Synthese (2007) 154:383–399
DOI 10.1007/s11229-006-9125-x

O R I G I NA L PA P E R

Quasi-truth, paraconsistency, and the foundations
of science

Otávio Bueno

· Newton C. A. da Costa

Published online: 9 February 2007
© Springer Science+Business Media B.V. 2007

Abstract

In order to develop an account of scientific rationality, two problems need

to be addressed: (i) how to make sense of episodes of theory change in science where
the lack of a cumulative development is found, and (ii) how to accommodate cases
of scientific change where lack of consistency is involved. In this paper, we sketch
a model of scientific rationality that accommodates both problems. We first provide
a framework within which it is possible to make sense of scientific revolutions, but
which still preserves some (partial) relations between old and new theories. The exis-
tence of these relations help to explain why the break between different theories is
never too radical as to make it impossible for one to interpret the process in per-
fectly rational terms. We then defend the view that if scientific theories are taken to
be quasi-true, and if the underlying logic is paraconsistent, it’s perfectly rational for
scientists and mathematicians to entertain inconsistent theories without triviality. As
a result, as opposed to what is demanded by traditional approaches to rationality,
it’s not irrational to entertain inconsistent theories. Finally, we conclude the paper by
arguing that the view advanced here provides a new way of thinking about the foun-
dations of science. In particular, it extends in important respects both coherentist and
foundationalist approaches to knowledge, without the troubles that plague traditional
views of scientific rationality.

Keywords

Rationality

· Inconsistency · Scientific change · Paraconsistent logic ·

Coherence

O. Bueno
Department of Philosophy,
University of Miami,
Coral Gables, FL 33124-4670, USA
e-mail: [e-mail]

N. C. A. da Costa (

B

)

Department of Philosophy,
Federal University of Santa Catarina,
Florianópolis, SC 88040-900, Brazil
e-mail: [e-mail]

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Synthese (2007) 154:383–399

1 Introduction

Among the various problems regarding the foundations of science, one is particularly
pressing: the problem of scientific rationality. This problem involves two main issues:
(a) How should we rationally understand theory change in science? In particular,
how should we accommodate, in rational terms, the apparent lack of a cumulative
development in scientific development? What is at stake here are the criteria of the-
ory selection, and the difficulties of rationally accommodating scientific development
given the presence of radical theory changes in science.

This immediately raises the second issue: (b) How should we make sense of episodes

that apparently challenge the rationality of scientific theorizing? For example, how
should we understand the various situations in which scientists, or even mathemati-
cians, entertain inconsistent theories? Are these simply cases of scientific irrationality?
By making sense of episodes where the lack of a cumulative development or of con-
sistency play a role, we obtain an account of scientific rationality more sensitive to
the complexities of scientific and mathematical practice; an account that is able to
understand important aspects of the foundations of science.

In this paper, we sketch a model of scientific rationality that accommodates these

two issues, namely, episodes that involve the lack of a cumulative development and
of consistency in scientific development. These issues are, in fact, closely connected.
After all, it’s often in those cases in which dramatic theoretical changes are involved
that we find scientists and mathematicians entertaining inconsistent theories. Spelling
out the role that inconsistent theories play in these episodes yields an account of
scientific rationality that is able to make better sense of scientific and mathematical
activity.

2 Puzzles about scientific rationality

Any discussion of scientific rationality involves making sense of the puzzling issues
about inconsistency and scientific revolutions just mentioned. Of course, the more
rigid the model of rationality we consider, the harder it will be to make sense, in
rational terms, of these features of scientific practice. In particular, any model of
scientific rationality that insists on the continuity of content between old and new the-
ories faces serious difficulties with the existence of scientific revolutions. For example,
a model of scientific rationality such as Karl Popper’s highlights the fact that new
theories should preserve all the content of past theories that has been successfully
“confirmed” (see Popper, 1963). But sometimes even confirmed parts of past theories
are rejected. Consider, for example, the fate of phlogiston.

Popper also emphasizes the point that rationality is a matter of criticism rather

than justification (see Popper, 1963). As an important component of rationality, this
is certainly correct. But it can hardly be the whole story, given that there’s more to
rationality than simply criticism. The assessment of evidence is also an important
component, and this includes determining the positive support that bits of evidence
provide to certain theories or hypotheses.

Should we then say that rationality is a matter of having good (perhaps conclusive)

reason to support one’s beliefs? This suggestion moves the discussion to the opposite
extreme. And it’s not clear that we will ever have conclusive reason for any belief
(see Miller, 1994). Of course, it doesn’t follow from this that everything is rationally

Synthese (2007) 154:383–399

385

admissible—otherwise the notion of rationality would have no normative role. And,
clearly, rationality has both normative and descriptive components. So, what we need
is a notion of rationality that is not so idealized that it’s unable to make sense of
our own practice as cognitive agents. But meeting the descriptive component is not
enough. After all, one needs to determine what is legitimate and what isn’t in this
practice. At this point, the normative component of rationality steps in. In the end,
both components need to be accommodated.

Moreover, any model of scientific rationality, again such as Popper’s, that insists that

inconsistencies trivialize any theory—in that every sentence in the theory’s language
can be derived from an inconsistency—will face serious difficulties to accommodate
the ubiquitous nature of inconsistencies in science and mathematics. Examples of
inconsistent theories abound, of course: from early formulations of the calculus to
Bohr’s atomic model; from Frege’s original logicist reconstruction of arithmetic to
current models of quantum mechanics and general relativity theory (taken simulta-
neously). Inconsistent theories are plentiful and widespread. How can one reconcile
this fact with scientific rationality? Are scientists and mathematicians simply irrational
when they entertain and articulate inconsistent theories?

Considerations such as these seem to motivate models of scientific rationality that

are more lenient. For example, Bas van Fraassen has developed a model according
to which rationality is bridled irrationality (see van Fraassen, 1989, pp. 171–172).
According to this model, what is rational to believe includes anything that we are
not rationally compelled to disbelieve. Clearly this is a much more indulgent model
of rationality than those that insist that it’s only rational to believe in what we are
compelled to believe. But the model is not too indulgent. After all, events that have
extremely low probability count among those that we are compelled to disbelieve.

It’s important to note that van Fraassen’s model does make room for revolutions

in science (see van Fraassen, 2002). After all, with the emergence of new information
and new ways of thinking about the world, what we are (or are not) rationally com-
pelled to disbelieve can also change. For example, although a Newtonian might have
been rationally compelled to disbelieve that the speed of light was constant, this was
certainly not the case once an Einsteinian framework is put forward. On the latter
framework, clearly we are no longer compelled to disbelieve that light travels at a
constant speed.

But van Fraassen’s model also faces difficulties: If we are compelled to disbelieve

events with low probability, clearly we are compelled to disbelieve inconsistent theo-
ries. After all, on traditional approaches to probability, such theories get probability
0. How can we reconcile this fact with the ubiquity of inconsistent theories in science?
Would we have to say that the physicists that adopted Bohr’s atomic model were being
irrational? How about the mathematicians who worked with the initial, inconsistent
formulations of the calculus? Was that a case of irrationality in mathematics? Simi-
larly, what is the status of the belief systems of contemporary physicists who believe in
both quantum mechanics and general relativity theory (given that they are mutually
inconsistent)? Should we take the contemporary physics community as irrational?
That seems hardly the case.

In other words, despite its leniency, van Fraassen’s model seems to characterize

important—and perfectly reasonable—episodes in science as cases of irrationality. In
each of the cases above, at the time in which the theories in question were formu-
lated, they provided the best available solution to the problems they addressed. The
solutions might have been defective in various ways. But this doesn’t diminish the fact

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Synthese (2007) 154:383–399

that these theories solved the problems they were meant to solve. It seems difficult to
take these episodes as cases of irrationality.

But is it acceptable to suggest that belief in inconsistent theories might be rational?

In this context, it’s important to note that we don’t take consistency to be either a
necessary or a sufficient condition for a belief to be rational. On the one hand, con-
sistency is not necessary for rationality, given that despite the inconsistency of Bohr’s
atomic model, the physics community was certainly rational in adopting and working
with that model: it was the best account of atomic behavior available at the time.
On the other hand, consistency is not sufficient for rationality, given that despite the
consistency of the phlogiston theory, it wouldn’t be rational to believe in the existence
of the latter. To make sense of this, it’s crucial to develop an account of rationality
that doesn’t link too closely rationality and consistency.

So, what we need is a model of scientific rationality that makes sense of both the

existence of revolutions in science and the widespread role played by inconsistencies
in scientific practice. But the model also needs to accommodate some more abstract
constraints, and they derive from four general components of scientific rationality
(see also da Costa, 1997):

(a)

The logical component. A crucial feature of rationality is the ability to make
derivations, to determine what follows from what. But this clearly requires some
logic. (Note that which logic is to be used is an open issue. This typically depends
on the context of application one considers.)

(b)

The inductive component. In order to obtain, for example, the premises from
which the deductions referred to above are made, some sort of non-deductive
procedure seems to be required. There are several such procedures, including
methods of statistical inference, analogies, simple induction and the hypothetic-
deductive method. The ability to generate and accommodate such procedures
is a significant feature of rationality.

(c)

The critical component. There’s no doubt that criticism is a crucial component
of rationality. Without criticism, scientific inquiry is not possible.

(d)

The goal-oriented component. One of the main features of rationality, especially
when we consider the behavior of certain agents, is the fact that rationality is a
goal-oriented activity. In this sense, a rational behavior clearly depends on the
goals one may have. In the context of science, the goal involves the search for
some sort of truth or regularity to make sense of experience.

Given that these four components are general features of scientific rationality, they

should apply, in particular, to various aspects of scientific practice. We will sketch
below a model of scientific rationality that meets these constraints while still making
room for scientific revolutions and the role of inconsistencies in science.

1

But to be

able to do that, we will need, first, to provide a framework in terms of which the model
of rationality will be articulated: the partial structures approach.

3 Quasi-truth and paraconsistency

The partial structures approach (as first presented in Milkenberg, da Casta, & Chuaqui,
1986; da Costa, 1986, and then extended in da Costa & French, 1989, 1990, 2003) relies

1

For a fascinating exploration of these issues, see also Granger (1998) and Meheus (ed.) (2002).

Synthese (2007) 154:383–399

387

on three main notions: partial relation, partial structure, and quasi-truth.

2

One of the

main motivations for introducing this proposal comes from the need for supplying
a formal framework in which the openness and incompleteness of information dealt
with in scientific practice can be accommodated in a unified way (see da Costa &
French, 2003). This is accomplished by two moves. First, the usual notion of structure
is extended, in order to model the partialness of information we have about a certain
domain. The notion of a partial structure is then introduced. Second, the Tarskian
characterization of the concept of truth for partial contexts is put forward, advancing
the corresponding concept of quasi-truth.

In order to introduce a partial structure, the first step is to formulate an appropri-

ate notion of partial relation. When investigating a certain domain of knowledge

,

we formulate a conceptual framework that helps us in systematizing and organizing
the information we obtain about

. This domain is tentatively represented by a set

D of objects, and is studied by the examination of the relations holding among D’s
elements. However, we often face the situation in which, given a certain relation R
defined over D, we do not know whether all the objects of D (or n-tuples thereof)
are related by R. This is part of the incompleteness of our information about

, and

is formally accommodated by the concept of partial relation. More formally, let D be
a non-empty set; an n-place partial relation R over D is a triple

R

1

, R

2

, R

3

, where

R

1

, R

2

, and R

3

are mutually disjoint sets, with R

1

∪ R

2

∪ R

3

= D

n

, and such that: R

1

is the set of n-tuples that (we know that) belong to R, R

2

is the set of n-tuples that

(we know that) do not belong to R, and R

3

is the set of n-tuples for which we do not

know whether they belong or not to R. (Note that if R

3

is empty, R is a usual n-place

relation which can be identified with R

1

.)

However, in order to represent the information about the domain under consid-

eration, we need a notion of structure. The following characterization, spelled out in
terms of partial relations and based on the standard concept of structure, is meant to
supply a notion that is broad enough to accommodate the partiality usually found in
scientific practice. Partial relations do the main work, of course. A partial structure S
is an ordered pair

D, R

i

i

∈I

, where D is a non-empty set, and (R

i

)

i

∈I

is a family of

partial relations defined over D.

3

Two of the three basic notions of the partial structures approach are now defined.

In order to spell out the last, and crucial one—quasi-truth—an auxiliary notion is
required. The idea is to use, in the characterization of quasi-truth, the resources sup-
plied by Tarski’s definition of truth. However, since the latter is only defined for full
structures, we have to introduce an intermediary notion of structure to “link” full to
partial structures. And this is the first role of those structures that extend a partial
structure A into a full, total structure (which are called A-normal structures). Their
second role is purely model-theoretic, namely to put forward an interpretation of a
given language and, in terms of that interpretation, to characterize basic semantic
notions. A-normal structures are defined as follows: Let A

= D, R

i

i

∈I

be a partial

2

Further developments and applications of the partial structures approach can also be found, for

example, in da Costa and French (1993a, b, 1995, 2003), da Costa, Bueno, and French (1998), Bueno
(1997, 1999a, b, 2000, 2002b), and Bueno and de Souza (1996).

3

Note that if partial relations and structures are partial, this is due to the incompleteness of our

knowledge about the domain under investigation. Given further information about this domain, a
partial relation may become total. Hence, the partiality modeled by the partial structures approach is
not understood as an intrinsic, ontological partiality in the world. In other words, we are concerned
here with an epistemic, not ontological, partiality.

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Synthese (2007) 154:383–399

structure. We say that the structure B

= D

, R

i

i

∈I

is an A-normal structure if (i)

D

= D

, (ii) every constant of the language in question is interpreted by the same

object both in A and in B, and (iii) R

i

extends the corresponding relation R

i

(in the

sense that each R

i

, supposed of arity n, is defined for all n-tuples of elements of D

).

Note that, although each R

i

is defined for all n-tuples over D

, it is known to hold for

some of them (the R

i1

-component of R

i

), and it’s not known to hold for others (the

R

i2

-component).

As a result, given a partial structure A, there may be too many A-normal struc-

tures. Suppose that, for a given n-place partial relation R

i

, we don’t know whether

R

i

a

1

. . . a

n

holds or not. One way of extending R

i

into a full R

i

relation is to look

for information to establish that it does hold, another way is to look for the contrary
information. Both are prima facie possible ways of extending the partiality of R

i

. But

the same indeterminacy may be found with other objects of the domain, distinct from
a

1

,

. . . , a

n

(for instance, does R

i

b

1

. . . b

n

hold?), and with other relations distinct from

R

i

(for example, is R

j

b

1

,

. . . , b

n

the case, with j

= i?). In this sense, there are too many

possible extensions of the partial relations that constitute A. We need then to provide
constraints to restrict the acceptable extensions of A.

In order to do that, a further auxiliary notion is introduced (see Mikenberg et al.,

1986). A pragmatic structure is a partial structure to which a third component has been
added: a set of accepted sentences P, which represents the accepted information about
the structure’s domain. (Depending on the interpretation of science that is adopted,
different kinds of sentences are introduced in P: realists will typically include laws
and theories, whereas empiricists will add certain laws and observational statements
about the domain in question.) A pragmatic structure is then a triple A

= D, R

i

, P

i

∈I

,

where D is a non-empty set, (R

i

)

i

∈I

is a family of partial relations defined over D,

and P is a set of accepted sentences. The idea is that P introduces constraints on the
ways that a partial structure can be extended (the sentences of P hold in the A-normal
extensions of the partial structure A

).

The conditions for the existence of A-normal structures can now be spelled out

(see Mikenberg et al., 1986). Let A

= D, R

i

, P

i

∈I

be a pragmatic structure. For each

partial relation R

i

, we construct a set M

i

of atomic sentences and negations of atomic

sentences, such that the former correspond to the n-tuples that satisfy R

i

, and the

latter to those n-tuples that do not satisfy R

i

. Let M be

∪

i

∈I

M

i

. Therefore, a pragmatic

structure A admits an A-normal structure if, and only if, the set M

∪ P is consistent.

Let’s assume that the conditions for the existence of A-normal structures are met.

We can now formulate the concept of quasi-truth. A sentence

α is quasi-true in A

according to B if (i) A

= D, R

i

, P

i

∈I

is a pragmatic structure, (ii) B

= D

, R

i

i

∈I

is an A-normal structure, and (iii)

α is true in B (in the Tarskian sense). If α is not

quasi-true in A according to B, we say that

α is quasi-false in A according to B. More-

over, we say that a sentence

α is quasi-true if there is a pragmatic structure A and a

corresponding A-normal structure B such that

α is true in B (according to Tarski’s

account). Otherwise,

α is quasi-false.

The idea, intuitively speaking, is that a quasi-true sentence

α does not describe the

whole domain to which it refers, but only an aspect of it—the one modeled by the
relevant partial structure A. After all, there are several different ways in which A can
be extended to a full structure, and in some of these extensions

α may not be true.

As a result, the notion of quasi-truth is strictly weaker than truth: although every true

Synthese (2007) 154:383–399

389

sentence is (trivially) quasi-true, a quasi-true sentence is not necessarily true (since it
may be false in certain extensions of A

).

4

An additional feature of the partial structures approach is that it allows for the

representation of more open-ended relations between theories. Sometimes, the rela-
tion between two theories, T

1

andT

2

, is one of embedding, where models of T

1

are

isomorphic to certain sub-models of T

2

. But often only a partial embedding can be

established between such theories, in which case models of T

1

are only partially iso-

morphic to certain sub-models of T

2

. The relevant notion of partial isomorphism,

which generalizes the usual notion of isomorphism, can be represented via partial
structures as follows (see Bueno, 1997; French & Ladyman, 1999).

Let S

1

= D, R

i

and S

2

= D

, R

i

be two partial structures, where for every

i

∈ I, R

i

= R

1

, R

2

, R

3

and R

i

= R

1

, R

2

, R

3

are, for example, binary partial rela-

tions. We say that a (partial) function f : D

→ D

is a partial isomorphism between

S

1

and S

2

if (i) f is bijective, and (ii) for every x and y

∈ D, R

1

xy

↔ R

1

f

(x)f (y) and

R

2

xy

↔ R

2

f

(x)f (y). (So, when R

3

and R

3

are empty, that is, when we are considering

total structures, we have the standard notion of isomorphism.)

There are cases, however, in which even partial isomorphism is too strong a relation

between the models of the theories in question. For example, the models in question
may not have the same cardinality. In this case, the notion of a partial homomorphism
is more adequate (see Bueno, French, & Ladyman, 2002). We say that a (partial)
function g : D

→ D

is a partial homomorphism between S

1

and S

2

if for every x

and y

∈ D, R

1

xy

→ R

1

g

(x)g(y) and R

2

xy

→ R

2

g

(x)g(y). (Again, if we only consider

total structures, the usual notion of homomorphism is obtained.)

To illustrate the use of these notions, let us consider a simple example. As is well

known, Newtonian mechanics is appropriate to explain the behavior of bodies under
certain conditions (say, bodies which, roughly speaking, have “low” velocity, are not
subject to strong gravitational fields etc.). But with the formulation of special relativ-
ity, we know that if these conditions are not satisfied, Newtonian mechanics is false.
In this sense, these conditions specify a family of partial relations, which delimit the
context in which the theory holds. Although Newtonian mechanics is not true (and
we know under what conditions it is false), it is quasi-true; that is, it is true in a given

4

It may be argued that because quasi-truth has been defined in terms of full structures and the stan-

dard notion of truth, there is no gain with its introduction. In our view, there are several reasons why
this is not the case. Firstly, as we have just seen, despite the use of full structures, quasi-truth is weaker
than truth: a sentence which is quasi-true in a particular domain—that is, with respect to a given
partial structure A—may not be true if considered in an extended domain. Thus, we have here a sort
of underdetermination, involving distinct ways of extending the same partial structure. Secondly, one
of the points of introducing the notion of quasi-truth is that in terms of this notion and the concept of
partial structure, a formal framework can be advanced to accommodate the openness and partialness
typically found in scientific practice (see da Costa & French, 1989, 1990, 1993a, 1993b, 2003). Bluntly
put, the actual information at our disposal about a certain domain is modeled by a partial (but not
full) structure A. Full, A-normal structures represent ways of extending the actual information which
are possible according to A. In this respect, the use of full structures is, basically, a semantic expedient
of the framework (in order to provide a definition of quasi-truth), but no epistemic import is assigned
to them. After all, typically, we are only dealing with partial information about the world. Thirdly,
full structures can be ultimately dispensed with in the formulation of quasi-truth, since the latter can
be characterized in a different way, in terms of quasi-satisfaction. This formulation preserves all the
features of quasi-truth and is independent of the standard Tarskian type account of truth (see Bueno
& de Souza, 1996). This provides, of course, the strongest argument for the dispensability of full struc-
tures (as well as of the Tarskian account) vis-à-vis quasi-truth. Therefore, full, A-normal structures
are entirely inessential; their use here is only a convenient device.

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Synthese (2007) 154:383–399

context, determined by a pragmatic structure and a corresponding A-normal one (see
da Costa & French, 1993a).

This example can be explored a little further (see Bueno, 2000; Post, 1971). We

can think of classical physics as constituted by two quite distinct parts: on the one
hand, Newtonian mechanics (formulated in a Hamiltonian way), and on the other
hand, Maxwell’s electromagnetic theory (which includes optics). Now, these two the-
ories are distinct because they are invariant under different transformations: whereas
Newtonian mechanics is Galileo-invariant, Maxwell’s theory is invariant under a
Lorentz transformation. Hence, it is not possible to unify these theories, since as
Post indicates, “unification in one coordinate system would be destroyed in another”
(1971, p. 223). It is therefore not surprising that, when Einstein formulated the spe-
cial theory of relativity, he had to reject part of the classical framework. In the end,
he articulated mechanics in a Lorentz-invariant way (bringing along, in particular,
the law that any curvature in the relevant structure implied time-dependence), and
rejected Galileo-invariance. However, his choice was by no means arbitrary, since the
Lorentz transformation degenerates into the Galileo transformation for v

/c → 0, but

not vice-versa (Post, 1971, p. 234). In other words, with the shift from classical physics
to special relativity, some structure is straightforwardly preserved (through Lorentz-
invariance), but only some, since Galileo-invariance holds only in limiting cases. So,
we have at best a partial embedding of classical physics’ models into models of special
relativity (the partial isomorphism preserves Newtonian mechanics structures only in
limiting cases).

5

An important feature to note here is that a sentence and its negation can be

both quasi-true. Of course, inconsistent sentences are not quasi-true in the same A-
normal structure, but they can still be both quasi-true—as long as they are true in
some A-normal structure. In other words, as defined above, if a theory is quasi-true,
it is consistent (given that it is true in some full A-normal structure). But in some
contexts, we may need to assert that an inconsistent theory is quasi-true. How can we
do that?

Here is a way. If a theory T is inconsistent, we say that T is quasi-true in a partial

structure A if there are “strong” subsets of T’s theorems that are true in some
A-normal structure. (We take ‘strong’ to be a pragmatic notion, involving theories
that are explanatory, have significant consequences, accommodate the relevant phe-
nomena etc.) In general, there are infinitely many “sub-theories” of T that meet this
condition. Of course, the interesting cases to consider are those in which A is a “good”
pragmatic structure, in the sense that it reflects well the informal counterpart of T.

5

As is well known, the symmetry group of Galilean space-time is a 10-parameter Lie group with ten

independent generators: it is just O

3

× T (which provide the translations and rotations) plus the addi-

tional generators of the Galilean transformations (see Friedman, 1983, pp. 87–92). Now, the symmetry
group of special relativistic kinematics is again a 10-parameter Lie group with ten independent gener-
ators: the same O

3

×T plus the additional generators of the Lorentz transformations (see Friedman,

1983, pp. 125–138). So, the transformation groups are indeed different (although they share seven of
the ten generators, namely O

3

× T). But what are we doing when we consider the limit when v/c → 0?

In a sense, we are providing conditions to constrain the models of Minkowski space-time, such that
under these conditions (in the limit), we have the desired agreement with Galilean space-time. The
Galilean and the Lorentz symmetry groups already share a great deal of structure; what these condi-
tions do is to constrain the structure that they don’t share. In the limit, part of the Galilean space-time
would be embedded into the Minkowski one—but only part of it (a slice of it, as it were). And that’s
why we only have a partial embedding.

Synthese (2007) 154:383–399

391

For example, let T be naïve set theory, formulated in first-order logic. In this case,

in the pragmatic structure A, the set P of basic statements is constituted by statements
that are typically taken to be unproblematic, such as the statement that asserts the
existence of the union of two sets and the statement that expresses the separation
schema restricted to a given set. In this case, there is only one relation in A, the mem-
bership relation, which is taken to hold for certain pairs of sets. Hence, T is quasi-true
in A, given that there are several sub-theories of T that are quasi-true, for instance,
Zermelo–Fraenkel set theory, Quine’s NF and ML, and von Neumann–Bernays–
Gödel set theory.

Of course, this construction presupposes, for the usual reasons, a metatheory that is

strong enough. Moreover, the construction is formulated in classical first-order logic,
but it can be easily extended to higher-order logics, as in Frege’s system, or to other
logics, using the theory of valuations (see da Costa, 1997).

Two points should be emphasized here: (i) The fact that inconsistent theories can

be quasi-true doesn’t entail that every sentence is quasi-true. After all, given a partial
structure A, there exist sentences that aren’t true in any A-normal structure. (ii) The
fact that inconsistent theories can be both quasi-true also doesn’t mean that every-
thing follows from the partial structures framework. After all, the logic of quasi-truth
is paraconsistent (see da Costa et al., 1998). And as is well known, in a paraconsistent
setting, it’s not the case that everything follows from an inconsistency. As a result, the
partial structures approach provides the right sort of framework to examine issues
regarding inconsistency in science. In terms of the approach, it’s possible to represent,
without triviality, inconsistent theories as being quasi-true.

Having said that, we can now return to the main issue under consideration, and

discuss how the partial structures approach allows us to consider it in a new way.

4 Modeling scientific rationality in inconsistent contexts

Using quasi-truth and partial structures, we can put forward a model of scientific
rationality with the following features (see also da Costa & French, 2003):

(a) Aim of science. Science aims at developing quasi-true theories. These theo-

ries are possibly true, given current accepted information about a particular domain.
But such theories may turn out to be false if we extend them beyond their proper
domains. A proper domain for a theory is a domain such that (i) the theory was orig-
inally constructed to accommodate it, or (ii) the theory has the right sort of structure
to accommodate the information about the domain, even though the theory wasn’t
originally formulated to accomplish that. (These domains are all described by partial
structures.) So, quasi-true theories accommodate their proper domains, but they may
fail when extended beyond them. For example, Newtonian mechanics is quasi-true,
given that it yields the correct results when we consider domains in which the objects
don’t move very fast (in comparison to the speed of light) and are not subject to strong
gravitational fields etc. These are the theory’s proper domains.

(b) Continuity without cumulative development. By partially embedding a quasi-

true theory into another, it’s possible to preserve some structures in scientific change
(see Bueno, 2000, 2002b). However, given that typically there’s only a partial embed-
ding between such theories, some structure is also lost. In this sense, the model accom-
modates the lack of cumulative developments in scientific development (losses are an
inevitable outcome of scientific change). But some structure is still preserved via the

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partial isomorphisms. In this sense, there is room for continuity without cumulative
development. And the case of Newtonian mechanics and relativity theory mentioned
above clearly illustrates this.

As a result, the partial structures approach provides a framework that allows one to

reassess the very notion of revolutionary change in science. There certainly are changes
in science, but they need not be taken to be revolutionary in the dramatic sense that
there is absolutely nothing in common between two rival theories. Despite the exis-
tence of losses in scientific development, some structure is typically preserved—if only
partially.

(c) Accommodating inconsistencies. Since a particular theory and its negation can

be both quasi-true, the approach has no difficulty in accommodating inconsistent
theories in science. To do that, as we noted, the underlying logic has to be paracon-
sistent, given that scientists don’t derive everything from an inconsistent theory. As
we pointed out, there’s no difficulty in accommodating these features in the present
framework, given that the logic of quasi-truth is paraconsistent.

The possibility of accommodating inconsistencies allows us to use the present

framework to generalize an important aspect of van Fraassen’s model of rationality.
Van Fraassen is certainly right in stressing that it’s rational to believe anything that
we are not rationally compelled to disbelieve. But, in some contexts, we might not be
rationally compelled to disbelieve inconsistent theories—as long as they are quasi-
true, and there’s no better consistent alternative available. We are certainly rationally
compelled to disbelieve what leads to triviality, given that even on a paraconsistent
framework, a trivial theory is completely uninformative. Clearly, there’s no reason
to accept such a theory, let alone believe in it. However, in a paraconsistent context,
an inconsistent theory need not be trivial. And so, in principle, we may have reason
to entertain inconsistent theories—particularly if they are quasi-true as well, and no
consistent alternatives have been presented. Moreover, it’s often the case that by
exploring further an inconsistent theory, consistent alternatives to that theory can
be developed (see Costa & French, 1993a, 2003). And so, the study of inconsistent
theories can be heuristically fruitful.

Given that the underlying logic of quasi-truth is paraconsistent, there’s no diffi-

culty in making sense of this situation. So, it’s indeed rational to believe anything that
we are not rationally compelled to disbelieve, and this may even include inconsistent
theories. In this way, the present approach can explain why it’s not irrational to pursue
inconsistent theories in science, provided such theories are quasi-true and no better
consistent alternatives are available. We take this to be a significant feature, given that
inconsistent theories are often pursued in scientific practice.

But does this mean that we now have reason to believe that inconsistent theories

are true? We don’t think so. As noted above, inconsistent theories are taken to be
quasi-true at best. And that’s why we talk about pursuing, exploring these theories
further, rather than claiming that they are true. Similarly, it’s not clear to us that
we have reason to believe in true contradictions about the world (Priest, 1987). The
best attitude here is to be agnostic about them: maybe there are such contradictions,
maybe there aren’t (see da Costa & Bueno, 1996). But nothing in the practice of
science requires the existence of true contradictions, and typically, scientists refrain
from asserting the truth of inconsistent theories. To assert their quasi-truth is enough.

Of course, by asserting only the quasi-truth of an inconsistent theory, we are not

committed to the existence of true contradictions, given that the truth of the the-
ory is never asserted. But it might be argued that the partial structures framework

Synthese (2007) 154:383–399

393

presupposes realism about partial relations and structures. Hence, the objection goes,
contradictions, even quasi-true ones, arguably express properties of inconsistent ob-
jects. In response, note that although a realist reading of the partial structures frame-
work is possible, such a reading is not required by the formalism, given that a perfectly
neutral interpretation is also available. For example, the framework could be inter-
preted modal-structurally, in the sense that the objects in terms of which partial
relations are characterized are just positions in structures, and the latter structures are
just ways of expressing modal claims. For instance, instead of asserting that there are
infinitely many prime numbers, the modal-structural interpretation states (roughly)
that: if there were Peano Arithmetic structures, then there would be infinitely many
prime numbers, and it’s possible that there are Peano Arithmetic structures (see Hell-
man, 1989). In particular, the objects in question (prime numbers) have a role only
in the context of such structures, and they have no independent existence beyond that
context. As a result, on this interpretation, no true contradictions and no inconsis-
tent objects are necessary, and the ontological neutrality of the present proposal is
preserved.

6

We can now return to the four main components of rationality that we discussed

above, and see how the approach developed here fares with respect to them.

(a) The logical component. As we saw, a crucial component of rationality is the

ability to make derivations, but this requires a logic. We don’t think, however, that
there’s one true logic that can be used in all contexts. Depending on the domain
under consideration, different logics are adequate for the task. In other words, we
favor a logical pluralist view (see da Costa, 1997; Bueno, 2002a). On our view, there
are different logical consequence relations, depending on the context in which logic
is applied. For example, in a context in which we aim at modeling constructive fea-
tures of mathematical reasoning, classical logic is not adequate, but one of various
intuitionistic logics might be. In inconsistent contexts, to avoid triviality and the loss
of information, one of several paraconsistent logics are adequate. These examples
illustrate that, in a given context, more than one logic can be appropriate. That’s the
content of logical pluralism as we conceive of it.

So, the logical component is satisfied in an “open-ended” way. Different contexts

will have different logics. But the fact that typically there is a logic is enough to meet
the logical component. The partial structures framework—with the introduction of
partial relations and partial structures—is, of course, well equipped to accommodate
this aspect of rationality.

(b) The inductive component. A variety of inductive procedures are often used to

generate additional information about a given domain. So, similarly to what goes on
with the logical component, we also favor a pluralist view. As is well known, there
are several distinct statistical theories, and the physicist should be able to choose
between them. The choice is, ultimately, articulated in terms of pragmatic factors, and
similarly to what goes on with regard to logic, there’s no unique statistical theory that
is adequate across all domains. Different packages are articulated—using different
statistical theories and the relevant scientific theories and data—and the packages
that solve more problems in a better way end up being preferred. However, typically,
more than one package does the trick. Hence, we have again a pluralist picture. This
pluralism can be readily accommodated in terms of the partial structures approach,
given that for the same partial structure A that represents, say, statistical information

6

We owe this point to an anonymous referee.

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Synthese (2007) 154:383–399

about a given domain, there are different A-normal structures that extend that partial
information to a total one. Different statistical theories are used to inform the con-
struction of different A-normal structures. And this generates the pluralism about the
inductive component typical of the partial structures approach.

7

(c) The critical component. Criticism is a crucial feature of rationality. But to criti-

cize any proposal, one needs to use logic. Given logical pluralism, which logic should
be used? Of course, for the logical pluralist, there’s no unique answer to this question.
Different contexts have different logics, and the choice between them is ultimately
made in terms of pragmatic factors as well. Again, we contrast a package that includes
a given logic plus the relevant scientific or mathematical principles with another pack-
age that contains the same principles but a different logic. By comparing the various
consequences of each package, it’s possible to determine the adequacy and limitations
of each of them. In this way, criticism can be articulated.

But if to choose a logic we will need to use logic, don’t we have a regress here? Well,

we would have a regress if we required—as the logical monist does—that there exists
only one true logic. Given that no such requirement is found in the logical pluralist
perspective, the regress can be easily stopped by simply accepting a logic in a given
context as a tool of criticism. Now if the use of such logic itself becomes contentious,
we have a new context, where a new tool of criticism (a new logic) is adopted to deter-
mine the adequacy (or not) of the previous logic for the task under consideration.
In this way, the main benefit of the logical monist picture can be obtained—namely,
always having an underlying logic to make derivations, and using this logic as a tool
of criticism. But the accompanying costs of logical monism are not to be found. In
particular, it’s not clear how the logical monist can justify the truth of that one true
logic. After all, given any logic that is taken to be true (or at least adequate to a
given domain), there are always rival logics that agree with the logic in the context
at hand but disagree elsewhere, in the sense that they yield different consequences
outside the domain. As a result, more than one logic turn out to be adequate for that
domain. Clearly, this is not a problem for the logical pluralist, given that the latter isn’t
committed to the existence of the one true logic. As a result, the critical component
can be met in the pluralist way recommended by the partial structures approach.

(d) The goal-oriented component. Rationality depends on goals. In science, such

goals involve some truth-related notion, or the discovery of some regularity about
the world. As noted above, this aspect of rationality can be accommodated by under-
standing the aim of science as the search for quasi-true theories. These are theories
that correctly describe certain aspects of a given domain, and that might well be true,
given current information about the domain in question. As a result, the stability of
the aim of science is obtained, given that, with respect to a given partial structure, a
quasi-true theory will always be quasi-true.

In this way, the partial structures approach provides a fruitful model of scientific

rationality, satisfying the main components of rationality, while still making sense of
scientific change and being able to accommodate inconsistencies in science.

7

This picture becomes still more interesting with the introduction of pragmatic probability: the

probability of the quasi-truth of a given theory, rather than its truth (see da Costa, 1986). But here is
not the place to develop this point further.

Synthese (2007) 154:383–399

395

5 Scientific rationality and the foundations of science

Having outlined the partial structures model of scientific rationality, we want to close
this paper by applying the model to provide a new approach to debates regarding the
foundations of science and of knowledge more generally. As is well known, there are
two main views to consider. According to foundationalism, knowledge has a founda-
tion to which all knowledge claims should be ultimately reduced. The main reason
for one to endorse foundationalism is usually to avoid the infinite regress of reasons.
Given that the items in the foundation are taken to be self-justified, or at least they
need not be warranted in terms of any other items, once the process of justification
reaches the foundation, as it were, it stops. Of course, the main challenge to founda-
tionalism is to provide an account of what is special about the foundation, and how
the latter actually has the self-justifying properties it is supposed to have.

Coherentism is then articulated as an alternative that doesn’t face the problems of

foundationalism. Instead of postulating a basis from which all knowledge should be
ultimately derived, the coherentist insists that knowledge should be conceptualized as
a network of interrelated beliefs, none of which being foundational. Warrant is then
the result of the mutual support that each belief in the network provides to each other.
Given that no particular belief has any special justificatory role, any belief is open to
revision. Justification is then an “emergent property” of the whole network. A belief
is then justified if it is part of a coherent belief system. In turn, as BonJour articulates
the view, a belief system is coherent if (i) it is consistent, (ii) its beliefs are logically
interconnected, (iii) it is explanatory, (iv) it is open to conceptual change, and (v) it is
empirically adequate (see BonJour, 1985).

It’s interesting to note that, in BonJour’s characterization, consistency is a neces-

sary condition for coherence. Although BonJour is certainly not alone in making this
demand, it is unclear to us that the demand is justified. Just as consistency is not a nec-
essary condition for rationality, it is not clear that it should be a necessary condition for
coherence either. For example, Frege’s reconstruction of arithmetic in second-order
logic was clearly coherent, even though it wasn’t consistent. There’s no doubt that in
Frege’s logicist system, the various components of the approach (definitions, principles
and theorems) are logically connected. For instance, Hume’s principle is derived from
Basic Law V, and from the former, several properties of natural numbers are obtained
(for references and further details, see Boolos, 1998). Frege’s system was also clearly
explanatory. As Frege argued, once arithmetic is reconstructed in terms of logic and
definitions, it becomes clear why our knowledge of mathematical truths as well as
the application of arithmetic is unproblematic. Frege’s system was certainly open to
conceptual change (as recent work by Wright, Boolos and others clearly show). And
it was “empirically adequate”, at least in the sense that it captures the right features of
natural numbers on a logicist basis. Clearly, Frege’s system was coherent, even though
it was not consistent.

The point is not restricted, of course, to Frege’s system. In fact, a number of incon-

sistent mathematical or scientific theories are coherent. This is the case, for example,
of the original formulation of the calculus, Bohr’s atomic model, and the conjunction
of general relativity theory and quantum mechanics. So, what is needed is an account
of coherence that doesn’t require consistency.

The partial structures approach allows us to meet this demand. In terms of this

approach, it is possible to provide a broader notion of coherence for which consis-
tency is not a requirement. We say that a belief system S is coherent if (i) S is quasi-true,

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(ii) the beliefs in S are logically interconnected, (iii) S has explanatory power, (iv) S
is open to conceptual change, and (v) S is empirically adequate. Let us elaborate on
each of these conditions.

(i) Quasi-truth. By requiring quasi-truth to be a condition for coherence, we high-

light the fact that the norm of scientific inquiry—the goal-oriented component of
scientific rationality—is also a requirement for a belief system to be coherent. There
are several advantages to this move. As noted above, by having quasi-truth as a norm,
it’s possible to make room for inconsistent theories in science—something that is ruled
out a priori with the consistency requirement. Moreover, it’s also possible to make
sense of the fact that inconsistent theories may be coherent, as the case of Frege’s
system clearly suggests. Thus, we can accommodate an important aspect of scientific
and mathematical practice.

(ii) Logical interconnection. A belief system needs to be logically interconnected

to be coherent. The logical interconnections between the various beliefs correspond
to the logical component of rationality. Different logics, of course, determine differ-
ent logical connections between beliefs. And the logical pluralist picture we favor
meshes nicely with this component of coherentism. An additional way of expressing
the logical interconnections between the various components of a belief system is
by identifying partial morphisms between the several components of the system (e.g.
partial isomorphisms or partial homomorphisms between the models that describe
the phenomena). The more such partial morphisms there are, the more coherent the
resulting system will be.

(iii) Explanatory power. It’s reasonable to require that the beliefs of a coherent sys-

tem be structured in such a way that they are explanatory. For the presence of explan-
atory relations clearly indicates an additional degree of interconnection between the
beliefs of a system. Interestingly, the notion of explanation can be characterized in
terms of partial structures by insisting that explanatory relations are, in part, relations
between models. In particular, by partially embedding some structures into others, at
least a partial explanation is provided. In other words, we can understand why a given
result holds given that it’s part of a broader set of (partial) relations. For example,
understanding explanation in terms of partial structures allows one to make sense
of why it’s possible to explain the success of Newtonian mechanics (in its domain)
in terms of relativity theory. After all, it’s possible to partially embed models of the
former into the latter, and so we can understand under which conditions the results
from Newtonian mechanics obtain and under which they don’t.

(iv) Conceptual change. The possibility of being open to conceptual change is an

important feature of a coherent system. After all, new evidence will always be gath-
ered and the system should have resources to accommodate it. It’s thus important
to highlight that a system of partial structures is always open to conceptual change
given that, with new evidence, partial relations can be “moved around”, and R

3

rela-

tions about which we don’t know whether they hold or not, are shifted to R

1

or R

2

relations. Moreover, the presence of various partial morphisms (partial isomorphism,
partial homomorphism) between the components of a belief system also makes room
for conceptual change. After all, given that such mappings are only partial, if new
relations are established, there is always room to reassign the mappings in question.
In this way, by partially mapping some partial structures into others, it’s possible to
accommodate changes in the relations under consideration.

(v) Empirical adequacy. Because fictions can be perfectly coherent, it’s crucial

to require that any coherent belief system be empirically adequate. Otherwise, it

Synthese (2007) 154:383–399

397

would be impossible for the coherentist to distinguish coherent facts from coherent
fictions. It’s important to note that the notion of empirical adequacy can be easily
characterized in terms of partial structures. A theory is empirically adequate if it is
quasi-true with respect to the partial structures that represent the phenomena (see
Bueno, 1997). Alternatively, a theory is empirically adequate if there is a partial iso-
morphism between the models of the phenomena and the empirical substructures of
a given model of the theory (see Bueno, 1997). This generalizes to contexts involving
partial relations the account of empirical adequacy provided by van Fraassen (see van
Fraassen, 1989, p. 64). Whereas van Fraassen insists on the existence of a full isomor-
phism between the empirical substructures and the phenomena as a requirement for
empirical adequacy, we think that the broader notion of partial isomorphism works
better. After all, requiring the existence of a partial isomorphism between the rele-
vant models yields an account of empirical adequacy that meshes nicely with scientific
practice, where partial information is the norm rather than the exception.

In this way, the partial structures approach provides a new, more general, char-

acterization of the notion of coherence. This characterization agrees with the more
familiar version of coherentism when we consider only full structures. After all, with
only full structures under consideration, all the notions invoked in the partial struc-
tures approach—partial relations, partial isomorphism, partial homomorphism etc.—
are reduced to the corresponding full notions. Moreover, the characterization also
accommodates inconsistent theories in science, given that, as we saw, a theory and
its negation can be both quasi-true. As a result, the characterization above allows for
inconsistent theories to be coherent, a crucial feature to accommodate the rationality
of scientific and mathematical practice.

But coherentism is not the only position that can be reformulated in terms of par-

tial structures. The same goes for foundationalism. In one interpretation of the partial
structures approach, it’s possible to capture an important aspect of foundationalism.
In a pragmatic structure A

= D, R

i

, P

i

∈I

, one can interpret the set of accepted sen-

tences P as being true—in the correspondence sense. The content expressed by the
sentences in P can then be interpreted as the foundations, in the sense that any accept-
able extension of A needs to preserve the truth of the sentences in P. In this way, the
role played by the foundations in the foundationalist picture—the stopping place on
which every justification has to rely—can be captured, in part, by the partial structures
approach. (Of course, this is only the case in a particularly strong interpretation of
the partial structures formalism.)

Hence, using partial structures and quasi-truth, it’s possible to capture crucial

features of foundationalism and coherentism in a single framework, identifying the
attractive aspects of each proposal. As a result, by developing a broader, more lenient,
approach to scientific rationality—in terms of partial structures and paraconsistent
logic—it’s possible to develop a broader, more lenient, view about the foundations of
science.

6 Conclusion

As argued above, the two main features of the problem of scientific rationality can
be accommodated in terms of the partial structures framework. (a) It is possible to
develop an account that makes sense of scientific revolutions, but which still pre-
serves some (partial) relations between old and new theories. The existence of the

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latter relations help to explain why the break between different theories, although dra-
matic in some cases, is never too drastic as to make it impossible for one to understand
the process in perfectly rational terms.

(b) Moreover, if scientific theories are taken to be quasi-true (and given that the

underlying logic is paraconsistent), we can see how scientists and mathematicians can
entertain inconsistent theories without triviality. In this way, as opposed to what the
traditional approaches to rationality demand, it’s not irrational to entertain inconsis-
tent theories—as long as they are quasi-true.

As a result, (c) the view that emerges provides a new way of thinking about the

foundations of science. In particular, it extends in important respects both coheren-
tist and foundationalist approaches to knowledge, without the troubles that plague
traditional views of scientific rationality.

Acknowledgements

Our thanks go to Steven French and Décio Krause for many illuminating

discussions on the issues explored here, and to two anonymous referees for their helpful comments
on an earlier version of this work.

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