
Degrees of Truth, Degrees of
Falsity
Toby Ord
In this paper I recall the reasons in favour of extending the
classical conception of truth to include degrees of truth as well
as truth value gaps and gluts, then provide a sketch of a new system
of logic that provides all of these simultaneously. Despite its power,
the resulting system is quite simple, combining degrees of truth and
degrees of falsity to provide a very flexible and elegant conception
of truth value.
One way to extend classical logic is to add new truth values. In classical
logic, we have but two: True and False. Sentences
are thus either absolutely true or absolutely false.
Fuzzy Logic
People have argued that this dichotomy is not warranted. For example,
consider sentences of the from, 'x is bald'. For some values
of x (i.e. for some people), this sentence is clearly true.
For others it is clearly false. However, there doesn't seem to be
any precise line between being bald and not being bald. For some people
it does not seem to be completely true to say that they are bald or
to say that they are not bald. Depending on one's conception of baldness,
this troublesome band might be thick or it might be quite thin, but
it certainly seems to exist.
One way to understand this situation is to claim that there are degrees
of truth. These are typically represented by the real numbers
in the interval [0,1]. The extreme points (0 and 1) represent absolute
falsity and absolute truth, while the values in between
represent intermediate truth degrees. We can then say that
the truth value of 'Michael is bald' is 1 (it is absolutely true),
the truth value of 'John is bald' is 0 (it is absolutely false) and
the truth value of 'Paul is bald' is 0.3. Or perhaps it is 0.4. We
do not need to be able to say precisely, but merely to insist that
there is some answer.
When a system of logic uses degrees of truth it is called fuzzy
logic. Such systems typically define the logical operations as
follows:
¬P

= 1  P 
P ∨
Q 
= max(P, Q) 
P ∧
Q 
= min(P, Q) 
Such systems thus allow us to assess sentences involving one or more
vague properties, such as warm, old, strong,
fit, happy, bright and so forth. This is
potentially of considerable philosophical use in tackling vagueness
and the related sorites paradox (although it is by no means a panacea).
Fuzzy logic also has more practical uses, having made a considerable
impact in engineering within the domain of control systems. It is
often convenient to program a series of logical conditions and corresponding
actions into a machine, and fuzzy logic allows the values of the propositions
involved to come directly from the machine's sensors (such as thermometers
or motion detectors). Fuzzy logic can then allow the machine's effectors
(such as the temperature of the machine's heating element) to vary
continuously as its inputs change, avoiding the discrete jumps that
would occur using classical logic. Of course such behaviour could
be directly programmed into the machines, but allowing the behaviour
to be governed by a system of logic has additional benefits such as
ease of design and the potential to prove results about its operation.
Truth Value Gaps
The restrictiveness of the truth values in classical logic is also
challenged on another front. Suppose that you have a radioactive atom
and want to know the truth value of the sentence 'The atom will decay
within the next halflife'. On the most common interpretation of quantum
mechanics, this sentence does not seem to have a determinate truth
value before the experiment is performed: it would appear to be neither
true nor false. This state is sometimes referred to as a truth
value gap. One might even think that this occurs in less exotic
statements about the future such as 'It will rain in Melbourne on
the first day of Autumn in 2077'.
Furthermore, one might think that sentences which suffer reference
failure are neither true nor false. For example, 'The greatest prime
number is not even' or 'Sherlock Holmes is wise'. The same might be
said about sentences that suffer from category mistakes, such as 'The
capital of Ireland is 3', or that are ungrammatical, such as 'Runs
capital violet the.' These sentences make different kinds of mistakes
and the best way to understand some of these might be to say that
the sentences are meaningless, or otherwise flawed, in such a way
as to prevent them from being true or false.
Alternatively, if you subscribe to an understanding of truth which
equates it with verifiability, then you might also run into truth
value gaps. This would occur when there is a sentence P such
that neither P nor ¬P can be verified.
None of these arguments are definitive, but there is at least a case
for introducing a new truth value for sentences that are neither true
nor false. Let us use the symbol 'n' for this truth value
and continue to use '1' and '0' for True and False respectively. We
could then use the following tables to define how the familiar logical
connectives operate on these values:
These tables seem to capture the relationships
involved and thus allow us to form and evaluate complex sentences
built up of statements that may be neither true nor false.
Truth Value Gluts
There is also another, quite different, reason for being attracted
to three valued logics. Consider the familiar troublesome sentence:
'This sentence is false'. If it is true, then by virtue of its meaning,
it must be false. If it is false, then since it claims this very
fact, it must be true. In either case, it would appear to be both
true and false. We might then be tempted to have a new truth value
to represent being both true and false. We could denote it by 'b'
for 'both'.
One objection to this is that it is a mistake to assume that the
liar sentence must be true or false — perhaps it is neither.
This would lead us once again to truth value gaps. Alternatively,
we might think that there is some other form of mistake here which
is not immediately analysable, but which will be solved at some
point in the future. This is quite plausible and thus philosophers
need not be committed to truth value gaps or gluts.
Perhaps surprisingly, the truth tables for three valued logic using
b turn out to be exactly the same as those using n.
However, the difference between the two logical systems comes in
the way that proofs are deemed acceptable. Classically, we think
that the hallmark of logical validity is truth preservation: for
an argument to be valid, it must be impossible for its premises
to be true without the conclusion also being true. There are two
obvious ways to generalise this notion to three valued logics and
these correspond to whether or not we think of the third value as
being true.
Thus, if our third value is n, the appropriate constraint
on truth preservation is that valid arguments cannot start with
premises valued 1 and deliver a conclusion valued 0 or n
— valid arguments cannot take us from something with truth
to something without. On the other hand, if the third value is b
then it possesses truth so, the truth preservation constraint dictates
that valid arguments cannot start with premises valued 1 or b
and deliver a conclusion valued 0.
In the first case, n acts much like 0 and in the second
case, b acts much like 1. We say that in the first case
1 is designated while n and 0 are not. In the
second case 1 and b are both designated while 0 is not.
Being designated corresponds to having the truth that we want to
preserve in logical implication. Thus we can say that valid arguments
can never take us from designated premises to nondesignated conclusions.
Another, particularly important difference stems from the role of
contradictions in these logics. In classical logic, it is well known
that contradictory premises can entail any conclusion — a
law known as ex falso quodlibet. This seems rather bizarre,
but is typically brushed over on the grounds that we should never
adopt contradictory premises in the first place. However, life is
not always that easy, and sometimes contradictory premises sneak
in unawares or are foisted upon us. For example, unbeknownst to
its creators, the first formulation of calculus (which used infinitesimal
quantities) was inconsistent. So too are the national laws that
govern us. Even small sections of such laws typically contain many
inconsistencies. For example, it can be shown in the British Immigration
act that certain people both can and cannot become citizens.
Nonclassical logics can deny the rule of ex falso quodlibet
and those that do so are known as paraconsistent logics.
An important example of such a logic is the three valued logic with
b. This allows us to reason formally about such systems
as the original infinitesimal calculus or the British Immigration
act without being reduced to gibberish. Contradictory conclusions
can be drawn, but they are limited to the the area in which the
contradictions arose and the paraconsistent reasoning can therefore
be more robust. Thus, even if one does not believe that statements
can be both true and false, it may still be tempting to use a logic
involving b for pragmatic reasons.
Generalised Theories
Given the reasons both for truth value gaps and gluts, one might
wonder why we shouldn't allow both. Indeed there is a system of
four valued logic known as FDE which allows sentences to
be true, false, both or neither. We can then define the standard
logical operations using the following diagram.
The negation of a value is the value found by reflection in the
line between n and b. Thus 1 and 0 swap places,
while n and b are unchanged by negation. The disjunction
of two values is their least upper bound and the conjunction of
two values is their greatest lower bound. Note that the partial
ordering used here ranks values according to increasing truth and
decreasing falsity. Thus disjunction can be said to maximise truth
and minimise falsity, while conjunction minimises truth and maximises
falsity. These definitions can be seen to agree with our intuition
and can also be represented as truth tables:
We can then define logical inference by saying
that 1 and b are designated while n and 0 are
not. Thus valid arguments are those that cannot take us from premises
valued 1 or b to conclusions valued n or 0. This
system has received considerable study and is a very natural generalisation
of the two kinds of three valued logic.
But why stop here? Is there some way to join up the degrees of truth
from fuzzy logic with the gaps and gluts? One way would be to take
the values [0,1] from fuzzy logic and to identify 0.5 as n.
By calculating the truth tables for fuzzy logic where propositions
take the values 0, 0.5 and 1, we can see that the familiar truth
tables for the three valued logic come out. It might thus seem that
we have unified fuzzy logic and truth value gaps.
However, there are some problems with this approach. For one thing,
consider that the truth tables for three valued logic with gluts
are the same as those for three valued logic with gaps. It is just
the rules of inference that change. Thus, the argument above should
also imply that we can interpret 0.5 as b, but this is
clearly leading us into trouble.
The problem is intimately related to the rules of inference for
fuzzy logic, which we have not yet discussed. As mentioned previously,
logical inference should preserve truth, but when truth comes in
degrees, how much should it preserve? A very natural answer is to
attach a real valued parameter, τ, to the notion of implication,
where τ represents the required degree of truth. Thus for an
inference to be valid, it must be impossible for the premises to
all have a degree of truth of at least τ whilst the conclusion
has a degree of truth below τ. In other words, values greater
than or equal to τ are designated.
We can then say that an argument is valid simpliciter (i.e.
without reference to a parameter) when it is valid for every value
of τ. This is the same as saying that the argument can never
lead to a conclusion that is less true than the weakest premiss.
Unfortunately, this conflicts with our understanding of n
and b, because n can never be designated and b
is always designated. Thus the identification of n with
0.5 works only for fixed values of τ greater than 0.5 and the
identification of b with 0.5 works only for fixed values
of τ less than or equal to 0.5.
Whilst this method might therefore have some limited success in
joining one or the other of n and b to the degrees
of truth from fuzzy logic, it is not altogether successful and could
never allow both values simultaneously. Moreover, there would seem
to be good philosophical reasons to think of each of them as distinct
from 0.5, which in fuzzy logic denotes something akin to half
true.
A New System
A more promising approach is to consider degrees of truth alongside
degrees of falsity. Let each truth value be represented by a pair
of real numbers from 0 to 1. We can thus represent truth values
as points on the following diagram:
In a truth value (p, q), p is the degree
of truth and q the degree of falsity. Thus the point at
the top (1,0) represents absolute truth while the point at the bottom
(0,1) represents absolute falsity. The point on the left (0,0) represents
the most extreme absence of truth and falsity, whilst the point
on the right (1,1) represents the most extreme excess of truth and
falsity. Note also the similarity of this diagram and the one used
earlier for four valued logic:
This new space of truth values provides us with a great degree of
flexibility. For example, consider the following regions:

Overdefined (truth value glut). 

Underdefined (truth value gap). 

Welldefined.
These are the values used in fuzzy logic. 

More true than false. 

More false than true. 

Equally true and false. 

More than half true. 

More than half false. 
As can be seen, this set of truth values takes us beyond those present
in fuzzy logic and four valued logic, but the extension is very natural.
The definitions of the logical operations also come very naturally,
combining the methods of fuzzy and four valued logics.

¬(p, q)
= (q, p)
Thus negation reverses the degrees of truth and falsity.
Note that negation is simply a vertical reflection in the lattice. 

(p1,
q1) ∨ (p2, q2) = (max(p1,
p2), min(q1, q2)) (p1,
q1) ∧ (p2, q2) = (min(p1,
p2), max(q1, q2))
Thus disjunction maximises truth and minimises falsity, while
conjunction does the reverse.
Note that disjunction is simply the least upper bound and conjunction
the greatest lower bound. 
Note that when restricted to classical values, these connectives give
the classical results; when restricted to 0, n, b,
1 they give the four valued logic results; and when restricted to
the welldefined values, they give the fuzzy logic results.
As with fuzzy logic, we can designate those values where the degree
of truth is greater than or equal to the value of a parameter, τ.
As with the three and four valued logics, n is never designated
while b is always designated. Other underdefined or overdefined
values may also be designated, depending on their degree of truth.
As per the three and four valued logics, the falsity of the value
is irrelevant when it comes to considering designation — only
the degree of truth matters. We can again define validity simpliciter
as validity for all values of τ.

The designated values.
Setting τ to equal 1 gives a particularly natural logic. 
On reflection, we can see a variety of potentially important properties
of these new truth values, such as definedness, net truth
and proportional truth:

Lines of equal definedness.
where the definedness of (p, q)
= p + q
This takes values from 0 to 2, with 1 representing 'welldefined'. 

Lines of equal net truth.
where the net truth of (p,
q) = p  q
This takes values from 1 to +1. 

Lines of equal proportional truth.
where the proportional truth of (p,
q) = p / q
This takes values from zero to infinity and is undefined when
the truth value is (0, 0). 
Note that there are also subsets of this space of truth values which
are of considerable independent interest, and for which the rules
above can be easily adapted. For example, there is the triangle formed
by removing the overdefined values. This leads to a natural unification
of the truth value gap based three valued logics with fuzzy logic.
There is also the corresponding triangle formed by removing the underdefined
values which leads to a unification of fuzzy logic with the truth
value glut based three valued logics.
One could also consider the union of the welldefined values (those
of fuzzy logic) with the points (0,0) and (1,1). This would give a
minimal unification of four valued logic with fuzzy logic. In this
case the conjunction and disjunction rules would have to be modified
so to take account of the missing intermediate values. It is not obvious
how best to do this.
Just as there are versions of fuzzy logic which have finitely many
truth values between 0 and 1, so we could do this in the present system.
One merely needs to posit a matching degree of falsity for every degree
of truth. Similarly, we could restrict degrees of truth and falsity
to rational values. Obviously these modifications could be combined
with the ones above.
Further Work
This paper is just a bare sketch of a theory of degrees of truth and
degrees of falsity, and there is much more work that could be done
on the topic.
For one thing, it is worth noting that I have brushed over the use
of conditionals in logics with more than two truth values. There are
two standard ways to construct the conditional in three valued logics
and at least two ways for fuzzy logic. I have not spent much time
considering the natural ways to do so in the unified system as I have
little intuition on the matter. Presumably there are at least two
natural ways to do this, which would each form their own logic. Developing
these would no doubt be an interesting project.
Similarly, I have not constructed a proof system and thus have no
soundness or completeness results. I don't imagine that this would
be too difficult and encourage others to do so.
Finally, and perhaps most importantly, I have not provided any example
domains where such truth values would naturally arise. I feel that
there are probably examples both philosophical and practical, but
have no 'killer applications'. On the practical side, there may be
applications where rules conflict (tempting us to move to a paraconsistent
logic to avoid the calamitous ex falso quodlibet) and where
these rules involve vague properties. I can envisage such fuzzy conflicts
occurring both in legal systems and in engineering control situations.
On the philosophical side, there may well be paradoxes which combine
vagueness and self reference, where the ability to provide independent
degrees of truth and falsity provides a way out. Whilst this space
of truth values is clearly of independent interest, it could certainly
benefit from some highly motivating examples.
References
Graham Priest, An Introduction to NonClassical Logic, (Cambridge:
CUP), 2001.
Note: — Since the writing of this piece I have become aware
of a book which seems to include a theory of degrees of truth and
falsity that is not unlike my own. I have not been able to examine
this book, but from what I can tell, its truth values only include
those whose definedness is less than or equal to 1. If so, it is a
theory that unifies fuzzy truth with truthvalue gaps, but does not
address truth values which are overdefined. This book is:
Atanassov K., Intuitionistic Fuzzy Sets, (Heidelburg: SpringerVerlag),
1999.

