G H Von Wright - Finnish-Soviet Logic Colloquium __ Truth, Negation, and Contradiction (1986) [10 2307_20116214]

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Truth, Negation, and Contradiction
Author(s): G. H. Von Wright
Source: Synthese, Vol. 66, No. 1, Finnish-Soviet Logic Colloquium (Jan., 1986), pp. 3-14
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G. H. VON WRIGHT 
TRUTH, NEGATION, AND CONTRADICTION 
1. 
The position of negation in "classical" logic is characterized by three 
fundamental laws or principles: the Law of Excluded Middle, the Law 
of (Non)Contradiction, and the Law of Double Negation. There are 
traditionally several ways 
- not necessarily all of them equivalent 
- of 
formulating these laws. One way is as follows: 
"Every proposition is true or false", for the Law of Excluded Middle. 
"No proposition is true and false", for the Law of Contradiction. And, 
"a proposition is true if, and only if, it is false that it is false", for the 
Law of Double Negation. 
In the object language of the classical calculus the laws are some 
times stated as follows: p v ~p, 
~ 
(p & ~p), and p<->-p. Thus, for 
example, the authors of Principia Mathematica say, in so many words,1 
that the first of the three formulas is the Law of Excluded Middle, and 
the second the Law of Contradiction. This, however, is a rather sloppy 
mode of expression. What is meant is that (the propositions expressed 
by) the formulas are logically true for all substitution instances of the 
variable. Thus the first formula, corresponding to the Law of Excluded 
Middle, would say that the disjunction of any proposition with its 
negation is logically or necessarily true 
- and the second, corresponding 
to the Law of Contradiction, that the conjunction of any proposition 
with its negation is logically or necessarily not true. Any given 
proposition and its negation form a pair of jointly exhaustive and 
mutually exclusive alternatives, one could also say. 
2. 
Although all three laws may be said to be "orthodox" and to have a 
"received" status in logic, they have also been contested. 
Aristotle, traditionally credited with the discovery of the first two 
laws, seems to have had difficulties with the Law of Excluded Middle, as 
Synthese 66 (1986) 3-14. 
? 1986 by D. Reidel Publishing Company 
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4 G. H. VON WRIGHT 
witnessed by the famous ninth chapter of De Interpretatione. The same 
difficulties were felt and discussed by logicians in the Middle Ages. 
In modern times, the Law of Excluded Middle has been an object of 
criticism from different quarters. -Lukasiewicz rejected it in the for 
mulation "every proposition is true or false", which he distinguished 
from the formulation which says that the disjunction of any proposition 
with its negation is necessarily true.2 For the first formulation he coined 
the useful name the Principle of Bivalence.3 -Lukasiewicz's criticism was 
a starting point for the development of so-called many-valued or 
polyvalent logics. 
Another consequential criticism of the Law of Excluded Middle 
stems from Brouwer. One target of Brouwer's criticism was the use of 
inverse proof in mathematics. Therefore his criticism also hit the Law of 
Double Negation. By disproving the antithesis of a given thesis one has 
not necessarily proved the thesis. The negation of the negation is not 
identical with the base. 
Brouwer's criticism, too, became the starting point of new develop 
ments in formal logic. In intuitionist logic neither pv ~p nor p<->-p 
are valid formulas. This is sometimes expressed by saying that in 
tuitionist negation is different from classical negation. 
3. 
The Law of Contradiction has been more immune than the other two to 
doubt or criticism within the classical tradition. But some peculiarities 
of contradictions have been noted - and startled logicians. 
One such peculiarity is known as Duns Scotus's Law or Principle, 
after the great medieval logician, the doctor subtilis of the Scholastics, 
who seems to have been the first to draw attention to it. It is sometimes 
identified, in the object language of the calculus, with the formula 
~ 
P~~* (P-* 4) or with the, in the classical calculus, equivalent formula 
p&~p->q. The first is sometimes read, rather inappropriately, "a 
false proposition implies any proposition" and the second, not very 
appropriately either, "a contradiction implies just any proposition". 
The more serious interpretation of the formulas is as follows. 
If a proposition and its negation can both be derived in a deductive 
system, then one can, using the above formulas as principles of 
inference or entailment, modo ponente derive in this system just any 
proposition (which can be expressed in its language). The appearance 
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TRUTH, NEGATION, AND CONTRADICTION 5 
of contradiction in a deductive system is a "catastrophe" since it 
"explodes" and therefore "trivializes" the system. These observations 
are the basis for the well-known definition of a consistent logico 
deductive system or calculus as one in which not just any proposition is 
derivable. 
Scotus's principle, however, has also worried logicians. The idea that 
a contradiction "entails" just any proposition may appear counterin 
tuitive. Entailment or logical consequence seems to presuppose some 
kind of "community of content" between the entailing and the entailed 
propositions. A motive force behind so-called relevance logic is a desire 
to circumvent the counterintuitive consequences of Scotus's law. 
If Scotus's principle is not accepted as a logical law of entailment, the 
appearance of contradictions in a system does not necessarily 
"explode" the system and need therefore not be considered a catas 
trophe. This attitude to contradictions underlies yet another recent 
development in formal logic known as paraconsistent logic. With it the 
study of logic in what may still be termed with some justification "the 
classical tradition" has arrived in the neighbourhood of another, 
eminently "nonclassical" tradition, viz., Hegelian or Dialectical Logic.4 
In Dialectical Logic there is an operation called Dialectical Syn 
thesis. It leads to something called the Unity of Opposites (coincidentia 
oppositorum). It can be described, in outline, as follows: 
A thesis is put forward, call it 0. It has an antithesis which is its 
negation, 
~ 6. It is then shown, one way or another, that the thesis is not 
true. Thus we have ?To, where the symbol "T" stands for the phrase 
"it is true that". It is also shown that the antithesis is not true, 
~ T ~ 0. 
Thus neither the thesis nor the antithesis is true. From this is concluded 
that both the thesis and the antithesis are true, T6&T 
~ 0. This is 
called DialecticalSynthesis. To illustrate, the arrow in Zeno's antinomy 
is neither moving nor at rest at a given point of its trajectory. Therefore 
it is both moving and at rest. 
4. 
I now set myself the following task. First I shall, with a minimum of 
departure from classical patterns in logic, try to do justice to some of 
the criticisms levelled against the Laws of Excluded Middle and of 
Double Negation. I think this can be done without indulging in what I 
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6 G. H. VON WRIGHT 
would regard as the "extravagances" of many-valued and intuitionist 
logic. 
Having done this, I shall discuss some questions relating to con 
tradictions and in particular the inferential move known as Dialectical 
Synthesis leading to the Unity of Opposites. This again I think can be 
done without resort to relevance or paraconsistent logic. Some criti 
cisms which I have against these unorthodoxies in modern logic I shall 
here suppress. 
For my purposes I shall construct a calculus or system which I 
propose to call Truth-Logic or the Logic of Truth, for short T-Logic or 
simply TL. The only "unorthodox" feature of TL is that it introduces 
the notion of truth in the object-language of the calculus. This happens 
in the form of a new symbol "T" (above p. 5). It is somewhat like a 
modal operator in that it is prefixed to schematic representations of 
sentences yielding new sentential sch?mas. With the aid of this symbol 
we can mark a distinction between two ways of negating a sentence: an 
external negation ~T to be read "it is not true that" and an internal 
negation T~ to be read "it is true that not". The internal negation 
signifies falsehood. It seems natural to say that the falsehood of a 
proposition means the truth of its negation (contradictory). 
In TL we thus distinguish between falsehood and not-truth. The 
second is the weaker notion. A false proposition is not true 
- but not any 
proposition which is not true is thereby false. The calculus caters, in 
other words, for the possibility that some propositions which are not 
true are not false either. They, as we say, lack truth-value, are neither 
true nor false. To cater for this possibility may be said to be the very 
"point" of our Truth-Logic. 
5. 
But are not all propositions, by definition, true or false? This has been 
said. The answer depends upon how one conceives of the concept of a 
proposition. Here I shall take the following view: 
The basic notion shall be that of a grammatically well-formed 
sentence. This I shall not try to define or otherwise explicate. A 
grammatically well-formed sentence expresses a proposition iff the 
sentence which we get by prefixing to it the phrase "it is true that" is 
also well-formed. For example, "it is raining" is well-formed. "It is true 
that it is raining" is also well-formed. Therefore "it is raining" expresses 
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TRUTH, NEGATION, AND CONTRADICTION 7 
a proposition. "Open the window" is well-formed. "It is true that open 
the window" is not well-formed. Therefore "open the window" does 
not express a proposition. 
Accepting this it is easy to find candidates for propositions which may 
be regarded as neither true nor false. For example, "prime numbers are 
green". It is well-formed, and so is "it is true that prime numbers are 
green". Hence it expresses a proposition. But is it true? Certainly not. Is 
it false? One may wish to say that it is not that either. Subject and 
predicate simply do not match. Colours do not apply to numbers. 
Therefore such predications are neither true nor false. 
Other, perhaps more serious, candidates for being propositions 
without truth-value are easily forthcoming. It is sometimes held that 
names and definite descriptions without a bearer 
- for example 
"Pegasus" or "The King of France" 
- do not yield true or false 
propositions when they occur as subject-terms in predications. Be this 
as it may, it is a possibility worth taking into account and therefore also 
worth accommodating in a system of logic. Other candidates are many 
"metaphysical" propositions, such as that ultimate reality is matter or, 
to take an example from the other extreme, that to be is to be perceived. 
It would be hard to deny propositional status to such claims; but it 
would not be unreasonable to think of them as "beyond truth and 
falsehood". 
What about definitions and norms (prescriptions)? "The length of the 
standard metre in Paris is 1 m."; "You must not park your car in the 
middle of the road". A common and, I think, correct view is that 
stipulations and prescriptions lack truth-value. But they are very often 
expressed in sentences to which it is correct to prefix the phrase "it is 
true that". 
6. 
I shall now describe, in the barest outlines, the calculus I call Truth 
Logic (TL). The vocabulary of TL is that of Propositional Logic + the 
operator T. The well-formed formulas of TL are defined as follows. 
A variable, or a truth-functional compound of variables, preceded by 
the letter T is an atomic T-sentence. An atomic T-sentence, or a 
truth-functional compound of atomic T-sentences and/or variables 
preceded by the letter T, is an atomic T-sentence (of higher order). A 
truth-functional compound of atomic T-sentences is a molecular T 
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8 G. H. VON WRIGHT 
sentence. Atomic and molecular T-sentences are the T-sentences. 
Examples: T 
~ 
p, T(p+* Tp), and Tp-> T 
~ T ~ p are T-sentences. 
Tp-> p is noi a (well-formed) T-sentence. 
The axioms of TL are: 
(AO) All tautologies of PL when T-sentences are substituted for 
the variables. 
(Al) Tp->~T 
~ 
p. "If a proposition is true then it is not false." 
(A2) Tp->T-p. "A proposition is true if, and only if, its 
negation is false." 
(A3) T(p&q)<r* Tp& Tq. "A conjunction is true if, and only if, 
all its conjuncts are true." 
(A4) T~(p&q)<r*T~pv T 
~ 
q. "A conjunction is false if, and 
only if, at least one of the conjuncts is false." 
(A5) T 
~ 
Tp+>~ Tp. "It is false that a proposition is true if, and 
only if, this proposition is not true." 
The last axiom calls for a comment. (A5) means, in effect, that 
whereas propositions are generally "three-valued" in the sense that 
they may be true or false or neither true nor false, propositions of the 
special form "it is true that" are two-valued, i.e., either true or false. 
The motivation for this is easy to see. If a proposition is true, then it is 
true to say that it is true. If it is false, it is false to say that it is true. And 
if it is neither true nor false, then it is also false to say that it is true 
(since it ?5 neither true nor false). 
The rules of inference of TL are: 
(Rl) Substitution of formulas of PL or of TL for the variables. 
(R2) Detachment (modus ponens). 
(R3) A Rule of Truth to the effect that if / is a theorem of TL, 
then Tf is a theorem too. 
The following me ta-theorems will be mentioned without proof: 
(Ml) If two formulas are provably equivalent in TL, they are 
intersubstitutable salva veritate (in formulas of TL). 
(M2) Formulas of TL of higher order, i.e., formulas containing 
occurrences of the symbol T within the scope of another T, 
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TRUTH, NEGATION, AND CONTRADICTION 9 
are provably equivalent with formulas of the first order, i.e., 
formulas inwhich no symbol T occurs within the scope of 
another. (In this regard TL may be said to resemble the 
system of modal logic (S5).) 
(M3) Any formula of TL is, in TL, provably equivalent with a 
truth-functional compound of atomic T-sentences of the 
simple form Tv or T 
~ 
v, where v is a variable. These 
atomic components will be called the truth-constituents of 
the given formula. 
Which truth-function of its truth-constituents a given formula 
expresses can be investigated and decided in a truth-table. In the table 
truth-values ("true" and "false") can be distributed in all possible 
combinations subject to the sole restriction, imposed by axiom (Al), to 
the effect that if a constituent Tv is given the value "true", the 
constituent T ~ v must be given the value "false", and vice versa. To 
one pair of constituents thus answers 3 possible combinations of 
truth-value; to n pairs answer 3n combinations. A formula of TL which 
turns out true for all allowed distributions of truth-value over its 
truth-constituents will be called a truth-tautology or T-tautology. 
(M4) All theorems of TL are truth-tautologies, and all truth 
tautologies are theorems of TL. 
7. 
It is sometimes said that the phrase "it is true that" is otious, redundant 
or superfluous when prefixed to a sentence. Tp+> p. "It is true that p if, 
and only if, p." Sometimes one even calls this an "identity". 
This idea of an equivalence or identity can be considered as an 
implicit argument against having the notion of truth figure in the 
object-language of a logical calculus 
- an argument against construc 
ting a "truth-logic" in fact. 
Let us therefore ask whether the suggested equivalence holds in TL. 
Is T(Tp**p) a T-tautology? After a series of transformations with 
which I shall not burden the present exposition, it may be shown that 
T(Tp++p) is, in TL, provably equivalent with the formula Tpv T 
~ 
p. 
This last formula is not a T-tautology since, as we know, both disjuncts 
may be false, i.e., the proposition that p itself lack truth-value. Hence 
T( Tp <-? p) is not a T-tautology either. 
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10 G. H. VON WRIGHT 
Tp v T 
~ 
p says that the proposition that p either is true or is false. 
This is the form of the Law of Excluded Middle which -Lukasiewicz 
called the Principle of Bivalence. (Above p. 4.) This is not valid in TL. 
What we have shown amounts to this: The idea that the phrase "it is 
true that" is redundant is logically equivalent with the idea that any 
given proposition is either true or false. Thus in a logic which accepts 
(or tacitly assumes) the Principle of Bivalence, the notion of truth is 
redundant (in the object-language). But TL does not accept the 
Principle of Bivalence and therefore does not accept the redundancy of 
truth either. 
8. 
What then of that version of the Law of Excluded Middle which says 
that the disjunction of any proposition with its own negation is a truth of 
logic? Is T(p v ~p) a T-tautology? The answer is No. It can, in fact, 
easily be shown that T(p v ~p) is logically equivalent (in TL) with 
Tpy T 
~ 
p, i.e., with the Principle of Bivalence. 
Here it should be noted, in passing, that no tautology of PL is a 
tautology of TL. This may sound like a paradox: no tautology is 
tautologically true. But what this means is that a sentential schema 
which, as for example p v ~p, has the form of a tautology is tautologi 
cally true only on condition that its atomic components stand for 
sentences which express true or false propositions. This certainly is as it 
should be. If it is neither true nor false that p then it is also neither true 
nor false that p v 
~ 
p. 
Tp\/ 
~ 
Tp, however, ?5 a T-tautology. It says that any proposition is 
either true or is not true. This version of the Law of Excluded Middle 
thus holds in truth-logic 
- and I find it hard to think of a reason why one 
would deny it. (Whereas I find it easy to think of reasons for denying the 
Principle of Bivalence.) 
As for the Law of Double Negation, neither the version Tp?>T 
~ 
T ? p which says that a proposition is true if, and only if, it is false that it 
is false, nor the version T(p*>-p) which says that it is true that any 
proposition is materially equivalent with the negation of its negation, is 
a T-tautology. What is a T-tautology, however, is Tp 
<r> T ~ ~ p or our 
axiom (A2). This too might be said to be a version of the Law of Double 
Negation. 
It should further be noted that Tp-*T 
~ T ~ p holds true in 
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TRUTH, NEGATION, AND CONTRADICTION 11 
truth-logic. If a proposition is true then it is false that it is false. In this 
regard the "intuitions" underlying our truth-logic agree with those 
underlying intuitionist logic, p??-p is intuitionistically valid, but not 
~~p-+p. ("~" here signifying intuitionist negation.) 
The reason why T 
~ T~ p^>Tp does not hold true should be 
obvious: If the proposition that p lacks truth-value then the assertion 
that it is false is itself false. 
Finally we note that 
~ 
T(p & 
~ 
p) is a T-tautology. It says that a 
contradiction is never true. This is a version of the Law of Contradic 
tion. It is equivalent (in TL) with the version which says that no 
proposition is both true and false: ~(Tp&T 
~ 
p). But the stronger 
version of the law which says T 
~ 
(p& 
~ 
p) does not hold in TL. It is 
thus a truth of truth-logic that no contradiction is true, but not a truth of 
truth-logic that every contradiction is false. The strong version of the 
Law of Contradiction is logically equivalent with the Principle of 
Bivalence - as has often been noted in the "classical" systems. 
9. 
Truth-logic opens interesting vistas on the antinomies. I shall define an 
antinomic proposition as follows: 
The proposition that p is antinomic if, and only if, assuming that it is 
either true or false, one can prove that, if it is true it is false, and if false 
true. This means that the proposition that p is antinomic if, and only if, 
one can prove Tpv T~p->Tp&T~p. But the consequent entails, 
by virtue of (Al), ~Tp&~T~p. This again is the negation of the 
antecedent. But an implication the consequent of which is the negation 
of its antecedent is logically equivalent with the consequent alone. Thus 
from the truth of p->~p follows the truth of ~p. This the Schoolmen 
called the consequentia mirabilis. Applying this schema to Tp v T' 
? 
p 
?>~Tp&~T~p one may conclude that, if a proposition is 
antinomic, then it lacks truth-value, is neither true nor false. 
Antinomic propositions do not "violate" the Law of Contradiction. 
Nothing is, as such, "wrong" with them. But it is their peculiarity that 
the assumption, should we happen to make it, that they are either true 
or false leads to self-refuting consequences and therefore cannot be 
true. And from this it follows that antinomic propositions are neither 
true nor false. 
This last is nothing new. Type-Theory, the Vicious Circle Principle, 
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12 G. H. VON WRIGHT 
various restrictions on the form of definitions are so many devices for 
establishing that antinomic propositions lack truth-value and for 
excluding them from discourse with true or false propositions. What is 
perhaps a new point of view is that none of these devices are needed 
(for that purpose). Truth-logic by itself does the job. 
Let the proposition be that this is false. What is false? This, i.e., the 
proposition that this isfalse. Call the sentence saying this "p". Thus we 
have defined "p" 
= df 
" 
T ~ p". 
Now asume Tpy T~p. Substituting "T~p" for "p" in the first 
disjunct we get TT 
~ 
p which is equivalent with T 
? 
p alone. Thus we 
have proved Tp?? T 
~ 
p. Making the same substitution in the second 
disjunct we get T ~T 
~ 
p which, by virtue of (A5), is equivalent with 
~T~p. Thus we have proved Tpy ~T~p which is equivalent with 
T ~ p-> Tp. We have thereby established that the sentence which says 
that the proposition which it expresses is false, itself expresses a 
proposition which is neither true nor false. 
10. 
Consider a process such as rainfall. It goes on for some time and then it 
stops. It does not stop suddenly, let us assume, but gradually. Let 
p ~p 
illustrate that, during a certain stretch of time it is first definitely raining 
(p), later definitely not raining (~p), and between these two states in 
time there is a "zone of transition" when a few drops may be falling 
- too 
few to make us say that it is raining then but too many to prevent us 
from saying that rain has definitely stopped. In this zone the proposition 
that p is neither true nor false. We can complete the picture as follows: 
Tp ~Tp&~T~p T~p 
--^ * -N 
One could, however, also take the view that as long as some drops of 
rain are falling then it is still raining 
- but also the view that when there 
are only a few drops of rain falling, then it is no longer raining. When 
viewing the situation from these points of view one includes the 
intermediate zone of transition or vagueness both under rain and under 
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TRUTH, NEGATION, AND CONTRADICTION 13 
not-rain, identifying raining with the state when it is not not-raining, 
and not-raining with the state when it is not raining. Then instead of 
saying that it is neither raining nor not-raining in the zone, one would 
say that it is both raining and not-raining in this area. 
I suppose that it is something like this which happens in Dialectical 
Synthesis. But it should be observed that a conceptual shift has now 
taken place in the notion of truth. It is not the same sense of "true" in 
which we say that it is neither raining nor not-raining and say that it is 
both raining and not-raining in the zone of transition. We could call the 
former a strict sense of "true" and the latter a liberal or more lax sense 
of "true". This liberal notion of truth I shall symbolize by V. It is 
defined in terms of strict truth as follows 
" 
T'p" =df 
"~ T ~ p". We now 
complete our picture of the process: 
Tp ~Tp&~T~p T~p 
We can also build a logic for the liberal notion of truth. We call this 
logic TL. In it the Law of Contradiction in the form ~(T'p8iT'~ p) 
is not valid. But the Law of Excluded Middle is valid in the form (of a 
Principle of Bivalence) T'p v T' 
~ 
p. The Law of Double Negation, 
finally, now holds in the version T' 
~ T' ~ p-> T'p. The falsehood of 
falsehood in the liberal sense implies truth in the liberal sense. But 
T'p-> T ~'T'~p does not hold. 
I find it of some interest to note this "duality" in the relation of the 
two truth-logics, TL and T'L. 
Dialectical Synthesis is a logically legitimate inference in certain 
cases but it involves a shift in the concept of truth from a stricter to a 
more liberal notion, both of which answer to common and natural uses 
of the words "true" and "false" when applied to propositions. This shift 
fits the facts particularly in situations when we are concerned with 
becoming or process, two ideas which are prominent in Hegelian and 
dialectical Logic. The liberal idea also has a natural application to cases 
of vagueness. 
Truth-logic thus seems to provide a kind of "bridge" between formal 
logic of the "classical" type to logic in the tradition of Hegel. Whether 
dialectical logicians will find this bridge safe enough to be worth 
crossing, thereby uniting the two traditions in logic, I leave to them to 
judge. 
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14 G. H. VON WRIGHT 
NOTES 
1 A. N. Whitehead and B. Russell: Principia Mathematica, 2nd ed., vol. I, pp. 101 and 
111. 
2 
J./Lukasiewicz: 'O Determinizmie', (in Z zagadnien logiki ifilozofii, ed. by J. Slupecki, 
Warzsawa 1961), sect. 9. 
3 
Ibid., sect. 11. 
4 
See Newton C. A. Da Costa and Robert G. Wolf: 1980, 'Studies in Paraconsistent 
Logic I: The Dialectical Principle of the Unity of Opposites', Philosophia 9. 
Department of Philosophy 
University of Helsinki 
Unioninkatu 40B 
00170 Helsinki 17 
Finland 
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	Issue Table of Contents
	Synthese, Vol. 66, No. 1, Finnish-Soviet Logic Colloquium (Jan., 1986), pp. 1-198
	Volume Information
	Front Matter
	Preface [p. 1-1]
	Truth, Negation, and Contradiction [pp. 3-14]
	A Computational Interpretation of Truth Logic [pp. 15-34]
	Boolean Algebra and Syllogism [pp. 35-54]
	Semantic Truth Conditionals and Relevant Calculi [pp. 55-62]
	Paraconsistent Structure inside of Many-Valued Logic [pp. 63-69]
	Logical Relations between Theories [pp. 71-87]
	'Ought' and 'Must' [pp. 89-93]
	Some Results on Dyadic Deontic Logic and the Logic of Preference [pp. 95-110]
	Hypothetical Imperatives and Conditional Obligations [pp. 111-133]
	Reasoning with Defeasible Principles [pp. 135-158]
	Mental Anaphora [pp. 159-175]
	Applying a Logical Interpretation of Semantic Nets and Graph Grammars to Natural Language Parsing and Understanding [pp. 177-190]
	Back Matter