Truthvalue: 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 truthvalues are
T and F or 1 and 0 for respectively true and
false or, sometimes, untrue. Also, it
should be noted that the truthvalues 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 truthvalue T or the truthvalue 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 nonstandard logic, such as
manyvalued logic, there may be more than two truthvalues, from 3 up to
infinitary many, in some systems.
The main reason to introduce a third
truthvalue 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 seabattle", of which the
definite truth or falsity will depend on what the world is like
tomorrow.
The main problems with more than two
truthvalues 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
truthvalues; and that it seems as if most people have fairly definite
and reliable intuitions about logic based on the standard two
truthvalues, but not in case there are more than two truthvalues.
Besides, at least part of the intended uses of logical systems with
more than two truthvalues 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 seabattle)
< 1, which is to say that the probability that tomorrow there is a
seabattle is neither certainly false nor certainly true.
Finally, there is an alternative way of
simulating three truthvalues 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 truthvalues 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
truthvalue semantics for such formulas: v(?q)=T iff 0<p(q)<1 and
v(?q)=F iff p(q)=0 V p(q)=1.
