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Problems and
Issues with Half-
Truths
By Tony Marmo
(Rio de Janeiro, December 18, 2018)
Tony Marmo
Abstract
Rescher'1967'discusses'the'pros'and'cons'of'having'hybrid'truth-values'vis-à-vis'
triviality'and'the'loss'or'gain'of'tautologies'and'antilogies.'We'argue'that'hybrid'
truth-values'are'not'an'option'for'systems'within'the'scope'of'the'hexagon'of'
oppositions,'and'try'to'clarify'certain'issues'of'many-valued'logics,'such'as'the'
basic'distinction'between'half-truths'and'teratologies.'
Problems and Issues with Half-truths
Page 3
Introduction.
Let us begin by overviewing some very general issues that motivate many-
valued logics:
To say that a glass is either full of water or
empty may be inadequate in a given situation,
where the volume of water in it corresponds only
to a half of its capacity. To say that in such cases
it is half-full or half-empty may be a matter of
perspective, or just two different manners of
saying the same thing. But can one claim that the
glass is both empty and full, if it’s neither?
Contrary to common sense, there have been affirmative and
negative answers to this question. Moreover, there may be situations where the
volume of water is below half of the capacity of the glass, and yet one cannot claim
that it’s empty. Or else, it can be the case that it exceeds half of that capacity and
yet the glass is not full. In those cases, the labels ‘half-full’ and ‘half-empty’ turn to
be descriptively inadequate as well.
We use these figures of thought to present the problems of the delimitation
between truth-like and false-like values in logics that are alternative to classical
logic. Though the solutions seem simple, they raise many questions that are not
easy to address. The overall picture is not too bad, on the other hand, for it is
possible to examine whether their consequences are the same or not.
In this work we intend to discuss some of the aspects of these logics that
embrace the possibility of half-truths (or half-lies), showing that the very definition
and the interpretation of the truth-values is constrained by concerns with triviality.
Tony Marmo
II. The Contrast between Clear and Unclear
Statements
We begin by analysing examples from natural language. Let us divide
statements into two kinds: clear and unclear ones. The difference between these
two is as follows:
Clear statements are such that can be considered either vraisemblable (ie,
verisimilar, accurate, true, plausible, probable, likely, etc.) or invraisemblable (ie,
inaccurate, false, unlikely, improbable, implausible, etc.). One feature of clear
statements is that negation can alter its truth or falsity. For example, the statement
below is clear and absolutely false:
(1) Spiders naturally make honey.
Its negation, on the other hand, is absolutely true and clear. This is not the
case with unclear statements, like:
(2) Johnathan’s love for David was more wonderful than that of women.
One cannot attest whether such statement is absolutely true or absolutely
false, and negation cannot change that. In fact, one feature of unclear statements
is that natural language negation cannot make them clear.
Many-valued logics developed from this type of concerns in the realms of
Mathematics, Philosophy and Science in General. They are part of approach that
recognises the insufficiency of postulating only two truth-values, namely, true or
false.
Indeed, many-valued logics embrace the possibility that some statements
are half-truths or half-lies, or that they may be partially or mostly true, or partially or
mostly false, etc. That is, they accept several degrees of evaluation of statements.
But they are not so ‘liberal’ as one might wrongly assume.
One key hypothesis behind these logics is that there is and there can be no
statement that is true and false at the same time. To wit, though a statement may
Problems and Issues with Half-truths
Page 5
be neither absolutely true, nor absolutely false, it will never be both. And this line
of argumentation is what we herein call protocanonical many-valued logics. Its
implementation can be explained by the aid of an hexagon of oppositions, as we
show in the next Section.
It is worthy to mention en passant that we have reasons to seek
deuterocanonical constructions of many-valued logics, as well. There are natural
statements the falsity of which is also resilient to negation. Take the example
below:
(3) The Pope is not a very good Protestant.
This sentence is intuitively deemed false, because it somehow suggests that
the Pope is Protestant. But is affirmative version is also false for the same reasons.
Besides these cases, there has been an aporia about how paradoxes should
be treated, at least since stoics and peripatetics debated issues of logic theory.
There are many possible ways to approach them: some may prefer to treat
paradoxical statements like any other fallacies or contradictions, whilst others
could try to take them as good examples of statements that are at the same time
true and false.
Everyday statements can be or seem paradoxical. Yet, people oftentimes
construe them in non-trivial ways. For instance, though it is not easy to analyse
statements like the following, they are not uncommon and hearers somehow find
ways to process them:
(4) Bob never buys Jane gifts, and when he does, he gives the same
thing to his secretary.
(5) Jane loves and doesn’t love Bob’s films.
Whether these sentences are plainly false or both true and false depends
on the kind of rationale is assumed, as we shall see hereinafter. Why they are not
usually construed in a way that they entail just any conclusion whatsoever, that is a
further topic.
Tony Marmo
III. The Interplay between truth-values and
connectives
A. The Issue of Indefinite Equivalences
We shall now deal with more specific issues:
Let us assume that statements can assume three different truth-values: true,
indefinite and false. Simple statements will be considered true, false or indefinite
depending on their assessment. But complex statements will also depend on how
the connectives involved are defined. Let us analyse this other natural language
statement:
(6) Economy is as deceptive as pornography.
One can construe it as an equivalence statement. Under the more
traditional two-valued doctrines, two statements are equivalent when they assume
the same truth-value. In this case, we may paraphrase the foregoing example like
this:
• Economy is deceptive if, and only if, pornography is deceptive.
Next, we ‘decompose’ our example into two simpler statements: ‘economy
is deceptive’ and ‘pornography is deceptive’. If both statements are false, then the
equivalence is true. If they are both true, the equivalence is also true. On the other
hand, if one of them is false and another true, then the equivalence is false.
Now, we have to ask the following: in which case the foregoing equivalence
can be considered an indefinite equivalence? If our three-valued approach
maintains the meaning of equivalences in two-valued logic, there will be no
circumstance in which the equivalence will be indefinite, for if both simpler
statements are indefinite, the equivalence will be true, but if one of them is
indefinite and the other is either true or false, the equivalence will be false.
We may represent this answer with the aid of a truth-table for an
equivalence connective of some propositional three-valued logic:
Problems and Issues with Half-truths
Page 7
Tony Marmo
Table 1
𝑎↔𝑏
𝑏
𝑎
T
I
F
T
T
F
F
I
F
T
F
F
F
F
T
A different approach on the same issue will provide cases in which an
equivalence can assume an indefinite truth-value if exactly one of its simpler
statements is indefinite. This alternative formulation of the equivalence of
statements may be represented with the aid of another truth-table for another
equivalence operator for some propositional logic:
Table 2
𝑎⇔𝑏
𝑏
𝑎
T
I
F
T
T
I
F
I
I
T
I
F
F
I
T
And many other equivalence operators may be defined for other
propositional logics.
With this very elementary remarks we aim to illustrate the idea that the
definitions of the logical connectives plays a crucial role in determining the truth-
values complex statements can assume.
Problems and Issues with Half-truths
Page 9
Let us recall some systems already proposed in the literature. First, we
construct a table for negation:
Table 3.
A
¬a
T
F
I
I
F
T
The three-valued system of Łukasiewicz, Ł3 is given by tables 2 and 3 above
and table 4 below:
Table 4
𝑎∧𝑏
𝑎∨𝑏
𝑎⇒𝑏
𝑏
𝑎
T
I
F
T
I
F
T
I
F
T
T
I
F
T
T
T
T
I
F
I
I
I
F
T
I
I
T
T
I
F
F
F
F
T
I
F
T
T
T
By modifying the tables above, we may get different systems. Accordingly,
another well-known system, due to Kleene, K3 consists of the following: negation
is given by table 3 and conjunction and disjunction is like the ones described in
table 4, but implication and equivalence are given in table 5 below:
Tony Marmo
We may also denote the same truth-values in the tables above by numbers,
like 1, ½ and 0, as usual.
These ideas also extend to logics having four, five or even infinite truth-
values, but we shall in this Section try to limit most of our observations to the
three-valued cases, for the sake of simplicity.
B. Tautologies and Antilogies
One interesting feature of these approaches is that some formulae
invariantly assume a certain truth-value v, that is, they are always true or always
false, regardless of the values its components may assume. In two-valued logics,
these formulae correspond to tautologies and antilogies (or contradictions). But it
is debatable whether tautologies should always be identified with truth-invariance.
The construction of three-valued logics is a way to argue against the alleged
tautological nature of certain formulae. This is oftentimes due to the fact that one
or more components of any formula ƒ may assume the value I, and when it does, ƒ
as a whole fails to assume the value T. We may then claim that a two-valued
tautology may not be a three-valued one.
The same happens in the case of the antilogical nature of certain formulae:
one or more components of any formula ƒ may assume the value I, in which case ƒ
as a whole will fail to assume the value F. We may henceforth argue that what is
Table 5
𝑎→𝑏
𝑎≡𝑏
𝑏
𝑎
T
I
F
T
I
F
T
T
I
F
T
I
F
I
T
I
I
I
I
I
F
T
T
T
F
I
F
Problems and Issues with Half-truths
Page 11
considered an antilogy from the standpoint of two-valued logic, is not anti-logical
from the standpoint of some three-valued logics.
Let us give some examples:
Table 6
C2
Ł3
K3
𝑎∨¬𝑎
Tautological
Non-tautological and non-antilogical
𝑎∧¬𝑎
Antilogical
Non-tautological and non-antilogical
𝑎⇒𝑎
Tautological
Tautological
Non-tautological and non-
antilogical
𝑎⇒¬𝑎∨¬𝑎⇒𝑎
Tautological
Tautological
Non-tautological and non-
antilogical
¬𝑎⇒¬𝑎∨¬¬𝑎⇒𝑎
Tautological
Non-tautological and non-antilogical
Above, C2 is the classical two-valued propositional logic.
Hence, if we accept the idea that a tautology is a formula that is invariantly
true, then K3 has no tautologies. In the same fashion, if we consider that an
antilogy is a formula that is invariantly false, then K3 has no antilogies either. And
these two features are important differences vis-à-vis Ł3 and C2. On the other
hand, though Ł3 and C2 are different systems, they have important similarities,
since the former still shares some tautologies with the latter.
Consequently, one may always claim that certain propositions are not
tautological, or not antilogical, by evoking the rationale of Ł3 or K3.
This sort of argument has counter-arguments, which we shall see in the next
Section. Basically, accordingly to another interpretation, the foregoing results
indicate that our conceptions of tautology and antilogy should be broadened, or
in a certain manner revised.
Tony Marmo
IV. Designation and Related Issues in Protocanonical
Logics
A. Value-designation within the scope of the hexagon of
oppositions
It is not difficult to imagine logics that have more than three truth-values.
Natural languages can provide a rich vocabulary for naming such values and we
can denote them by numbers: completely accurate (1), very accurate (¾), indefinite
(½), very inaccurate (¼) and completely inaccurate (0). We may even dispense with
the names and adopt just numbers, if we think of system with a large or infinite
number of values.
However, the task is much more complex than it looks. One cannot simply
postulate a number of truth-values, without specifying how these values are
related to each other. Fundamentally, there ought to be a scheme to organise and
classify them. In order to do so, one must look back at the notions underlying the
formulation of the values. Here, we shall do so by resorting to the old hexagon of
oppositions.
Regardless of what definition of truth is available, if there is any, one
important issue is its relation to the notion of falsity. Intuitively they are in
opposition, but it is necessary to explain in which sense they oppose each other.
Two-valued logic is based on the impression that truth and falsity are contradictory
notions, to wit, one of them has to be the case with regard to any statement. If a
statement is not true, then it is false and if it is not false, it is true. But this view does
not specify what should be the contrary notion of truth, nor does it tell us anything
about what should be the subcontrary of falsity.
On the other hand, when we broaden the notions of truth and falsity, and
think in terms of verisimilitude (or likelihood) and of unlikelihood, we may also
consider a hypothesis according to which these notions are contrary. The truth-
values belonging to the domain of verisimilitude are the so-called designated
values, whilst those falling in the domain of unlikelihood are the antidesignated.
As the notions underlying these values are contrary, no value can be both
designated and antidesignated, but may be neither one of them.
Problems and Issues with Half-truths
Page 13
Let us illustrate it with the aid of the figure hereinafter:
Non-designated truth-values do not necessarily coincide with the
antidesignated one. And the same truth-value may be both non-designated and
non-antidesignated, as the underlying notions of non-unlikelihood and non-
verissimilitude are subcontrary.
The classification of truth-values into designated and anti-designated is a
matter of interpretation. Whether it is a matter of arbitrary choices or not is an
issue open to debate, but it is not inconsequential, for the reasons hereunder.
Now, it is possible also to change our views on tautologies and antilogies. A
tautology is a formula that always assumes a designated value, whilst an antilogy is
a formula that always assumes an anti-designated value. Outside of these two
cases, formulae are considered contingencies.
We may indicate which values are considered designated by the sign +,
and the antidesignated ones by –. Let us assume that the values T and I are
designated for K3. Hence, K3 will now gain some tautologies. For instance,
consider the formula:
(7) 𝑎⇒¬𝑎∨¬𝑎⇒𝑎
It is clear by tables 3 and 5 that the formula in (7) always assumes either a T
or an I value, and never F. Inasmuch as we now hold T and I as designated values,
(7) can now be deemed a tautology of K3.
Next, let us examine this other formula:
Tony Marmo
(8) ¬𝑎⇒𝑎
By the same tables 3 and 5, the formula in (8) assumes either an F or an I
value, and because of that it cannot be considered an antilogy of K3, since I is a
designated value. We may verify this same fact with regard with to other formulae
that are antilogies in C2, but not in K3.
Briefly, if we designate both T and I, then K3 will have some tautologies, but
will lack antilogies. However, if we change this radically, by considering both I and
F antidesignated values, K3 will now gain antilogies and loose its tautologies.
How could we interpret the truth-values of K3 in a way that it could have
both tautologies and antilogies? One solution would take the value I to be both
designated and anti-designated (in other words, a hybrid value). But this
hypothesis is out of the scope of the hexagon of oppositions formulated
hereinbefore.
Let us entertain the following hypothetical cases:
A. A logic system S has no nondesignated truth-values;
B. A logic system S has only nondesignated truth-values;
C. A system S has only antidesignated truth-values.
A system like that in hypothesis A is said to be trivial (or absolutely
inconsistent), for its formulae are all tautologies. A system like that in hypothesis C
is said to be void, for its formulae are all antilogies. Both hypotheses are not of
great interest for logicians in general.
A system like that in hypothesis B, on the other hand, lacks truth and will not
necessarily have antilogies, but is more interesting. If the system also lacks
antidesignated values, then it will be said contingent.
In sum, what these considerations indicate is that the consistency of a logic
system is a construction that departs from the choice and classification of its truth-
values.
Problems and Issues with Half-truths
Page 15
B. Matters of consistency
A logical system S is said to be negation-consistent, to wit, consistent with
regard to its negation operation, if for every formula ƒ, it is not the case that ƒ and
¬ƒ are both tautologies of S. On the other hand, we say that S is absolutely
consistent if there is one formula ƒ such that ƒ is not a tautology of S.
One key idea is that negation-consistency and absolute consistency
coincide under certain conditions. For instance, if a system S has the following
rule:
(9) Modus ponens. Where 𝑝→𝑞∈𝑇𝑆 and 𝑝∈𝑇𝑆, then 𝑞∈𝑇𝑆,
𝑇𝑆 being the set of tautologies of S.
And if it additionally has the formula 𝛼→¬𝛼→𝛽 among its tautologies,
then S is absolutely consistent if, and only if, it is negation-consistent. In other
cases, where these conditions do not obtain, nor anything similar to them,
absolute consistency and negation consistency are separate issues.
Let us consider the system 𝑆!
∗, which is given by tables 3 above and 7 and 8
below:
The system 𝑆!
∗ is negation-inconsistent, since clearly ⊺𝑎∈𝑇𝑆!
∗ and
¬⊺𝑎∈𝑇𝑆!
∗. Notice that the formula 𝛼→¬𝛼→𝛽 is not among its tautologies,
for if
α
assumes the value I, so ¬
α
, and if
β
assumes F, then ¬𝛼→𝛽 assumes F, and
Table 7
Table 8
𝑎∧𝑏
𝑎→𝑏
𝑎
⊺𝑎
𝑏
𝑎
T
I
F
T
I
F
+T
I
+T
T
I
F
T
I
F
+I
I
+I
I
I
F
T
T
F
–F
I
–F
F
F
F
T
T
T
Tony Marmo
so 𝛼→¬𝛼→𝛽. Thus, the system is absolutely consistent, for there is a formula
that is not among its tautologies.
IV. Deuterocanonical Approaches
The impossibility of having hybrid values is a characteristic of what we call
protocanonical many-valued logics. Such characteristic is related to the idea that
the same statement or formula of a given language cannot at the same time be
tautological and antilogical (ie, contradictory) within the same system.
The idea that a certain formula could be a tautology and an antilogy within
the same logic is a hypothesis that we shall herein baptise ‘teratology’.
Nuel Belnap has proposed a well known four-valued system containing the
following truth-values: T for true, B for both true and false, N for neither true nor
false and F for false. This system can be understood in terms of product logics,
such that T corresponds to the pair (1, 1), B to (1, 0), N to (0,1) and F to (0, 0).
In his system:
1. The negation ~𝑎 is defined as 1−𝑎!,1−𝑎!, and
2. The conjunction 𝑎&𝑏 is defined as min 𝑎!𝑏!,min 𝑎!𝑏!;
3. The disjunction 𝑎∨𝑏 is defined as max 𝑎!𝑏!,max 𝑎!𝑏!.
We can check these rules by simply looking into two tables he proposes:
Table 9
Table
10
𝑎&𝑏
𝑎∨𝑏
𝑎
~𝑎
𝑏
𝑎
(1,0)
(1,1)
(0,0)
(0,1)
(1,0)
(1,1)
(0,0)
(0,1)
±(1,0)
(0,1)
±(1,0)
(1,0)
(1,0)
(0,0)
(0,0)
(1,0)
(1,1)
(1,0)
(1,1)
+(1,1)
(0,0)
+(1,1)
(1,0)
(1,1)
(0,0)
(0,1)
(1,1)
(1,1)
(1,1)
(1,1)
Problems and Issues with Half-truths
Page 17
These aspects are unexceptional. What is more interesting about it is that
Belnap proposes a scheme for these four values that is apparently out of the scope
of the hexagon. In doing so, he seems to suggest a system in which there is or
could be at least one value that is both designated and antidesignated, namely B.
Nevertheless, it is not evident that Belnap’s four valued logics were
designed in a manner to allow that (at least some) teratologies. His desire was to
engender non-trivial reasonings from inconsistent data. But there is nothing in his
system that suggests that it will provide us such odd results.
Indeed, his system is very akin to classical logic. For a formula to be a
teratology it would necessary that it always assumed a value v, such that v was at
the same time designated and antidesignated. Suppose B was such value in
Belnap’s system: then any formula that invariantly assumed the value B would be
teratological. But there is no operation or set of operations or of rules in this
system that makes a formula always assume such value. It would be necessary to
add new operator or new rules to the system in order to obtain such result, to wit,
a radical modification would have to take place.
V. Conclusions
–(0,0)
(1,1)
–(0,0)
(0,0)
(0,0)
(0,0)
(0,0)
(1,0)
(1,1)
(0,0)
(0,1)
(0,1)
(1,0)
(0,1)
(0,0)
(0,1)
(0,0)
(0,1)
(1,1)
(1,1)
(0,1)
(0,1)
Tony Marmo
The foregoing Sections were motivated by a discussion presented in
Rescher 1967, dealing with the apparent loss of tautologies and antilogies from
one system to another. Given the considerations we made hereinbefore, we can
now respond to it:
Rescher appeals first to the redefinition of tautologies and antilogies, by
moving from the original two-valued conception to two broader versions thereof:
! An antilogy always assumes an antidesignated truth value;
! A tautology always assumes a designated truth value.
But this is not sufficient to recover all classical tautologies and antilogies for
some systems. Hence, he cogitates the possibility of hybrid truth-values, namely,
values that are both designated and antidesignated at the same time and for the
same system.
We find this cogitation cumbersome. And the reason is not limited to the
fact that hybrid values are not possible within protocanonical logics. It is just that is
much simpler to think of another more suitable, more direct and even broader
redefinition of the two concepts in cause:
! An antilogy always assumes a non-designated truth-value;
! A tautology always assumes a non-antidesignated truth-value.
Under this solution, tautologies and antilogies may assume the value I in
odd-numbered logics, and we do not need to interpret it as a hybrid value.
But we should not even be concerned with this. For there is no real problem
in the fact that different systems have different tautologies and antilogies: that is
precisely why those systems represent different rationales!
In the development of may-valued logics, there have been some different
intuitions that guided the construction of the systems: systems like K3 or Ł3 have
been designed within the scope of the hexagon of oppositions for the notions of
truth (or similitude) and falsity (or unlikelihood). Their concern was clearly to make
something different from two valued logics (C2), but at the same time preserving
absolute consistency, and this reflected in the choice of values and their
designation.
Problems and Issues with Half-truths
Page 19
Belnap’s system has been conceived as something out of the scope of the
hexagon. Systems that have hybrid values like Belnap’s allow us to think of
teratologies, i.e., formulae that are at the same time both antilogies and
tautologies. So far, we know no system that produces them.
Bibliography
Belnap, N. (1975). A Prosentential theory of truth. Philosophical Studies , 27, 73–125.
Béziau, J.-Y. (2012b). The Power of the Hexagon. Logica Universalis , 6, 1-43.
Blanché, R. (1966). Structures intellectuelles. Essai sur l’organisation systématique des
concepts . Paris: Vrin .
Łukasiewicz, J. (1910). On the Principle of Contradiction in Aristotle. Bulletin international
de l'Académie des sciences de Cracovie , 15-38.Rescher, N. (1969). Many-Valued Logic. UK:
McGraw-Hill.
(Belnap, 1975)
