Walter BARK 1

Ex Contradictione Non Sequitur Quodlibet

Walter A. Carnielli and João Marcos

We summarize here the main arguments, basic research lines, and results on

the foundations of the logics of formal inconsistency. These involve, in

particular, some classes of well-known paraconsistent systems. We also

present their semantical interpretations by way of possible-translations

semantics and their applications to human reasoning and machine

reasoning.

1. Do we need to worry about inconsistency?

Classical logic, as we all know, cannot survive contradictions. Among the

principles that were gradually incorporated into the “properties of correct

reasoning” since Aristotle, the Principle of Pseudo-Scotus (PPS), also known

since medieval times as ex contradictione sequitur quodlibet (and also called the

Principle of Explosion by some contemporary logicians), states that in any theory

exposed to the enzymatic character of a contradiction A and A one can derive

any other arbitrary sentence B, so that the theory would turn out to be trivial.

Another principle called the Principle of Non-Contradiction (PNC) states that

there should be theories from which no such contradictions are derivable. To

those principles, one could add the Principle of Non-Triviality (PNT), stating

that there should be at least one theory and one sentence B such that B is not

derivable from this theory.

In order to fully understand what those principles mean, what the

relationship is between them and what their importance is for modeling the

concept of inconsistency let us introduce some formalism. This formalism will be

apt for the syntactical approach to the logics of formal inconsistency we discuss in

the first three sections of this paper. However, the reader should be aware that it

is possible to start from a purely syntactical account, as we do in Sections 4 and 5.

Let For be a collection of formulas of a certain language, and call a theory

any subset of For. Let a consequence relation over For be a relation between

theories and formulas of For, that is,

⊆

(

℘

(For) × For), where

℘

(For)

denotes the power set of For. If Γ is a subset of FF

, we write Γ A when

Γ, A ∈ . We write Γ A when it is not the case that Γ A . We define a

logic L to be a structure constituted of For and the relation . The consequence

relation of a given logic is often defined by its axioms and rules, or else from

some semantical interpretation associated with the logic.

This material was discussed during the II World Congress of Paraconsistency (WCP’2000) in

Juquehy, SP, Brazil, and will appear more fully in W. A. Carnielli, and J. Marcos, "A taxonomy of

C-systems."

Bulletin of Advanced Reasoning and Knowledge 1 (2001) 89–109.

90 Walter A. Carnielli and João Marcos

Some basic assumptions on what the relation would have to obey in order

to be considered a consequence relation, known as the Tarskian conditions, are the

following:

(1) Reflexivity If A ∈ Γ, then Γ A .

(2) Monotonicity If

Γ A and Γ

⊆

∆

, then

∆

A .

(3) Transitivity If

Γ A and {

∆

, A} B, then Γ ∪

∆

B .

These conditions allow for the characterization of an immense number of distinct

logics, but they still can be made more permissive (as, for example, weakening the

requirement of monotonicity in order to characterize non-monotonic logics). In

our present study, we will stick to these three basic assumptions and consider the

effect of some additional properties of the relation .

Fix some logic L for the following discussion. A theory Γ of L is said to be:

(1) Contradictory with respect to (or simply

contradictory

)

If there exists a formula A such that Γ A and Γ A .

(2) Trivial

If for every A, we have Γ A .

(3) Explosive

If for every A, we have Γ ∪ { A, A} B .

These definitions are important for distinguishing theories from their underlying

logic: A logic L is contradictory, trivial, or explosive if, respectively, all of its

theories are contradictory, trivial, or explosive.

We can now restate PNC, PNT, and PPS in more formal terms.

• The Principle of Non-Contradiction (PNC) for a logic L

L should have non-contradictory theories, that is, there should be some theory

such that for no A, Γ A and Γ A .

• The Principle of Non-Triviality (PNT) for a logic L

L should have a non-trivial theory, that is, there must exist a theory

Γ and some

formula A such that

Γ A .

• The Principle of Pseudo-Scotus (PPS) for a logic L

L should have only explosive theories, that is, for every theory

Γ, Γ ∪ { A, A}

is trivial.

For classical logic, of course, all these principles hold. It seems intuitively

acceptable that PNT should be taken as the most important of these three principles

—after all, if PNT does not hold for a certain logic, then every

Γ would entail every

A and the relation would be total, that is,

(

℘

(For) × For). In that case,

would not be an interesting deductive relation, for it would stop “making the

difference,” failing to give, we argue, any special meaning to the notion of

derivability. We shall, accordingly, accept PNT throughout this study, avoiding the

consideration of trivial logics.

It is also interesting to remark that, using the property of monotonicity

Ex Contradictione Non Sequitur Quodlibet 91

mentioned above, it is sufficient to consider PPS valid for just the empty theory:

Call PPS

∅

the statement “{A, A} is trivial.” Then, in view of monotonicity, we

conclude that PPS

∅

implies PPS. The converse (PPS implies PPS

∅

) is obvious.

Similar reasoning applies to PNC and PNT. Call PNT

∅

the statement “the empty

theory is not deductively trivial” and call PNC

∅

the statement “the empty theory is

non-contradictory.” By monotonicity, PNT implies PNT

∅

, and the converse is

obvious, and the same for PNC and PNC

∅

. Thus, for monotonic logics we do not

need to care about the distinctions between PPS, PNT, PNC and their respective

counterparts, PPS

∅

, PNT

∅

, PNC

∅

. Dealing with non-monotonic logics, the

distinctions would have to be taken into consideration, but our primary treatment of

the question would still apply.

We intend to discuss here the logics of formal inconsistency, which constitute

a large class of paraconsistent logics where the notion of inconsistency can be

linguistically expressed. For those logics, we may adopt or not a cautious position

to the effect that not only PNT but also PNC should be required—although we want

our logics to be able to support contradictory theories, we may not want that our

logics derive contradictions.

Paraconsistent logics are often misunderstood as logics that inevitably derive

contradictions. This is clearly a mistake. Although there exist some logics (the so-

called dialectical logics, or logics of impossible objects) that violate both PPS and

PNC and have theses which are not classical theses, this particular case of

paraconsistent logics will not be studied here.

The logics surveyed in this study

just support contradictions and permit reasoning with them, but neither engender

contradictions nor validate any bizarre form of reasoning. To the contrary: As we

shall see, the logics of inconsistency are, in a sense, “more conservative” than

classical logic.

While PNT and PNC can be regarded as ontological principles in that they

presuppose the existence of certain theories inside our logics, PPS can be seen as a

kind of flexible, operative principle: It describes how the logic works when its

theories are exposed to contradictory formulas, and thus it seems likely to be

changed or even discarded by some logics and situations in which the trivializing

operation that PPS describes is not justified. So, the starting point to our approach to

paraconsistency will be the cautious one, and we will ask ourselves: (1) While

maintaining PNT and PNC, is there any good reason why we should challenge PPS?

(2) If so, should we modify or simply abandon this principle?

The first is a factual question: We should try to challenge PPS if there is

sufficient demand, and we will argue that there is. The second is a foundational

question of logico-mathematical character, and we will also argue that it is plainly

possible to construct a great variety of interesting logics alternative to (propositional

and quantified) classical logic maintaining PNT and PNC, while modifying only

PPS.

Compare C. Mortensen, "Aristotle’s thesis in consistent and inconsistent logics” and N. C. A.

da Costa, and R. G. Wolf, "Studies in paraconsistent logic I: The dialectical principle of the unity

of opposites".

92 Walter A. Carnielli and João Marcos

An interesting example of an intellectual activity where holding contradictory

or inconsistent hypotheses is more the rule than the exception is abductive reasoning,

conceived of as reasoning that looks for explanatory hypotheses and the evaluation

of such hypotheses. Scientific activity, and in particular medical diagnosis, generally

uses abduction when looking for explanations. According to P. Thagard, and C.

Shelley, this is inevitable:

We are not urging inconsistency as a general epistemological strategy, only

noting that it is sometimes necessary to form hypotheses inconsistent with

what is currently accepted in order to provoke a general belief revision that

can restore consistency.

A second example of how contradictions can easily be incorporated into

reasoning (independent of any ontological commitments to the actual existence of

concrete inconsistent objects in the world) concerns the domain of machine

intelligence and the efforts to find rules for automated reasoning. The following

example given by A. Rose in “Remarque sur les notions d’indépendance et de non-

contradiction” was intended to show that the concepts of independence and triviality

in a formal system are themselves independent of each other, but it can also be used

as an example of how certain rules and procedures, widely used in the formalization

of machine reasoning, must face the problem of contradictions. One such procedure

is the closed-world assumption, largely used in databases and logic programming,

which proposes that, if from a certain (knowledge-based) system S one cannot infer

information A and know nothing about its negation A, then one would be entitled to

add A to S by default. Consider now the following fragment S of classical

propositional logic closed under modus ponens and the substitution rule:

1. (((A→ A) → A) → A)

2. (A → (B→ A))

3. (( B→ A) → (A→ B))

4. ((A→ B) → ((B→ C) → (A→ C))

Consider also the formula D = ( A→ A)→ A. It is not hard to show that S

does not entail either D or D, but if D is adjoined to S the resulting system

becomes trivial. Since the closed-world assumption has exactly the effect of

adjoining D to S, it is clear that that assumption may produce contradictions,

which will result in triviality due to PPS. It is, in fact, not hard to see that PPS

holds in S:

a. From A and A, by axiom 2 derive ((B→ B) → A)

and ( B→ A) .

b. From ( B→ A) by axiom 3 derive (A→ B) .

c. From ((B→ B)→ A) and (A→ B) by axiom 4 derive

((B→ B)→ B) .

d. From ((B→ B)→ B) by axiom 1 derive B .

Hence for every Γ, A, and B, we have Γ ∪ {A, A}

B .

"Abductive reasoning: logic, visual thinking, and coherence."

Ex Contradictione Non Sequitur Quodlibet 93

As we shall see, such a trivialization does not occur in the logics of formal

inconsistency that we will consider, for axiom 3 above will in all cases be valid

only under the proviso that A is known to be not inconsistent.

2. The contradictory and the inconsistent

The concepts of contradiction and inconsistency need not be taken to be

equivalent. That classical logicians take them as equivalent is explained by two

facts: first, for classical logicians, “contradiction,” “inconsistency” and “triviality”

are usually seen as synonymous, and, second, PNC and PNT are indeed

equivalent for several logics. Yet this equivalence is far from necessary.

It is clear that trivial theories, closed or not, are necessarily contradictory (if

there is a symbol for negation in the language) since they derive all sentences,

negated and non-negated. From the definitions of PNC and PNT above we

obtain the following result, which will hold in all of our logics of formal

inconsistency:

Metatheorem I Every trivial theory is contradictory, that is, PNC implies PNT.

Proof: Suppose PNC holds. Then, there exists a theory Γ such that, for every A,

either Γ A or Γ A . Hence there exist Γ and A such that Γ A, and this is

what PNT states.

The converse, however, does not hold, because an appeal to PPS is necessary

in order for a contradictory theory to be trivial. We may add, as our second founding

result:

Metatheorem II Assuming PPS, contradictory theories and trivial theories

coincide; that is, PPS implies: PNT if and only if PNC.

Proof: Suppose that PPS holds. It follows then from the transitivity of that if

a theory is contradictory it is trivial, that is, that PNT implies PNC.

This last metatheorem can be understood as ex contradictione non sequitur

quodlibet, in the sense that what follows from a contradiction depends on other

underlying principles (in this case, it depends on PPS). So, controlling the

explosive power of contradictions is necessary to gain control over the destructive

effects of trivialization. But is that enough?

It is clear what contradictory and trivial theories are, as those concepts can

be defined in terms of purely set-theoretic conditions. It is also clear what the role

of PPS is, since, if PPS holds, every contradictory theory turns out to be trivial.

If we can only modify PPS, the sets of consequences of contradictory

theories will not necessarily be trivial. But what is a consistent theory? While

contradiction and triviality are definable, respectively, in terms of negation (hence

as linguistic) and in set-theoretical terms (hence as metamathematical),

consistency is better seen as a semantic concept.

The novelty of our approach is that, although acknowledging the semantic

94 Walter A. Carnielli and João Marcos

character of the notion of consistency, we endeavor to internalize it in the

language and treat it then from a purely abstract point of view, independently

from contradiction and triviality. Considering that it is PPS that forces negation

to behave in such absolute terms that contradictory theories collapse into triviality,

it is possible, as we shall see, to define forms of “careful” negation in several

ways so as to avoid such collapse.

In traditional logic, inconsistency and contradiction are taken to be just two

ways of referring to the same concept. For example, neither Aristotle nor

ukasiewicz, in his celebrated analysis of Aristotelian logic, ever mentioned

consistency or inconsistency, but always contradictions, since in classical logic

the distinction is immaterial—for, in this case, the metamathematical concept of

inconsistency is internalized into the object language by means of conjunctions

of contradictory statements.

This is so, however, due to PPS (Metatheorems I

and II). If PPS is not to be accepted as dogma, for the reasons already presented,

such an internalization will not be guaranteed.

It is possible, nonetheless, to conceive of the concept of inconsistency in

such a way that, while contradictory theories are certainly inconsistent, the reverse

might not be true. In a similar way to the concept of point, which is taken in

geometry as a primitive notion only describable through its relationship to other

concepts, inconsistency can be taken in logic to be a primitive notion as well.

From this point of view, inconsistent and contradictory theories do not coincide,

as our logics of formal inconsistency bC and Ci will make clear.

Contradictory theories depend strictly on negation and its properties, while

inconsistent theories do not. So, for example, if we say “It is raining in Ghent”

and “It is not raining in Ghent,” those are contradictory statements, but should not

necessarily lead to trivialization. They may lead to trivialization if we add some

extra information, for example, that we are talking about the same instant of time,

and that the concept of instant of time is sharp enough so as to exclude the

possibility of rain and absence of rain at a given instant of time. Or we may say

“It is raining at 11h 34m22s” and say “It is not raining at 11h34m22s,” which

again are contradictory statements, but do not necessarily lead to trivialization. In

this case, they may become so if we add extra information, for example that we

talking about the same point in space, and that the concept of point is strict

enough so as to exclude rain and the absence of rain.

Von Wright suggests that the Kantian conceptions of space and time in the

Critique of Pure Reason

(though with different aims) are not alien to this kind

of intuition:

If this representation [he refers to time] were not an a priori (inner) intuition,

no concept, no matter what it might be, could render comprehensible the

possibility of an alteration, that is, of a combination of contradictorily

opposed predicates in one and the same object, for instance, the being and the

not-being of one and the same thing in one and the same place. Only in time

J. ukasiewicz, “On the principle of contradiction in Aristotle.”

Compare I. Kant, Critique of Pure Reason, Transcendental Aesthetic II, §4.

Ex Contradictione Non Sequitur Quodlibet 95

can two contradictorily opposed predicates meet in one and the same object,

namely, one after the other.

The point here is not whether or not one accepts the transcendental ideality

of time, but the role of time (and also space) as examples of entities that bind

contradictory statements together and make them inconsistent. In other words,

contradictory statements A and A by themselves will not be sufficient to entail

any other statement, unless we require an extra condition—in our approach, that A

is consistent. The consistency here could be understood as introduced by a

Kantian rendering of space and time as concepts which exclude the possibility of

the concomitant existence of opposed predicates at the same point of space or

time.

Based on this intuition, as an addendum to the results in the Metatheorem I

and Metatheorem II, we propose the following as our basic metaprinciple:

Metaprinciple I

No contradictory theory is consistent, but a contradictory non-consistent theory

need not be trivial.

To this metaprinciple we can consider the addition of another (a kind of

converse of Metaprinciple I):

Metaprinciple II

Every inconsistent theory is contradictory, but not necessarily trivial.

As we will show, distinct classes of paraconsistent calculi arise, depending on

whether we take Metaprinciple II in conjunction with Metaprinciple I. It is

noteworthy that virtually all known paraconsistent systems in the literature do

assume Metaprinciple II.

It is possible to give models for inconsistent theories, even if those might be

regarded as epistemologically puzzling. Obtaining models and understanding

their role is an extraordinarily important mathematical enterprise: It required

enormous efforts of the most brilliant minds, and more than twenty centuries, until

mathematicians would allow themselves to consider models where, given a

straight line S and a point P outside of it, one could draw not just one line, but

infinitely many or no parallel lines to S passing through P, as in the well-known

case of non-Euclidean geometries.

3. A logic for the illogical?

The challenge is to find mathematically interesting systems that can provide a

foundational sense for what contradictions and inconsistency are and suggest an

acceptable semantic interpretation with which people would feel comfortable

while reasoning with contradictions. The case of (imaginary) complex numbers

seems to make a good comparison: Even if we do not know what they are, and

may even suspect there is little sense in insisting on which way they can exist in

the “real” world, the most important aspect is that it is possible to calculate with

G. H. von Wright, “Time, change and contradiction”.

96 Walter A. Carnielli and João Marcos

them. Girolamo Cardano, who first had the idea of computing with such numbers,

seems to have seen this point clearly—he failed, however, to acknowledge the

importance of it. In 1545 he wrote in his Ars Magna:

Dismissing mental tortures, and multiplying 515+− by 515−− , we

obtain 25 15−−(). Therefore the product is 40. . . . And thus far does

arithmetical subtlety go, of which this, the extreme, is, as we have said, so

subtle that it is useless.

His discovery that one could operate with a mathematical concept independently

of what our intuition says, and that utility (or something else) could be a guiding

criterion for accepting or rejecting experimentation with mathematical objects,

certainly contributed to the proof of the Fundamental Theorem of Algebra by

C. F. Gauss in 1799, before which complex numbers were not fully accepted.

The idea of proposing logics that enable one to operate with what does not

appear to be “rational” goes in the same direction: Good underlying mathematical

theory plus usefulness would have to constitute the only criteria to evaluate a

mathematical formalism that deals with inconsistency or contradictions.

It is time now to give a more precise definition for the logics of formal

inconsistency (LFI-systems): An LFI is any logic where a syntactic notion of

formal consistency can be defined in a syntactical way in such a way that this new

notion of formal consistency and the already known notion of contradiction can be

related in the light of Metaprinciples I and II. In particular, as we discuss below,

in many cases this can be done by endowing the language with a new connective

and considering new appropriate axioms.

Among LFI-systems it is possible to identify a subclass of the so-called

C-systems that preserve the positive fragment of some other logic and in which

consistency or inconsistency are expressible by means of new connectives. As a

subclass of the C-systems, we define the dC-systems to be those in which the

notion of formal consistency can be introduced as a defined connective. The dC-

systems include several classes of paraconsistent systems, including the ones in

the hierarchy C

of N. C. A. da Costa.

The main axioms we will consider here for the study of an interesting class

of C-systems based on classical logic are the following: Call C

min

an appropriate

axiomatization of the positive fragment of classical propositional logic in the

language , → , ∧ , ∨ , , plus the axioms ( A→ A) and (A∨ A) and closed

under the rules of

modus ponens

and substitution.

Define the basic logic of

formal inconsistency, bC, as C

min

plus the following deduction scheme, where

is a new unary operator meant to model “A is formally inconsistent”:

The Gentle Principle of Explosion A, A, A B

Note that this axiom is in line with Metaprinciple I: A contradictory theory (one

containing A and A) is not consistent and is not necessarily trivial. It would

See J. O’Connor, and E. Robertson, "The MacTutor History of Mathematics Archive".

See N. C. A. da Costa, Inconsistent Formal Systems.

This system was studied in our "Limits for paraconsistent calculi".

Ex Contradictione Non Sequitur Quodlibet 97

become trivial if, besides being contradictory, it were formally consistent. In such a

case its very consistency becomes contradictory, and this situation leads to triviality.

Examples of dervived consequences of bC are:

(1) A, A

If A is contradictory, then A is not formally consistent.

(2) A (A∧ A)

If A is formally consistent, then A is non-contradictory. (1st form)

(3) A ( A∧ A)

If A is formally consistent, then A is non-contradictory. (2nd form)

It is clear that the theorems above are variations on Metaprinciple I. A very

important observation is that in bC the notions of “not consistent” and

“inconsistent” do not coincide. Indeed, even if we introduce the concept of

“consistent” as internal to the language through a new symbol , understanding

A to model “A is formally inconsistent,” A and A, and A and A would

not be interdefinable, contrary to what one might assume. This will be further

clarified below.

In bC a new negation, called strong negation, can be defined as:

~A ≡

Def

A ∧ A

This recovers several features of classical negation, though not all. We have, for

instance, A, ~A

B , and thus PPS holds relative to this strong negation (that

is, this negation is explosive). But

(A∨ A) and

(A→ ~~A) .

Consequently, the strong negation ~A is not classical, even though it is explosive

(intuitively, ~A is somehow analogous to intuitionistic negation). But another

strong negation, this one having all the properties of classical negation, is

definable in bC by letting:

.A ≡

Def

A→ (A∧ ~A)

Yet another logic of formal inconsistency, Ci, is defined by first defining

A ≡

Def

A and adjoining to bC the deduction scheme:

A (A∧ A)

In the system Ci, inconsistency and contradictions are equivalent to each

other, due to the fact that (A∧ A)

A (as noted before) and the axiom just

introduced, plus the definitions. Omitting the definitions, however, we obtain

intermediate logics between bC and Ci that realize Metaprinciple II.

Moreover, in Ci both strong negations mentioned above are equivalent and

acquire all the properties of classical negation—but it is still possible to define

non-classical strong negations in Ci, via for example ~A and

.A . Some

properties of this logic are:

In many systems of paraconsistent logic, although (A∧ A) and ( A∧ A) are equivalent,

(A∧ A) and ( A∧ A) are not. See, e.g., . João Marcos, Possible-Translations Semantics.

98 Walter A. Carnielli and João Marcos

(4) A, A

B and A, A

The

Principle of Explosion

holds for formally consistent (or formally

inconsistent) formulas.

(5)

A and

Both consistent and inconsistent formulas are consistent.

(6)

If a formula is consistent, its negation is also consistent.

(7)

A formula is inconsistent if its negation is inconsistent.

What doesn’t hold in this logic? PPS still does not hold, that is, A, A

for some A and B. De Morgan’s Laws and the rule of contraposition only hold in

restricted forms, e.g., (A→ B)

( B→ A), though (A→ B)

( B→ A).

Also, Ci does not prove any formulas to be consistent, unless they already refer to

consistency or inconsistency. That is, A is provable in Ci if and only if A is

itself of the form B, B, B or B .

The converses of (2) and (3) are still not valid, and thus we may consider the

addition of some other deduction schema to Ci, as for example:

Levo-based scheme for contradictoriness (1):

(A∧ A) A .

Dextro-based scheme for contradictoriness (d):

( A∧ A) A .

Bi-directional scheme for contradictoriness (b):

(A∧ A) ∨ ( A∧ A) A.

Global scheme for contradictoriness (g):

(B↔(A∧ A)) → ( B↔ (A∧ A) .

In formal terms, a dC-system is any C-system where and can be defined in

terms of the usual connectives of the language , ∧ , ∨ , → . In the case of da

Costa’s C

, the strongest calculus of his hierarchy of paraconsistent logics in his

Inconsistent Formal Systems, A is defined as (A∨ A) and A as A, so that

this logic is an extension of Cil, the logic obtained by the addition of the axiom (l)

to Ci.

Several other distinct dC-systems can be defined by choosing adequate

axioms of “propagation” for consistency and inconsistency. Many different

choices are possible. Depending on the particular ones we choose, we may obtain

finite many-valued paraconsistent logics or infinite-valued paraconsistent logics.

Some choices which have been tried for the axioms of propagation are the

following:

First Choice ( A∧ B) → (A#B) for every binary connective # .

Second Choice (

A∨ B) → (A#B) for every binary connective # .

This axiom holds for the system C

of da Costa, for example, as discussed in W. A. Carnielli,

“Possible-translations semantics for paraconsistent logics” and J. Marcos, Possible-Translations

Semantics.

Ex Contradictione Non Sequitur Quodlibet 99

Third Choice ( A# B) for every binary connective # .

Fourth Choice

(A∧ B) ↔ ( ( A∧ B) ∨ ( B∧ A))

(A→ B) ↔ ( B∧ A)

Fifth Choice

(A#B) ↔ ( A∧ B) for every binary connective # .

Sixth Choice

( A) for every formula A .

Seventh Choice

A for every formula A .

The first choice plus (l) above (p. 97) defines the calculus C

of the

hierarchy of C

; the second choice plus (l) defines the calculus C

of N. C. A.

da Costa, J.-Y. Béziau, and O. Bueno, "Aspects of paraconsistent logic." Neither

of these are many-valued. The third and sixth choices plus (b) define the three-

valued maximal logic P

, introduced in A. M. Sette, “On the propositional

calculus P

.” The third choice plus (b) and (A→ A) defines another maximal

three-valued logic.

The fourth and fifth choices define, respectively, the three-

valued maximal logics LFI1 and LFI2 studied in W. A. Carnielli, J. Marcos, and

S. de Amo, “Formal inconsistency and evolutionary databases.” And finally the

seventh choice defines classical propositional logic. Many other combinations are

possible.

The fourth choice gives yet another axiomatization for the three-valued

paraconsistent calculus J

, bearing some resemblance to the axiomatization given

in R. Epstein’s Propositional Logics. There, the axiom (A∧ ( A∧A))→ B is

considered among ten other axioms, where A is defined as ( A∧ A) for a

primitive modal operator . The matrix interpretation of A in that book

coincides with our matrix interpretation for A, as discussed in the next section.

This logic also coincides with the logic CLuNs, to be found, for instance, in

D. Batens’ “A survey of inconsistency-adaptive logics,” and it has appeared quite

often in the literature.

So much for the syntactical part of the logics of formal inconsistency.

Semantical interpretations are a complicated issue for paraconsistency in general.

The first C-systems were introduced only in proof-theoretical terms, and only

some years later were semi-truth-functional bivalued semantics proposed for their

interpretation. Those semantics, however, offer a very weak and debatable

“meaning” to paraconsistent logics, and we describe in the next section an

attractive alternative semantics called possible-translations semantics.

4. The Rosetta stone analogy

Found by Napoleon’s troops in 1799 near the town of Rashid (Rosetta) in Lower

Egypt, the Rosetta Stone is a piece of basalt which after several unclear episodes

happens to be found today at the British Museum in London, containing

See C. Mortensen,”Paraconsistency and C

” and J. Marcos, Possible-Translations

Semantics.

See W. A. Carnielli and J. Marcos, “A taxonomy of C-systems” and J. Marcos, “8K solutions

and semi-solutions to a problem of da Costa”.

100 Walter A. Carnielli and João Marcos

inscriptions that were the key to deciphering Egyptian hieroglyphic writing.

The deciphering was possible only due to the inscriptions appearing in three

forms: hieroglyphic, Demotic, and Greek. By comparing the hieroglyphic and

Demotic scripts with the Greek version, and assuming that they contained the

same text, the British physicist Thomas Young and the French Egyptologist Jean

François Champollion were able to decipher the hieroglyphic and Demotic

versions in 1822. Moreover, by further comparing the hieroglyphic text to

equivalents of the better known Coptic language they could attach a phonetics to

the hieroglyphic writings, which were supposed to be only symbolic. Our

semantic approach to paraconsistent logics, known as possible-translations

semantics, is in many aspects similar to the deciphering of the Rosetta stone.

In very general terms, a translation from a logic system L into a logic

system L´ is just a language homomorphism that preserves derivability, that is, if

A is provable in L from premises Γ, and * is a translation from L into L´, then A*

should be provable in L’ from premises Γ* ={ B*: B∈Γ}, that is, if Γ A then

Γ* A*. Several specializations and variations of this notion have been studied in

the literature,

but this general definition is adequate to our present purposes.

In intuitive terms, the idea is to project a given “hieroglyphic” logic by

means of translations of it into simpler (usually many-valued) systems, and

combine their respective forcing relations in order to obtain a sound and complete

semantical interpretation to the initial complicated system. The simpler systems

would thus play the role of Greek and Coptic in the Rosetta stone analogy. We

may think of this process as working in two distinct directions

: When

analyzing a complicated logic in terms of simpler components, we call the process

splitting logics; but it is also possible to think of this process in the direction of

synthesis, by defining a complex logic starting from simpler ones, and in this case

we call the process splicing logics. Possible-translations semantics can be seen

as a kind of distributed semantics where the meaning of a sentence in the

“hieroglyphic” logic is made clear by way of a suitable combination of all the

translations of that sentence into the component logics.

We give here an example of how this kind of semantics can help to give

meaning to contradictions. For the basic C-systems bC and Ci the logics playing

the role of auxiliary languages in the Rosetta stone analogy will be copies of the

three-valued logic pictured below, with truth-values T, t and F, of which T and F

are absolute “true” and “false,” while t can be understood as “provisionally true.”

See E. A. W. Budge, The Rosetta Stone.

As in W. A. Carnielli, “Possible-translations semantics for paraconsistent logics” and J.

Marcos, Possible-Translations Semantics. The analogy is due to J. Marcos, who presented it first

in his Master Thesis defense at the University of Campinas, Brazil.

See W. A. Carnielli and I. M. L. D’Ottaviano, “Translations between logical systems: a

manifesto”.

As introduced in W. A. Carnielli and M. E. Coniglio, “A categorial approach to the

combination of logics”.

Ex Contradictione Non Sequitur Quodlibet 101

Tt F

Tt t F

tt t F

FF FF

∧

Tt F

Tt t

tt t

∨

Tt F

Tt t F

tt t F

ttt

→

TFF

tt F

w s

TTT

tTF

For Ci, as we show below, the meanings of ∧ , ∨ , → are fixed, but the meanings

of and vary: Each of them will be assigned two distinct interpretations,

namely a weak and a strong one. For negation , the weak interpretation

regards the value t as careful truth, and assigns to the negation of t also the value t .

On the other hand, the strong interpretation

makes no distinction between t and

T, and assigns F to the negation of t . For , the weak interpretation

forgets the

distinction between t and T, while the strong interpretation

recognizes the value

t as “provisionally true,” and thus potentially inconsistent.

For the system Ci the set of all recursive possible translations to be

considered is definable by the following clauses, to be obeyed by any translation *

in this set:

Tr1 For atomic p, p* = p .

For atomic p, ( p)* =

p .

Tr2 For non-atomic A, either ( A)* =

A or ( A)* =

A .

Tr3 For # any of ∧ , ∨ , , (A # B)* = A* # B* .

Tr4 If ( A)* =

A*, then ( A)* =

A .

If ( A)* =

A*, then ( A)* =

A .

As an example, a formula of the form A will have eight possible distinct

translations, according to the above clauses: If ( A)* =

( A)*, then

( A)* will be either

A*, or

A* or

A* . If ( A)* =

( A)*, then ( A)* will be either

A*,

A*, or

A* or

A* .

In other words, the syntax will be interpreted in different semantic scenarios,

and here of course we have infinitely many distinct translations interpreting the

formulas of CC

into distinct fragments of the above three-valued logics, according

to the choices for the interpretation of the connectives ∧ , ∨ , → , , and .

Possible-translations semantics are a powerful tool for combining logics,

complementary to other methods such as fibring. In what concerns the two main

directions for combining logics cited before (splitting logics and splicing

logics

) possible-translations semantics initially seem to be more apt for

splitting while fibring methods are more apt for splicing.

Possible-translations semantics have already been given for the calculi in the

See W. A. Carnielli and M. E. Coniglio, “A categorial approach to the combination of logics”.

102 Walter A. Carnielli and João Marcos

hierarchy C

and for a slightly stronger version of C

, offering thus a solution to

the difficult problem of finding good semantics for paraconsistent logics.

It is

also possible to give such semantics for some many-valued logics.

Based on

these semantics, connections to other views on logics like dialogical logic could

also be established, as suggested in Shahid Rahman and Carnielli, “The dialogical

approach to paraconsistency.” A categorial treatment of possible-translations

semantics is presented in W. A. Carnielli and M. E. Coniglio, “A categorial

approach to the combination of logics,” where completeness with respect to

possible-translations semantics is characterized by means of limits of categorial

diagrams.

5. Modeling human and computer reasoning, or why we can’t reason

without contradictions

A good example of why the logic of ordinary language reasoning would have to

abhor the Principle of Explosion is the following. Suppose that, in the course of

an investigation, you receive information on some given subject (for example, as a

response by two or more people to “Does Dick live in Arizona?”) in the form of

two sentences A and B of type “yes” or “no.” Now there are exactly three

possibilities (in a two-valued logic) concerning the truth-values of those

sentences: Either they are both true, or both false, or they are contradictory, which

occurs only in case they are distinct. Now it happens that the contradictory

answer is very opportune, because it is the only case in which you’re sure you

received wrong information! Human reasoners profit from this possibility, and

we would not exaggerate in saying that this ability is even essential for survival:

In many cases, signs of danger are recognized in this way. So, reasoning with

contradictories, instead of deriving anything else from them, seems to be an

essential trait of human thinking. By itself, this would be a strong case in favor of

developing a logic where PPS is not accepted as universal.

Interesting applications of paraconsistent logics can be found in the domain

of automated reasoning and knowledge-based systems. We mention two

examples of applications. The first refers to automated proof methods and logic

programming. The resolution method and the tableau method are known to be

equivalent for classical logic, and the programming language PROLOG which is

based on such methods permits computer programs to be written almost

axiomatically, directly in a sort of logic programming language, though for non-

classical logics those methods do not necessarily coincide or have comparable

computational complexity. In W. A. Carnielli and M. Lima-Marques, “Reasoning

under inconsistent knowledge” we proposed a signed tableau method for the

paraconsistent calculus C

of da Costa (and the method is immediately

generalizable to other calculi of the C

hierarchy) which differs from the classical

See J. Marcos, Possible-Translations Semantics and W. A. Carnielli, “Possible-translations

semantics for paraconsistent logics”.

A particular case of possible-translations semantics called “society semantics” has been

studied in W. A. Carnielli and M. Lima-Marques, “Society semantics and multiple-valued logics”.

Ex Contradictione Non Sequitur Quodlibet 103

signed tableaux with respect to the rules for negation (all other notions, like closed

signed tableaux, etc, being the same as the classical ones).

Recalling that in C

the formula A

(which plays exactly the same role of

the formula A in the axiomatization presented in Section 3 above) abbreviates

(A∧ A), the new rules at the propositional level are the following, where #

stands for any of the binary connectives ∧ , ∨ , → . As usual in tableau proofs,

rules containing the symbol | represent an “or” tree, with branching nodes.

T A)

F(A) | F(A )

(

If A is true, then either A is false or it is false that A is consistent, i.e.,

A is inconsistent/contradictory.

F( A)

T(A)

If A is false, then A is true.

FAB)

F(A ) | F(B )

(( # )

If A

# B is inconsistent, then either A or B is inconsistent.

Notice that the rule for T( A) differs from the classical one just by adding

an alternative branch F(A

). The rule for F((A#B)

) does not exist in the classical

case, however both could be added as redundant rules to the classical tableau

rules.

Completeness of these rules for the propositional calculus C

(and for its

first-order version, C

* with additional rules) was proven in W. A. Carnielli and

M. Lima-Marques, “Reasoning under inconsistent knowledge,” where other

examples for automated reasoning were also given.

A simple yet illustrative example is the well-known “Nixon Diamond,”

often repeated in the literature. Suppose we have the following statements:

Nixon is a Quaker: Q(n).

If Nixon is a Quaker, then Nixon is a pacifist: Q(n) → P(n).

Nixon is not a pacifist: P(n) .

These statements are contradictory, assuming that there is some person

having all such properties. At this point, a human reasoner would suspect that one

of them should be disqualified, but this maneuver would be blocked if all of the

claims had equal confidence status. If all of them are to be taken as true, there is

no other rational possibility besides some predicate being inexact, or vague, or

subject to contradictions. Let’s suppose that it can always be clarified who is a

Quaker and who is not. Then the only candidate for inexactness or possible

contradictoriness is “pacifist.” This is exactly the conclusion given by our

system

: We can run a tableau for the set of propositions S = {T(Q(n)),

In W. A. Carnielli and M. Lima-Marques, “Reasoning under inconsistent knowledge.”

104 Walter A. Carnielli and João Marcos

T(Q(n) → P(n)), T( P(n))}. The system concludes, instead of becoming

blocked, that F(P(n)

), that is, P is contradictory.

The second example is for designing an implementation of databases that are

robust enough to work in the presence of contradictions. There are several ways

in which a database can be contradictory. Different users having equal access to

some given database may introduce new claims, and even new rules or

constraints, which, despite being consistent from the point of view of each user,

can still be globally contradictory. Traditional databases may detect contradictory

information and then start a complicated, and computationally extremely

inefficient, procedure for “restoring consistency,” but by no means can they afford

modification of constraints, for reasons explained below. It is thus natural to

embed databases in logical environments that permit reasoning with contradictory

information, while maintaining all other desirable features of traditional logic,

such as reasoning with the law of excluded middle, reasoning by cases, and

reasoning by means of quantifiers.

In general, information stored in databases must be checked to verify some

previous conditions (called integrity constraints) in order to be safely integrated in

the database. Integrity constraints are expressed by (fixed) first-order sentences.

For example, a database storing information about books may contain the

requirement that no book in the collection can have more than one title, a

condition that could be expressed by the following first-order formula, where

Title(x, y) means that y is a string that is the title of book x :

∀

( (Title(

x, y

) ∧ Title (

x, z

)) →

)

Updates in traditional databases are only performed if the new database would

satisfy the integrity constraints; if not, the database maintains its previous state.

So, in a traditional database system, an imperative control never allows

contradictory information to be considered.

The situation would be worse for traditional databases if integrity constraints

themselves could be changed in time, instead of remaining fixed forever. Such

evolving databases, which we call evolutionary databases, seem very interesting

for the domain of artificial reasoning. The fact that, traditionally, integrity

constraints are defined by the database designer and remain fixed for the users

during the database lifetime is a severe limitation on databases, due solely to the

logical foundations of classical database theory.

In W. A. Carnielli, J. Marcos, and S. de Amo, “Formal inconsistency and

evolutionary databases” we have introduced new logics that axiomatize a formal

representation of inconsistency in classical logic, starting from a purely

semantical standpoint. Though in that paper we take inconsistency to be

equivalent to contradictoriness, that is not necessary according to the observations

made in Section 3. That simplification does not affect the treatment of

information. Starting from an intuitive semantic account of what contradictory (or

inconsistent) data should be, and taking into consideration some basic

requirements, we provided two distinct sound and complete axiomatics for such

Ex Contradictione Non Sequitur Quodlibet 105

semantics, LFI1 and LFI2, as well as for their first-order extensions, LFI1* and

LFI2*, depending on which additional requirements are considered.

These two formal systems are examples of Logics of Formal Inconsistency

(LFI) and form part of a much larger family of maximal paraconsistent three-

valued logics.

It is important to note that LFI1* and LFI2* are proper

subsystems of classical logic extended with a consistency operator, and they entail

thus, in intuitive terms, fewer theorems than this extended form of classical logic.

We have shown, however, that they can codify any classical or paraconsistent

reasoning, because there exist grammatically faithful conservative translations

from classical and paraconsistent first-order logics into LFI1* and LFI2*.

We repeat here an example of an evolutionary database considered in W. A.

Carnielli, J. Marcos, and S. de Amo, “Formal inconsistency and evolutionary

databases.” Suppose that a claim P is proposed by a certain source. It may enter

the database either with the token √ or the token ××

××

appended to it. In case not-P

is proposed, the claim P enters with the token ××

××

or does not enter at all. In case

that we know nothing about P, nothing is added to the database. As a

consequence, in case P and not-P are simultaneously proposed (for instance, by

different sources), then P enters the database with the token ××

××

As a concrete example, suppose we have a database schema DS containing

three relations: Author(Name, Country), Title(Book, String), and

Translated(Book, Language). Suppose also that two different sources, I and II,

provide information to our database, telling us:

Source I

1. Joaquim Maria Machado de Assis was born in Brazil.

2. Gabriel García Marquez was born in Colombia.

3. Machado de Assis is author of Dom Casmurro.

4. Dom Casmurro has not been translated into Polish.

Source II

5. Gabriel García Marquez was not born in South America.

6. Gabriel García Marquez is author of One Hundred Years of Solitude.

7. One Hundred Years of Solitude has been translated into Polish.

Now claim (4) is negative,and may be stored in DS with a ××

××

or not stored at

all. On the other hand, claims (2) and (5) are contradictory, given certain already

stored knowledge about geography, and in this case those claims are added with

the token ××

××

appended to it. The remaining positive information may be added

liberally either having √ or ××

××

as a suffix.

So we have already at least one piece of inconsistent information stored in

DS. Besides that one, another one may appear if, for instance, Source I adds a

new constraint asserting that “No South American author has ever been translated

into Polish.” After DS has been updated taking this new constraint into consider-

See J. Marcos, “8K solutions and semi-solutions to a problem of da Costa”.

See W. A. Carnielli, J. Marcos, and S. de Amo, “Formal inconsistency and evolutionary

databases”.

106 Walter A. Carnielli and João Marcos

ation, the relation “Translated” will also contain contradictory information. While

traditional databases cannot support this situation, our model permits us to reason

with these contradictions, even taking advantage of this controversy to get better

knowledge about the sources, much in the same way a human reasoner would do.

6. Closure

In this paper we have tried to defend the idea that reasoning with contradictions is

not only useful but perfectly well-founded from the logico-mathematical

standpoint. We have also discussed some underlying questions of

paraconsistency, indicating some applications of it to human and automated

reasoning and to database models. We have shown how several axiomatic

systems of paraconsistent logics can be formulated, and how possible-translations

semantics can be assigned to some of them. We also have tried to show how

those interpretations are intuitively clear and philosophically appealing, despite

some common objections that could be raised.

Often philosophers make predictions about what is mathematically possible.

Some writers, in their efforts to find definitive arguments in favor of their own

beliefs, try to blame logic, or more widely, mathematics as a whole, risking their

reputation on what could or could not be done in the domain of mathematical

possibilities. Some “throw the baby out with the bathwater,” prematurely

dismissing the possibility of constructing logic systems which are robust enough

to carry a good portion of classical reasoning. Karl Popper, while objecting

against dialectics, acknowledges the intrinsic interest of contradictions:

Dialecticians say that contradictions are fruitful, or fertile, or productive of

progress, and we have admitted that this is, in a sense, true.

However, he cautiously warns the reader that from a pair of contradictory

premises any conclusion may be deduced in classical logic, and he asserts:

The question may be raised whether this situation holds good in any system

of logic, or whether we can construct a system of logic in which

contradictory statements do not entail every statement. I have gone into this

question, and the answer is that such a system can be constructed. The

system turns out, however, to be an extremely weak system. Very few of the

ordinary rules of inference are left, not even the modus ponens which says

that from a statement of the form ‘If p then q’ together with p we can infer

q. In my opinion, such a system is of no use for drawing inferences,

although it may perhaps have some interest for those who are specially

interested in the construction of formal systems as such.

Popper’s remarks can be seen as a good defense of our position: He

concludes not only that contradictions can be seen positively, but also that other

systems of logic could be constructed so that PPS, at least as taken for granted in

classical logic, would not be valid. Unfortunately, he stops too soon, blocked by

Conjectures and Refutations: The Growth of Scientific Knowledge, p. 316.

Ex Contradictione Non Sequitur Quodlibet 107

his own misfortune.

But the fact that a great philosopher did not succeed in

the mathematical enterprise of constructing a system of this sort, powerful enough

to incorporate most of ordinary rules of inference including modus ponens,

certainly does not mean that this task is impossible. Several mathematicians and

philosophers in the course of history have experienced similar difficulties

imagining non-Euclidean geometries, complex numbers and the like, but

fortunately science does not survive on the imagination (or lack of it) of one

individual.

By taking seriously the task of constructing logics for formal inconsistency

we may also be helping to understand other philosophical questions: according to

S. Knuuttila and A. I. Lehtinen in “Change and contradiction: a fourteenth-century

controversy,” the possibility of accepting that contradictory sentences could be

true at the same instant of time was already considered (for theological

arguments) almost seven centuries ago. Instead of blaming logic or mathematics

and risking sibylline prophecies on what cannot be done, it is often better to in-

vestigate what can be achieved—even if what is achieved in the end are negative

results, as the mathematical impossibilities of solving the classical Greek

problems of elementary geometry, the theorems of Gödel, the indecidability of

first-order logic and many other results familiar to contemporary logic.

Bibliography

Batens, D.

2000 A survey of inconsistency-adaptive logics, in D. Batens, C. Mortensen and

G. Priest, eds., Frontiers in Paraconsistent Logics, Proceedings of the I

World Congress on Paraconsistency, Ghent, King’s College Publications,

pp. 49–73.

Budge, E. A. W.

1989 The Rosetta Stone, Dover Publications, reprint edition.

Carnielli,W. A.

2000 Possible-translations semantics for paraconsistent logics, in D. Batens,

C. Mortensen and G. Priest, eds., Frontiers in Paraconsistent Logics,

Proceedings of the I World Congress on Paraconsistency, Ghent, King’s

College Publications, pp. 149–163.

Carnielli, W. A. and M. E. Coniglio

1999 A categorial approach to the combination of logics, Manuscrito, vol. 22,

pp. 64–94.

Carnielli, W. A., and I. M. L. D’Ottaviano

1997 Translations between logical systems: a manifesto, Contemporary Brazilian

research in logic, Part II, Logique et Analyse, vol. 40, no.157, pp. 67–81.

Carnielli, W. A. and M. Lima-Marques

1992 Reasoning under inconsistent knowledge, The Journal of Applied Non-

Classical Logics, vol. 2, no.1, pp. 49–79.

His system was introduced in K. R. Popper, “On the theory of deduction. I. Derivation and its

generalizations” and “On the theory of deduction. II. The definitions of classical and intuitionist

negation”.

108 Walter A. Carnielli and João Marcos

Carnielli, W. A. and M. Lima-Marques

1999 Society semantics and multiple-valued logics, in W. A. Carnielli and

I. M. L. D’Ottaviano, eds., Advances in Contemporary Logic and Computer

Science, Proceedings of the XII Brazilian Logic Meeting, American

Mathematical Society, series Contemporary Mathematics, vol. 235,

pp. 33–52.

Carnielli, W. A. and J. Marcos

1999 Limits for paraconsistent calculi, to appear in Notre Dame Journal of

Formal Logic, vol. 40, no. 3. [To appear in 2001.]

Carnielli, W. A. and J. Marcos

2001 A taxonomy of C-systems, to appear in 2001, lecture delivered at the II

World Congress on Paraconsistency, Juquehy, SP, Brazil, May 2000 under the

title "The C-Systems: Paleontology and Futurology".

Carnielli, W. A., J. Marcos, and S. de Amo

2000 Formal inconsistency and evolutionary databases, to appear in Logic and

Logical Philosophy (Torun, Poland), vol. 7/8.

da Costa N. C. A.

1963 Inconsistent Formal Systems (in Portuguese), Thesis, Universidade Federal

do Paraná, 1963. Reprinted by Editora UFPR, Curitiba, 1993, 68 p.

da Costa, N. C. A. and R. G. Wolf.

1980 Studies in paraconsistent logic I: The dialectical principle of the unity of

opposites, Philosophia (Philosophical Quarterly of Israel), vol. 9,

pp. 189–217.

da Costa, N. C. A., J.-Y. Béziau, and O. Bueno

1995 Aspects of paraconsistent logic, Bulletin of the IGPL, vol. 3 1995, no.4,

pp. 587–614.

Epstein, R. L.

2001 Propositional logics: The semantic foundations of logic, with the assistance

and collaboration of W. A. Carnielli, I. M. L. D'Ottaviano, S. Krajewski and

R. D. Maddux. Second edition, Wadsworth-Thomson Learning.

Kant, I.

???? Critique of Pure Reason. Translated by N. Kemp-Smith (second edition,

1787), Macmillan Press.

Knuuttila, S. and A. I. Lehtinen

1979 Change and contradiction: a fourteenth-century controversy, Synthese, vol. 40

no. 1, pp. 189–207.

ukasiewicz, J.

1910 O zasadzie sprzecznosci u Arystotelesa, translated as “On the principle of

contradiction in Aristotle”, Review of Metaphysics, vol. 24 (1971),

pp. 485–509.

Marcos, J.

1999 Possible-Translations Semantics (in Portuguese), Master’s Thesis, Unicamp,

xxviii+240 p., 1999. <ftp://www.cle.unicamp.br/pub/thesis/J.Marcos/>

2001 8K solutions and semi-solutions to a problem of da Costa, to appear.

Lecture delivered at the Joint Austro-Italian Workshop on Fuzzy Logics and

Applications, Università degli Studi di Milano, Milan, Italy. December

2000.

Ex Contradictione Non Sequitur Quodlibet 109

Mortensen, C.

1984 Aristotle’s thesis in consistent and inconsistent logics, Studia Logica, vol. 43,

pp. 107–116.

1989 Paraconsistency and C

, in G. Priest, R. Routley, and J. Norman, eds.,

Paraconsistent Logic: essays on the inconsistent, Philosophia Verlag,

pp. 289–305.

O’Connor, J. and E. Robertson

2000 The MacTutor History of Mathematics Archive (August 2000 edition).

<http://www-history.mcs.st-andrews.ac.uk/history/index.html>

Popper, K. R.

1948 On the theory of deduction. I. Derivation and its generalizations,

Indagationes Mathematicae, vol. 10, pp. 44–54.

1948b On the theory of deduction. II. The definitions of classical and intuitionist

negation, Indagationes Mathematicae, vol. 10, pp. 111–120.

1963 Conjectures and Refutations: The Growth of Scientific Knowledge, Routledge

& Kegan Paul Limited.

Rahman, Shahid and Walter A. Carnielli

2000 The dialogical approach to paraconsistency. Festschrift in honor of Newton C.

A. da Costa on the occasion of his seventieth birthday. Synthese, vol. 125,

no. 1–2, pp. 201–231.

Rose, A.

1951 Remarque sur les notions d’indépendance et de non-contradiction, Comptes

Rendus de l’Academie de Sciences de Paris, vol. 233 , pp. 512–513.

Sette, A. M.

1973 On the propositional calculus P

, Mathematica Japonicae, vol. 18

pp. 173–180.

Thagard, Paul and Cameron Shelley

1997 Abductive reasoning: logic, visual thinking, and coherence, in Logic and

scientific methods (Florence, 1995), Synthese Library, vol. 259, Kluwer

Acad. Publ., Dordrecht, Netherlands 1997, Maria Luisa dalla Chiara, Kees

Doets, Daniele Mundici, Johan van Benthem, von Wright, G. H., eds.

pp. 413--427.

von Wright, G. H.

1983 Time, change and contradiction, in Philosophical Logic– Philosophical

Papers, vol. II, pp. 115–131, Cornell Univ. Press.