Truth-value: In logic, what is assigned by a rule of valuation to a statement in order to express how it relates to the facts in the Universe of Discourse.
In standard logic, the truth-values are T and F or 1 and 0 for respectively true and false or, sometimes, untrue. Also, it should be noted that the truth-values assigned to statements in logic are often purely hypothetical, and are assigned, for example, simply to cover all possible cases, and that the notions of true and false are used but not defined in elementary logic: One supposes a statement has the truth-value T or the truth-value F, and calls these by the names true and false, respectively, or similar terms, but does not analyse or define these in elementary logic.
In non-standard logic, such as many-valued logic, there may be more than two truth-values, from 3 up to infinitary many, in some systems.
The main reason to introduce a third truth-value is to cover the cases of statements that are intuitively neither true nor false. One class of examples of such statements, already noted by Aristotle, are contingent statements about the future. Aristotle's example was: "Tomorrow there is a sea-battle", of which the definite truth or falsity will depend on what the world is like tomorrow.
The main problems with more than two truth-values are that it produces more possible cases to analyse; that it turns out that people don't find it easy to agree on the meanings of the standard logical connectives even in case of just three truth-values; and that it seems as if most people have fairly definite and reliable intuitions about logic based on the standard two truth-values, but not in case there are more than two truth-values.
Besides, at least part of the intended uses of logical systems with more than two truth-values can be also served by probability theory, the formulas of which are simply true or not, but which also attribute a probability to propositions that is a number between 0 and 1 inclusive. Thus, the example of Aristotle quoted above can be rendered in probability theory as: 0 < p(Tomorrow there is a sea-battle) < 1, which is to say that the probability that tomorrow there is a sea-battle is neither certainly false nor certainly true.
Finally, there is an alternative way of simulating three truth-values by using the standard two, namely by using prefixes for propositions like +, - and ?, which may be read - for example - respectively as "verified", "falsified" and "undetermined", while still having each statement either true or not, as in standard logic. This is done in EPL, and one good analogy is that one does not need more than two truth-values to deal with e.g. the three alternatives that one is short or long or neither short nor long. And one can use what was said in the previous paragraph to set up a truth-value semantics for such formulas: v(?q)=T iff 0<p(q)<1 and v(?q)=F iff p(q)=0 V p(q)=1.