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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)