, the matter being referenced has to be a matter which is not settled; it has to be a matter which is to some extent indeterminate. To say that something is possibly true is also to say that it is possibly false; to say that something is possibly true is to say that there is indeterminateness with regards to how the referenced matter is (to be) settled or determined. 
However, what is incoherent is to claim that a statement about something is both a truth and possibly true. This is because it is incoherent to claim that something is both true (or a truth) and possibly false – which is precisely what would be claimed were something ever to be described as both true and possibly true or as a truth and a possible truth. 
To claim that something is a truth is to claim that it is a determinate matter.
To claim that something is a determinate matter is to claim that alternatives are not compossible.
To claim that something is a determinate matter is concurrently to claim that it is concurrently impossible for alternatives to be true.
To claim that something is true is to deny the truth of alternatives.
It is only in indeterminate conditions that alternative possibilities are compossible, but, so long as there is indeterminateness, none of the alternative possibilities by themselves are true (or truths).
Determinateness is a characteristic necessary for truth.
 Even indeterminate states or conditions can be expressed as truths. For instance, the statement, “It is possible that Bob wears a hat on Wednesday, and it is possible that Bob does not wear a hat on Wednesday” can be determinate – and, therefore, a truth – so long as the conjoined contraries exhaustively account for the possibilities and definitely describe the constituent indeterminateness.
 It is important to note that possibility and contingency as well as possibly and contingently are not identical or interchangeable pairs of terms, despite the fact that many discussions in modal logic or possible worlds terms might make it seem that these words are necessarily identical and interchangeable. As discussed in the text, it is incoherent to claim that something is both true and possibly true; however, there is no such incoherence when something is claimed to be both true and contingent or contingently true.