Quasi-truth and defective knowledge in science: a critical examination

Abstract

Quasi-truth (a.k.a. pragmatic truth or partial truth) is typically advanced as a framework accounting for incompleteness and uncertainty in the actual practices of science. Also, it is said to be useful for accommodating cases of inconsistency in science without leading to triviality. In this paper, we argue that the formalism available does not deliver all that is promised. We examine the standard account of quasi-truth in the literature, advanced by da Costa and collaborators in many places, and argue that it cannot legitimately account for incompleteness in science. We shall claim that it conflates paraconsistency and paracompleteness. It also cannot properly account for inconsistencies, because no direct contradiction of the form S ∧ ¬S can be quasi-true according to the framework; contradictions simply have no place in the formalism. Finally, we advance an alternative interpretation of the formalism in terms of dealing with distinct contexts where incompatible information is dealt with. This does not save the original program, but seems to make better sense of the apparatus.

Keywords:
Quasi-truth; Pragmatic theories of truth; Inconsistency; Incompleteness

1 Introduction

Scientific theories and scientific knowledge, in general, may be said to be defective in a multitude of ways. Our abilities to generate knowledge are known to be less than perfect, and we are frequently found to be holding mistaken views about the nature of reality. Of course, science itself is self-correcting, but even the optimistic hopes for a final correct and true theory cannot avoid the fact that our situation is less than perfect on what concerns our actual theories.

One of the senses in which current science is defective concerns the fact that it is not complete, in an important sense of ‘complete’. There is much that still needs to be investigated, many open questions, and many tentative claims and theories that are not known to be true, or that are certainly false. It seems that a formal framework expecting to represent with such situations should accommodate this kind of incompleteness or openness of current knowledge. As da Newton Costa and Steven French [¹⁰, pp.13-14] put it, considering the possibility that there is an ideal limit to which scientific knowledge may converge,

If the final conception is taken to be complete or total, then our conception at any given time prior to the realization of this limit may be said to be partial. And because it is, at any given time, partial, it is, at that time, open in the sense that it may be completable in a variety of ways.

Philosophically, this incompleteness is thought of as challenging because most of our accounts of scientific theories rely on classical apparatuses that cannot straightforwardly be said to accommodate incomplete knowledge. In a nutshell, the semantic view of scientific theories, for instance, which is taken by many to be the current orthodoxy, regards scientific theories as classes of set theoretical structures (see Balzer et. al. ² and also Krause and Arenhart ¹⁷ for details on different takes on the semantic view). These structures, on their turn, are total, not partial, in the sense that properties classify each entity as either having such property or as not having it (and the same holds for relation); no openness or incompleteness allowed. This also invites some trouble when it comes to discuss the appropriate epistemic attitudes we may have towards such theories, given that in the classical framework they are treated as either completely true or else completely false. It is a matter of all or nothing, it seems.

There is, however, an alternative framework that was advanced precisely in order to deal with these situations: the concept of pragmatic truth or quasi-truth, as introduced by Irene Mikenberg, Newton C. A. da Costa, and Rolando Chuaqui ¹⁹. The account was further explored and applied by da Costa and French ¹⁰, among many others. As da Costa and French [(¹⁰, p.14] put it, the framework was advanced precisely to model the current situation in science involving lack of complete knowledge, because “[i]t is precisely this sense of partiality and openness that our account attempts to capture and further explore”. Pragmatic truth, then, promises to account for incomplete information in science, and as a byproduct deliver the benefits that are expected to result from such a more realistic characterization of science.

However, there is more than that in pragmatic truth. The view is said to deal with another kind of defective situation in science: the ones involving contradictions. As Bueno and da Costa [⁷, p.385] claim, an account of scientific rationality must explain how is it possible that scientists entertain inconsistent theories. That inconsistent theories have been frequently entertained is typically argued for by the presentation of examples of theories that were indeed entertained, and were found out to be inconsistent: the early formulation of the calculus, Bohr’s atomic model, Frege’s original logicist reconstruction of arithmetics, naïve set theory, quantum mechanics and general theory of relativity (when taken together), to mention a few. Again, the major problem consists in explaining our epistemic attitudes towards such theories, and also accounting for their apparent non-triviality. Well, pragmatic truth is said to accommodate also cases of inconsistency in science without entailing triviality (Bueno [⁶, p.275]; da Costa and French [¹⁰, chap.5]; Bueno and da Costa [⁷, p.392]). Inconsistent theories may not be true, but still be quasi-true, without logically implying everything. That makes of quasi-truth a nice formal tool for philosophers to approach current science, no?

In this paper, we shall argue that quasi-truth falls short of delivering what is promised.¹ Basically, our claim is that defective situations in science such as incompleteness and inconsistency cannot be clearly accounted for by the use of the conceptual tools provided by the notion of pragmatic truth. The view fails in providing the appropriate views of incompleteness and also of inconsistency. More than that, we shall argue that it conflates inconsistency and incompleteness, which are clearly distinct phenomena when it comes to knowledge and information. The source of such problems, we argue, comes basically from a too flexible use of the concept of quasi-truth, which in most informal uses of the conceptual framework in hand simply ignore that the concept is restricted to a given structure. Once it is recalled that a sentence may be quasi-true in a total structure relatively to a given partial structure, then, a weaker reading of quasi-truth may be provided for. This weaker reading does not do everything that quasi-truth was expected to do by its proponents, but it does a better job in making sense of the formalism.

The paper is structured as follows. In section 2 we recall the formalism of pragmatic truth as it is typically presented by da Costa and collaborators. We then present in section 3 the main problems for the view, and how it is found wanting on its own terms. One of the diagnosis of what has gone wrong is that the formalism has conflated inconsistency and incompleteness; it attempts to capture incomplete knowledge through the exhibition of contradictions. In section 4, we argue that the original formalism by da Costa may be open for a distinct, pragmatic interpretation, that is better suited for its purposes. The new account is no longer seen as admitting inconsistencies or contradictions - in fact, it avoids them -, but it is closer to what the formalism originally introduced actually does. We conclude in section 5.

2 Quasi-truth: the basic definitions

Let us begin by reviewing the basic concepts of the definition of quasi-truth or pragmatic truth, as employed in discussions by da Costa and collaborators. Basically, this kind of approach requires a detour through the Tarskian definition of truth, so that the latter plays a central role in accounting for partial truth (and this will be important for our discussion).

When investigating a domain of knowledge ∆, we formulate a conceptual scheme in order to deal with the entities in this domain and their properties and relations. This requires setting a set D, the domain of entities, which may have as elements both concrete entities, such as tracks in a cloud chamber, as well as non-directly observable posits of our theories, such as quarks and strings. Accounting for the behavior of these entities also involves devising a family K of relations and properties that hold for these entities, on which we are interested in. This will give rise to a set theoretical structure $\mathfrak{A}$ = ⟨D, R _k ⟩_k∈K (following the literature on quasi-truth, we use standard Zermelo-Fraenkel set theory with the axiom of choice in the metalanguage).

In classical structures (that is, structures in classical model theory), we typically define each R _k as being a total relation, that is, if R _k is an n-ary relation, for each n-tuple of elements of D, either R _k holds of this n-tuple, or else it doesn’t; no further options available. In science, however, there are situations in which our knowledge about the relations R _k is incomplete, so that it may not be completely specified whether some n-tuple of elements of D holds of R _k or not. That is, the relation R _k is only partially defined, because for some n-tuples it is left open whether they are related by R _k or not. The concept of partial relation was devised precisely to accommodate such cases of incomplete information.

In formal terms, a partial n-ary relation R over D is defined as a triple R = ⟨R ₁, R ₂, R ₃⟩ of sets of n-tuples of elements of D such that these three sets are mutually disjoint and their union is D ⁿ (that is, the sets are mutually exclusive and jointly exhaustive of D ⁿ ). The explanations for the division are typically framed in terms of our lack of knowledge or lack of definition (as we did, following the literature, in the previous paragraph): R ₁ is the set of n-tuples of elements of D for which we know that R holds, R ₂ is the set of n-tuples which we know not to hold of R, and R ₃ is the set of n-tuples for which we do not know whether they are R-related or not (see Bueno [⁶, p.279]). In other places, the triple division in a partial relation is explained not in epistemic terms, but in terms of ‘definition’, so that “R ₁ is the set of n-tuples that belong to R, R ₂ is the set of n-tuples that do not belong to R, and R ₃ is the set of n-tuples for which it is not defined whether they belong or not to R” (Bueno [⁶, p.279], see also da Costa and French [¹⁰, p.18]). Sometimes, the idea is further explained as follows (for binary relations, where A is used as the domain set):

R₁ is the set of ordered pairs which are satisfied by those sentences expressing the relationship between the entities concerned, R ₂ is the set of ordered pairs not satisfied by these sentences, and R ₃ is the set of ordered pairs for which it is left open whether they are satisfied . . . It is precisely this which is meant when we say that R _k is “not necessarily defined for all n _k -tuples of elements of A” (da Costa and French [¹⁰, p.19])

That is, a relation ‘not being defined’ for some n-tuples is thought to be ‘precisely’ explained in terms of a relation being ‘left open’ for those entities. Our point is that such characterizations are not clearly equivalent, given that they shift from epistemic to semantic. The intuitive interpretation that is expected to be captured by the formalism is important, however, because it reveals the intended meaning conferred to the third component of a partial relation, and because it is thought that it is precisely this component which accounts for the incompleteness of scientific knowledge.

A partial structure is a structure $\mathfrak{A}$ = ⟨D, R _k ⟩_k∈K , where D is the domain of the structure and R _k is a family of partial relations over D. But that is not enough. We also need to enrich partial structures with a set P of sentences of the language to be interpreted in the partial structure (a language of the same similarity type as that of the structure), sentences to be accepted as true in the Tarskian sense. Which sentences are to be found in P? It depends on which sentences of the theory one is willing to countenance as true in the classical, Tarskian sense. Empiricists may be willing to accept as true only empirically decidable sentences, describing the results of experiments; scientific realists may go a step further and include in P statements or laws concerning unobservable entities. The resulting structure ⟨D, R _k , P⟩_k∈K is called a simple pragmatic structure.

The rationale behind the introduction of P is simple. Given a partial structure $\mathfrak{A}$, and a partial n-ary relation R, for any n-tuple $\overrightarrow{d}$ = ⟨d ₁, . . . d _n ⟩ of elements of D, if $\overrightarrow{d}$ is in R ₃, we may extend R in two directions: according to one extension, $\overrightarrow{d}$ is in R ₁, according to another extension, $\overrightarrow{d}$ is in R ₂. That provides for too many possible extensions of a structure, given that this process may be repeated for each n-tuple and for each relation in the structure. In pragmatic structures, then, the role of P is to limit the number of extensions: “P introduces constraints on the ways that a partial structure can be extended” (Bueno and da Costa [⁷, p.388]). Informally, an extension makes for a completion of the relations, leading to a complete structure, and these extensions are allowed only in the cases where they are consistent with the sentences in P. In other words: a partial structure is legitimately converted into a complete structure only in the cases in which such process of making the third component of partial relations empty generates total structures consistent with the set P.

Now, we are almost ready to define quasi-truth. In order to do that, we need the concept of a $\mathfrak{A}$-normal structure, which is a complete or total structure associated with a partial structure $\mathfrak{A}$, in which the notion of truth is just the classical, Tarskian notion.

Given a first-order language ℒ of the same similarity type as of that of a simple pragmatic structure $\mathfrak{A}$ = ⟨D, R _k , P⟩_k∈K , in which ℒ is interpreted, and a simple pragmatic structure $\mathfrak{B}$ = ⟨D′, R′_k , P⟩_k∈K , we say $\mathfrak{B}$ is an $\mathfrak{A}$-normal structure if²:

1. D = D′,

2. every individual constant in the language is interpreted in the same object in both $\mathfrak{A}$ and $\mathfrak{B}$ ,

3. R′_k extends the corresponding R _k for a total relation,

4. $\mathfrak{B}$ is a model of P, in the sense of Tarski.

A sentence S of ℒ is quasi-true in$\mathfrak{A}$ according to $\mathfrak{B}$ iff

1. $\mathfrak{A}$ = ⟨D, R _k , P⟩_k∈K is a simple pragmatic structure,

2. $\mathfrak{B}$ = ⟨D′, R′_k , P⟩ is an $\mathfrak{A}$ -normal structure,

3. S is true in $\mathfrak{B}$ in the Tarskian sense.

If S is not quasi-true in $\mathfrak{A}$ according to $\mathfrak{B}$, then S is quasi-false in $\mathfrak{A}$ according to $\mathfrak{B}$.

Also, S is quasi-true in $\mathfrak{A}$ if there is an $\mathfrak{A}$-normal structure $\mathfrak{B}$ such that S is quasi-true in $\mathfrak{A}$ according to $\mathfrak{B}$. Otherwise, S is quasi-false in $\mathfrak{A}$.

Notice that there are two related concepts of quasi-truth being defined:

• (I) a sentence S is quasi-true in a structure $\mathfrak{A}$ according to $\mathfrak{B}$ (respectively, quasi-false in $\mathfrak{A}$ according to $\mathfrak{B}$ ), and

• (II) a sentence S is quasi-true in $\mathfrak{A}$ - no mention of which $\mathfrak{B}$ is being considered.

Bueno introduces a further definition

• (III) “we say that a sentence S is quasi-true if there is a partial structure $\mathfrak{A}$ and a corresponding $\mathfrak{A}$ -normal structure $\mathfrak{B}$ such that S is true in $\mathfrak{B}$ (according to Tarski’s account). Otherwise, S is quasi-false” (Bueno [⁶, p.280], with notation adapted for the sake of terminological uniformity).

Notice the differences between the three cases. In (I), a sentence S may be quasi-true in $\mathfrak{A}$ according to $\mathfrak{B}$, but ¬S cannot be quasi-true in $\mathfrak{A}$ according to $\mathfrak{B}$, because $\mathfrak{B}$ is a Tarskian classical structure. In (II), both S and ¬S may be quasi-true according to$\mathfrak{A}$: it is enough that S is quasi-true according to a given total structure $\mathfrak{B}$, and ¬S is quasi-true according to a distinct total structure $\mathfrak{C}$. This seems to accommodate contradictions in $\mathfrak{A}$ (more on this soon). Definition (III) introduces a predicate of quasi-truth simpliciter, not relative to any specific structure. As a result, notice, a sentence S may be quasi-true, with its negation ¬S also quasi-true: S may be quasi-true because it is quasi-true in a given partial structure $\mathfrak{A}$, and ¬S may be quasi-true because it is quasi-true in another partial structure $\mathfrak{B}$.

In summary, we have two notions of quasi-truth in a model (I and II), and a notion of quasi-truth which does not mention any specific model (III). The fact that there are three distinct concepts under the same name invites confusion when it comes to discuss the idea that partial structures and the accompanying notion of quasi-truth account for incompleteness and contradictions in science. One must always keep in mind which of these three definitions of quasi-truth one is talking about in discussing the philosophical applications of quasi-truth and whether it achieves its goals. Let us do that now.

3 Discussion

3.1 Partiality and incompleteness

Once these definitions are in place, let us check how the partial structures apparatus is supposed to deal with incompleteness and inconsistency of our theories. We begin with incompleteness and partiality (but this is directly related with inconsistency, as we shall see soon). According to da Costa and French ¹⁰, p.19] “the incomplete and imperfect nature of the majority of our representations of the world is, we claim, represented by the simple pragmatic structures just provided”. That is, the hope is that pragmatic structures somehow accommodate incomplete knowledge by the fact that some relations are partial, which lead to some sentences of the interpreted language being quasi-truth, and not the whole truth.³ However, that the partial structures approach does indeed provide for a good account of such incompleteness is something that must be argued for, and not just claimed. Let us try to enlighten this issue.

Let us begin by following how Bueno and da Costa [⁷, p.388] describe the workings of the conceptual machinery of pragmatic truth in order to capture partiality and incompleteness. Intuitively, they claim, the idea is that a quasi-true sentence S does not describe the whole domain in which it is being interpreted, but only an aspect of it, the aspect modelled by the relevant partial structure. The explanation for this claim is the following: there are different ways a partial structure may be extended, and in some of them the target sentence S may be classically true, in others classically false. Incompleteness means, then, that completion of the partial relations may be performed in distinct, incompatible, ways.

To begin with, notice that there seems to be two somehow incompatible claims about incompleteness here: first, that a quasi-true sentence is a model of a partial part of the domain, not of the whole of it; second, that incompleteness is accommodated in the account by the possibility of distinct incompatible extensions of the partial relations. Let us examine each of these claims.

The meaning of the first claim is not completely clear. The idea seems to be that a quasi-true sentence accommodates partiality because it models only the aspects of the domain that are known or defined, not the undefined or unknown parts. That is, given a partial structure $\mathfrak{A}$, a sentence involving a relation symbol R interpreted in ⟨R ₁, R ₂, R ₃⟩ describes only the behavior of those entities for which it is defined, i.e., for entities in the corresponding R ₁ or in R ₂. The behavior of the entities in R ₃ is not accounted for by the sentence, because their status is unknown or undefined. If this is what is being claimed, then, notice that this is far from incomplete or uncertain. That is, the aspects being said to be modelled by the sentence are precisely the well-known, well-defined aspects of the structure. The sentence is a complete description of that part of our knowledge of the domain of investigation that just cannot change anymore. So, it cannot be claimed that partial truth is partial because of this aspect of the model.

This kind of certainty also spreads to some of the entities that are in the R ₃ part of an interpretation of a relation symbol R, in some cases. Notice that it may well be the case that, once the set P of sentences is added to the partial structure, then, the sentences in P constrain the $\mathfrak{A}$-normal structures in such a way that, for some partial relations R, the elements of R ₃ may not be allowed to be extended consistently both to R ₁ in some of them, and to R ₂ in others. In that case, there is only one kind of extension for R consistent with P, and this is no longer a legitimate case of incomplete knowledge. That is, there are some situations in which P adds such constraints to the structure so that part of the incompleteness previously available just disappears.

As a result, the specific partiality and incompleteness that quasi-truth and partial structures are said to capture cannot concern the well-defined, well-known aspects of the relations and properties. There is nothing uncertain or partial there. In other words: in cases where we are dealing with a formula that is not making reference to entities in the R ₃ part of the model, our knowledge is not partial or incomplete. Suppose we are interested in a relation R in a partial structure $\mathfrak{A}$, and the individual constants in the sentence are interpreted by an n-tuple of objects in the domain which lies in R ₁ or in R ₂. Then, the corresponding sentence is true or false in any $\mathfrak{A}$-normal structure extending $\mathfrak{A}$. No uncertainty or incompleteness is involved here.

Let us focus on the second claim above, viz. the one to the effect that partiality is accounted for in those cases where it is left open that, for some $\mathfrak{A}$-normal structures, an n-tuple in R ₃ is extended to R ₁, and in another $\mathfrak{A}$-normal structure, to R ₂. Is it a good representation of a case of incomplete knowledge? Is incompleteness accounted for in the formal apparatus provided for?

To make our discussion simpler, and without losing generality, let us restrict ourselves to the case of a predicate symbol P, interpreted in a pragmatic structure $\mathfrak{A}$ as $P^{\mathfrak{A}}$ = ⟨P ₁, P ₂, P ₃⟩, and a single element d in the corresponding P ₃, which is the denotation of an individual constant a. It seems that the claim that our knowledge of whether the corresponding sentence Pa is the case is uncertain is modelled by the fact that one may have an $\mathfrak{A}$-normal structure where the corresponding sentence Pa is quasi-true, and another $\mathfrak{A}$-normal structure where ¬Pa is quasi-true. We just don’t know which one is the structure describing reality correctly. So, we have incomplete information, it is said.

But now, let us consider whether that really represents a case of incomplete knowledge or lack of information in the formal apparatus just presented. We begin by considering the definition of quasi-truth given by (I). When plugged to this definition of quasi-truth, the plan seems to be as follows: the incompleteness is represented in terms of the lack of information or knowledge to determine which, among at least two incompatible $\mathfrak{A}$-normal structures, is to be chosen. We know that Pa is quasi-true in an $\mathfrak{A}$-normal structure $\mathfrak{B}$, and we also know that ¬Pa is quasi-true in an $\mathfrak{A}$-normal structure $\mathfrak{C}$, but we don’t know which one is to be taken as a representation of reality; this reflects our epistemic situation of uncertainty and incomplete information, and this is represented in the framework by the fact that we have statements “Pa is quasi-true in $\mathfrak{A}$ according to $\mathfrak{B}$” and “¬Pa is quasi-true in $\mathfrak{A}$ according to $\mathfrak{C}$”.

However, if that is what is being taken to be a representation of incomplete or partial knowledge, then it is not easy to see that putting the problem in terms of quasi-truth gives us any advantage over classical logic. The classical logician could put the same situation in the same terms: we know that the sentence Pa is true (in the Tarskian sense) in a total structure $\mathfrak{B}$, and that ¬Pa is true (idem) in a total structure $\mathfrak{C}$. We just don’t know which is the case, i.e. which of the structures should be chosen or adopted. Then, the situations of uncertainty or incompleteness as described by quasi-truth (according to definition (I)) and as described by classical logic are completely parallel. There is nothing essentially different in the classical case from the quasi-truth case. Then, it could be claimed, classical logic also advances an account of our incomplete information!

But that would pervert the purposes of introducing the quasi-truth apparatus to begin with. It is preferable to claim that quasi-truth fails to account for incomplete knowledge under definition (I). Certainly, the problem is that the uncertainty or incompleteness is not represented inside the model or conceptual apparatus of quasi-truth, when the definition of quasi-truth is stated in terms of definition (I), because that definition leads us to the problem of choosing a classical, Tarskian, structure, which is a problem the classical logician also faces. The diagnosis for that may be put as follows: typically, lack of knowledge is represented inside a framework by a failure of the law of excluded middle (LEM), either in its syntactical formulation, viz. α ∨ ¬α, or else in a semantical formulation, stating that a sentence and its negation cannot both be false. One could hope that, if our knowledge of whether P holds for a in $\mathfrak{A}$ is uncertain, then, neither Pa nor ¬Pa is the case in the model. That is, LEM, in some version of it, should fail in some cases. That is typically how lack of information is accounted for.⁴ However, Pa ∨ ¬Pa is quasi-true in every pragmatic structure $\mathfrak{A}$ in relation to any $\mathfrak{A}$-normal structure (quasi-true according to definition (I)). Also, Pa ∨ ¬Pa is quasi-true in any pragmatic structure $\mathfrak{A}$ (definition II). Finally, this sentence is also quasi-true in the third sense (III). Other versions of LEM are also valid, such as the following version for a quasi-truth predicate: it is the case that for any $\mathfrak{A}$-normal structure, Pa is quasi-true, or else ¬Pa is quasi-true in $\mathfrak{A}$. Here is yet another version of LEM in semantic terms, which holds for the quasi-truth approach: for each $\mathfrak{A}$-normal structure, Pa is quasi-true or quasi-false. How can that accommodate incompleteness? The incompleteness is only accounted for outside of the model, in the terms we have already explained: we don’t know which $\mathfrak{A}$-normal structure should be chosen, the one in which Pa is true, or the one in which ¬Pa is true. But that problem is also available for the classical logician, and if that kind of problem represents incompleteness, then, the classical logician can also ‘represent’ such situations.

Perhaps the idea that one may represent incomplete information is better accounted for by definitions (II) and (III) of partial truth. It could be claimed that they represent incomplete information inside the model by allowing inconsistent sentences to be both quasi-true. For the definition (II) (quasi-truth in a pragmatic structure $\mathfrak{A}$), it is possible to say that, in cases of elements d of P ₃ named by the individual constant a,⁵ both Pa and ¬Pa are quasi-true in a pragmatic structure $\mathfrak{A}$. For (III), one ends up being able to say that both Pa and ¬Pa are quasi-true tout court. Then, for every case where information is claimed to be lacking about a sentence S, one ends up discovering that both S and ¬S are quasi-true, either in the same partial structure (definition II), or else just quasi-true in distinct partial structures (definition III). This, it could be claimed, represents the incompleteness of information.

There are good evidences that this is the intended reading of the goals of the concept of quasi-truth. It should be noticed, for instance, that da Costa and French, for instance, are really willing to allow that switching between incompleteness and inconsistency. Regarding the Bohr model of the atom, which delivered inconsistencies by selectively applying fragments of classical mechanics and Planck’s formula, they claim:

Structurally, these were both inconsistent and incomplete in that with the inconsistency we do not know whether the relevant properties and relations hold in the domain or not . . . this openness or lack of closure . . . is representable model theoretically by a partial structure. (da Costa and French [¹⁰, p.105])

That is, incompleteness is present because there is inconsistency. However, that is again a bad idea. According to this explanation, every incomplete relation generates a kind of contradictory information, in the sense that we have in fact over-determined information, not lack of information. If this is the intended representation in the partial structures approach of incomplete and uncertain information, then, it seems, it misses the target by providing a representation of cases where we have in fact a lot of information. This could be intuitively explained in terms of truth values: the account confuses cases where we have gaps (lack of truth value) with cases where we have gluts (abundance of truth values). These situations must be distinguished, because they give rise to distinct kinds of treatments. To wit, systems of logic such as FDE⁶ may be seen as having a four-valued functional semantics, with gaps and gluts clearly distinguished. Even those who prefer to deal with inconsistency in purely epistemic terms, not in terms of additional truth values, as Carnielli and Rodrigues ⁸, hold that incompleteness of information and inconsistent information are distinct phenomena. Incompleteness is thought to be accounted for by paracompleteness in the logical consequence relation, not by paraconsistency. Incompatible evidence, on the other hand, generates some kind of paraconsistency regarding the logical consequence relation. Violation of a version of the law of non-contradiction (LNC) is the wrong way to represent lack of information; in these cases, it is some version of LEM that should be violated.

Notice also that if such inconsistency as saying that Pa is quasi-true and that ¬Pa is also quasi-true represents incomplete information, then, according to the definition (III) above, the classical logician also has the resources to express this kind of incompleteness. Clearly, for most sentences S (except for logical truths and logical falsehoods), there is a classical structure which makes S true, and another classical structure which makes ¬S true. Then, classical logic would also be able to express the same situation. But that simply shows that the account, following definition (III), is inadequate.

In order to avoid such direct conflation between inconsistency and incompleteness, one could claim that definition (II) really leads us back to definition (I), given that it is presented in terms of (I), and making the definitions explicit we see that there is no real contradiction, but a kind of uncertainty over which normal structure to choose. This avoids confusing inconsistency and incompleteness, but then the claim that we have incomplete knowledge just comes back to the claim that we are uncertain about which $\mathfrak{A}$-normal structure to choose, and that is just the same situation as in classical logic, as we have already argued.

So, our preliminary conclusions may be stated as follows: in the case of definition (I) of quasi-truth, there is no representation of incomplete knowledge inside the framework, and there is no advantage over classical logic; in the case of definitions (II) and (III), the representation is inadequate for its purposes, because instead of modelling lack of information, it models excess of information by contradictory sentences being both quasi-true; (III) also has the disadvantage of being excessively general, so that if this kind of definition is allowed to account for incomplete knowledge, then classical logic may also represent incomplete knowledge.

3.2 Inconsistency

This leads us to the treatment of inconsistency by the apparatus of pragmatic truth. We have already seen that quasi-truth leads to some kinds of apparent contradictions, with both a sentence and its negation being quasi-true in some structures. Perhaps this is what da Costa and French [¹⁰, p.85], have in mind when claiming that “we offer a model theoretic account in which regarding theories in terms of partial structures offers a straightforward and natural way of accommodating inconsistency”.

In order for us to check whether this claim holds good of the pragmatic truth approach, we must first make clear what kind of contradiction or inconsistency is being dealt with here. As Priest [²⁴, p.144] has remarked, there are at least three types of inconsistency in empirical science:⁷

1. inconsistency between theory and observation,

2. inconsistency between theories,

3. inconsistency internal to the theory.

The case i), inconsistency between theory and observation, is probably not the case one is aiming at with quasi-truth. No one really wants to accommodate or entertain such cases; when observation contradicts the predictions of a theory, one typically revises the theory, or else provides for an explanation that accounts for the incompatibility, showing that what was once perceived as a conflict between theory and observation is not really so. For instance, the orbit of Uranus was not in complete agreement with Newtonian theory; however, the discrepancy was later explained with the discovery of Neptune. What matters for us here is that quasi-truth adds no special ingredient on the relation between theory and observation. Sentences of the theory are quasi-true or quasi-false, and their relation to observation is accounted for empirically, not by the apparatus of the framework.

Case ii) is also not a case to be dealt with by quasi-truth, although some examples of this kind of inconsistency have been used to motivate quasi-truth, as the case of the incompatibility between quantum mechanics and general relativity, one of the most mentioned examples of inconsistency in science. In this situation, distinct theories are considered, not the same theory. The contradiction comes from distinct sources, and the problem is not a matter of extending the same structure in distinct ways, but of unification of theories. Then, quasi-truth is of no help here too.

The only case in which quasi-truth could really be of any help is case iii), internal contradictions. However, it seems that quasi-truth is not the correct tool for that too. As Bueno and da Costa [⁷, p.390] remark (with notation modified for the sake of terminological uniformity):

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$\mathfrak{A}$ -normal structure, but they can still be both quasi-true - as long as they are true in some $\mathfrak{A}$ -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 $\mathfrak{A}$ -normal structure).

This remark is not explicit on which notion of quasi-truth is involved. Remember: when it comes to definition (I), a sentence S cannot be quasi-true in a pragmatic structure $\mathfrak{A}$ relative to a $\mathfrak{A}$-normal structure $\mathfrak{B}$ with its negation being quasi-true in the same$\mathfrak{A}$-normal structure $\mathfrak{B}$. In relation to definition (II), however, S and ¬S may be both quasi-true in the same pragmatic structure, because they are classically true in distinct$\mathfrak{A}$-normal structures. In relation to (III), S and ¬S may be both quasi-true, each in its own structure, and with distinct structures for each sentence, of course.⁸

However, despite the lack of specification, the above quote seems to put quasi-true contradictions in the correct perspective. As Bueno [⁶, p.281] indicates, if we use Q as a predicate of sentences representing quasi-truth, we may have Q(S) ∧ Q(¬S) (when quasi-truth is understood according to definitions II and III), however, we do not have Q(S ∧ ¬S) (and this holds for the three definitions of quasi-truth). The first indicates merely that a pragmatic structure $\mathfrak{A}$ may be extended in incompatible ways, so that in a $\mathfrak{A}$-normal structure $\mathfrak{B}$ it may be the case that S is quasi-true, while in another$\mathfrak{A}$-normal structure $\mathfrak{C}$, ¬S is the case. A quasi-true sentence is consistent, in the classical sense. The idea that contradictions can be accommodated in the quasi-truth apparatus comes from ignoring that a sentence and its negation are quasi-true in distinct$\mathfrak{A}$-normal structures. However, Q(S ∧ ¬S) cannot be the case, because no $\mathfrak{A}$-normal structure models a contradiction; they are Tarskian, classical structures. Then, if a theory provides for a sentence S ∧ ¬S, there is no hope of accommodating it in the apparatus of pragmatic truth.

But that means that the quasi-truth framework cannot really accommodate inconsistencies. On the one hand, a contradiction of the form S ∧ ¬S is never quasi-true, and this is precisely the kind of inconsistency one finds in cases of theories internally inconsistent. On the other hand, it is an exaggeration to claim that Q(S) and Q(¬S) is a contradiction, or lead to a contradiction. Both are true according to different$\mathfrak{A}$-normal structures.⁹ It is easy to introduce parameters to account for the apparent incompatibility: S is true according to an $\mathfrak{A}$-normal structure $\mathfrak{B}$, and ¬S is true according to an $\mathfrak{A}$-normal structure $\mathfrak{C}$. A legitimate contradiction would require Q(S ∧ ¬S) or Q(S) ∧ ¬Q(S). The first case is not allowed for, as we have already commented. Q(S) ∧ ¬Q(S) is also not possible. Given that the metalanguage is classical, ¬Q(S) means that S is not quasi-true, that is, S is quasi-false, in a given $\mathfrak{A}$-normal structure. However, no sentence can be quasi-true and quasi-false in the same $\mathfrak{A}$-normal structure. That is, the apparatus of quasi-truth fails in accommodating inconsistencies.

However, Bueno and da Costa seem to acknowledge that limitation, and attempt to overcome it. As they claim [⁷, p.390]: “But in some contexts, we may need to assert that an inconsistent theory is quasi-true. How can we do that?” That is, it is recognized that one cannot have a contradiction being quasi-true, but, anyway, some inconsistent theories (in the sense of internal inconsistency) should be said to be quasi-true. Bueno and da Costa present a solution to the problem that, curiously, does not require that we admit a contradiction S ∧ ¬S as quasi-true. Their solution to the problem is presented in a very informal way (again, adapting the notation):

If a theory T is inconsistent, we say that T is quasi-true in a partial structure $\mathfrak{A}$ if there are “strong” subsets of T′s theorems that are true in some $\mathfrak{A}$ -normal structure. (Bueno and da Costa [⁷, p.390])

The plan is explained as follows: one just selects consistent “strong” subsets of T′s theorems and check for some $\mathfrak{A}$-normal (that is, complete, Tarskian) structure that satisfies them. The idea of “strong” subsets of theorems is to be understood pragmatically, but that only adds to the mystery, of course. As an instance ([⁷, p.391]), they discuss the case of naive set theory, which clearly derives a contradiction as one of its theorems (Russell’s paradox, let us say). In order to deal with that theory in the pragmatic truth approach, it is said, we should select a consistent set P of restrictions (the unproblematic postulates, union axiom, power set axiom, and so on), and let membership be the only partial relation of the structure. This membership relation, it is said, may be extended in distinct ways, provided that they are consistent with P. They claim then, for instance, that the membership relation may be extended to obtain ZFC, or to obtain Quine’s NF or ML, or to obtain von Neumann-Bernays-Gödel set theory (NBG) ([⁷, p.391]).

Now, although such theories are thought to be consistent, and to be related to naive set theory, the example is very implausible. These set theories are formulated in quite distinct languages,¹⁰ and they cannot be seen as literally extending a common core of the membership relation. Obviously, the theories must have something in common, but to claim that they are precisely the same in some rather undetermined part is unreasonable. Although they are intended to be theories about the same thing, they are quite distinct theories.

More than that: notice that the apparatus of quasi-truth is performing no essential job in this strategy. A classical logician may also restrict her interest to a class of consistent theorems of the naive set theory and reconstruct the theory in a consistent way. That throws no light on the original theory’s truth or quasi-truth. Also, nothing is made of the Russell contradiction in the original theory. Why is Cantorian set theory said to be quasi-true, given that it is trivial? No explanation is given. The classical logician seems even to have an advantage. There is an explanation for why such re-constructions are pursued: because they are admittedly consistent. Cantor’s theory is trivial, and must be fixed. No need for quasi-truth. Something similar may be said of Frege’s arithmetic. The development of Frege’s theorem indicates that the inconsistency may be eliminated, and arithmetic developed in a second order logic with Hume’s Principle (Zalta ²⁹). No need to say that the original theory is quasi-true.

From these discussion, it results that the claim by Bueno and da Costa [⁷, p.391] seems unjustified:

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.

As we noticed, the quasi-truth approach does not represent contradictions, and the strategy actually advanced to deal with inconsistent theories does not appeal to quasi-truth at all.

4 Re-interpreting quasi-truth

So far, we have argued that the traditional definitions of quasi-truth do not deliver what was promised, that is, an account of incomplete knowledge, which may also accommodate inconsistencies in science.

What can we do with the formalism of quasi-truth? Here, we shall argue that the formalism provided by the quasi-truth approach models a much more restricted class of situations.¹¹

What we have in mind is the contextual approach to inconsistency defended by Brown in ⁴,¹² and critically discussed by da Costa and French [¹⁰ chap.5]. In a nutshell, according to Brown, some inconsistent theories are not to be accepted as true when taken as a whole. We accept only that distinct parts of the theory apply to distinct kinds of contexts or situations, avoiding to bring into each context incompatible claims. As da Costa and French [¹⁰, p.88] comment, the plan consists in breaking an inconsistent theory into sub-theories, each sub-theory dealing with distinct contexts in such a way that incompatible principles are not brought into play together.¹³ So, if a theory somehow endorses both a proposition S and its negation ¬S, we break it into a context where S is accepted, and in another context, where ¬S is accepted. We never apply the theory using both S and ¬S (this interpretation of quasi-truth, as a case of a paraconsistent treatment of contradictions, was also advanced in Arenhart ¹; see Omori and Arenhart ^{22), (23} for further discussions).

This approach is best illustrated by the example of the Bohr ‘model’ of the atom (as discussed by Brown ⁴). It is widely accepted that the model involved some tensions between classical and quantum principles, and the theory is frequently cited as a case of inconsistent theory. However, as Brown argues, the inconsistent principles are never applied together. On Bohr’s model, an electron in a Hydrogen atom, for instance, is always in some discrete orbit, which forms its so-called stationary states. In these cases, classical mechanics is employed to account for the dynamics of the electron in the stationary state it finds itself in. When, however, one wishes to account for the transition of the electron between the distinct discrete stationary states, quantum principles are called for, Planck’s formula giving the relation between the amount of energy and frequency of radiation emitted.

As a result of such a confinement, no real contradiction obtains. Classical principles hold in one context; quantum principles in another.¹⁴ Although da Costa and French [¹⁰, p.89] are rather critical of Brown’s approach, we believe that the formal apparatus of quasi-truth is a model of precisely this kind of situation, where distinct contexts are applied in order to accommodate incompatible sentences. In order to see why, consider the informal discussion of a modal logic of quasi-truth, as discussed by da Costa [⁹, pp.135-136]. Given a partial structure $\mathfrak{A}$, the $\mathfrak{A}$-normal structures extending $\mathfrak{A}$ may be seen as possible worlds in a Kripke semantics for S5. A sentence is quasi-true in $\mathfrak{A}$ if there is a world where it is true. A sentence is strictly valid if it is true in every world. Then, obviously, each world (i.e., $\mathfrak{A}$-normal structure) operates as a context, completely classical, where no contradiction is admitted. Of course, a sentence may be quasi-true, and its negation too, but in distinct possible worlds. This captures the idea of a confining of consistent principles in a context.¹⁵

The main problem with this approach, as da Costa and French [¹⁰, p.89] see it in their criticism of Brown, is that it has rather limited application, given that it is not clear that every inconsistent theory will allow for such a division into consistent contexts. However, that criticism applies to the quasi-truth approach also, due to its relation to the Kripke semantics of S5, as briefly discussed above, and, which is equivalent to the fact already discussed that a sentence in the form of S ∧ ¬S is never quasi-true, so that such a contradiction must be broken in two contexts, one verifying S, another one verifying ¬S.

In this interpretation, then, quasi-truth is not about incompleteness and inconsistency, but rather, more pragmatically, about assuming incompatible sets of commitments according to our needs. Distinct total structures represent the commitments one temporarily has assumed to account for a given context. This has a much less ambitious aim than the one originally proposed by da Costa and French ¹⁰, but it is, we believe, closer to what the formalism really presents us with.

As a further minor remark, this interpretation also points to another inadequacy of the typical rendering of the formalism of the pragmatic truth approach: the idea that scientific knowledge progresses or improves by choosing this or that filling of gaps in our knowledge (that is, by a choice of this or that total structure). However, the real problem, as it is illustrated in the case of the Bohr model, is not that we are ignorant of which total structure we should choose (classical mechanics or quantum principles), but rather, we are asked to provide for a new, unifying framework, which accounts for both situations under the same set of principles (which the ‘new’ quantum theory of Heisenberg and Schrödinger did). Then, while this temporary use of incompatible information in distinct contexts may be accommodated by our interpretation of quasi-truth, the interpretation also allows us to remark that unification of incompatible models is what is typically sought (for further discussion of a related more recent case, see ²⁰).

5 Concluding remarks

Quasi-truth is a formal framework advanced to accommodate more realistically the actual incompleteness and inconsistencies of science; it is part of a highly ambitious plan on the philosophy of science. We have argued that on what concerns accommodation of incompleteness and inconsistency in science, the approach falls short of delivering the promised goods. Let us review shortly what was achieved.

On what concerns the definition of quasi-truth advanced directly by da Costa and collaborators, it is implausible to claim that incompleteness is accommodated by the framework. If one considers that a sentence is quasi-true in a structure $\mathfrak{A}$ with relation to a $\mathfrak{A}$-normal structure $\mathfrak{B}$, then, incompleteness reduces to the possibility of choosing between distinct complete models ($\mathfrak{A}$-normal structures) where a sentence may be false or true in the Tarskian sense. This is just the same situation as in the classical case, and no real gain is obtained by this approach. The incompleteness is not codified in the language of the framework, it is an extra-systematic issue. However, when one adopts the other two definitions of quasi-truth (definitions II and III), then, it seems, the mark of incompleteness in the approach consists in the fact that a sentence may be quasi-true, while its negation may also be quasi-true in some pragmatic structure. However, we have argued, this is the wrong path to incompleteness, because a contradiction (true or quasi-true) may be better understood as representing excess of information, not lack of information.

When it comes to deal with contradictions and inconsistency, quasi-truth seems to fail again. The fact is that even though a sentence and its negation may be both quasi-true, they are quasi-true in distinct normal structures. A parametrization strategy accounts for the apparent contradiction. As we have discussed, this is not really and accommodation of contradictions, but rather their elimination (see also Arenhart ¹ and Omori and Arenhart ^{22), (23} for further accounts of inconsistency in classical settings). Also, a contradiction of the form S ∧ ¬S cannot be quasi-true; however, this is precisely what one needs in order to account for most cases of internally inconsistent theories.

Finally, we have advanced a more modest reading of the formal apparatus of quasi-truth as a kind of contextualization process. Quasi-truth models the workings of sentences in distinct contexts, with a common core of sentences. We have argued that this idea can be associated with the Chunk and Permeate technique, something to be pursued further in other works. The suggestion by da Costa ⁹ according to which one may interpret quasi-truth in possible world semantics for S5 gave further evidence to the plausibility of this interpretation (paraclassical logic could be used here as an alternative). In our interpretation, there are no contradictions or incompleteness accommodated, but it is possible to make sense of the formalism in realistic situations, in which distinct sets of (complete and consistent) assumptions are held in distinct contexts, for distinct purposes. If this does not carry forward the program initiated by da Costa and collaborators, at least it provides for a much more plausible understanding of the formalism, it seems to us.

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  » https://plato.stanford.edu/archives/sum2019/entries/frege-theorem/

• 1
  Notice that there are other accounts of scientific theories that also promise to deal with incompleteness and inconsistency in science. The reader will find discussion of a variety of such views in ^{18), (26}. In this paper we concentrate on pragmatic truth in the tradition originated with Mikenberg, da Costa and Chuaqui ¹⁹.
• 2
  Notice that the set P is the same in both structures
• 3
  So that the language gets interpreted in the partial structure. Notice that without a language, it makes no sense to say that we are in fact defining an alternative conception of truth, given that quasi-truth, just as complete truth, is a predicate of sentences of a language.
• 4
  See ^14),(8 for further discussion on how lack of information is traditionally modelled in the philosophical logic literature.
• 5
  We are still restricting our discussion to the case of a single predicate letter P.
• 6
  First-degree entailment; for details, see Priest [²⁵, chap.8] and Omori and Wansing ²¹.
• 7
  Things become even more difficult in cases where not only the empirical theory is inconsistent, but the underlying logic itself already derives contradictions, while being non-trivial. One can check a discussion of some of such logics in ^{27), (28}. Given that our focus in this paper is on empirical science, we shall not enter in the details of such more radical cases.
• 8
  Not being logical truths or logical falsehoods, there are structures modeling them.
• 9
  As we shall discuss in section 4, this may be interesting indeed for it seems to accommodate typical “Bohrian” complementary situations, where we need to have both situations in order to fully understand a phenomenon (say the particle and the wave behaviour of matter), but they cannot be taken together in a same situation. See also ¹¹.
• 10
  As it is known, NF and ML require a stratification procedure in the language (a kind of typified language), where formulas must obey some constraints in order to be well-formed. This does not happen in ZFC or NBG. NBG, on its turn, allows for a distinction between classes and sets, which ZFC and NF do not. Many other differences in language and in the theories themselves could be pointed ou. The reader is referred to Fraenkel, Bar-Hillel and Levy for a very useful introductory account ¹⁵).
• 11
  In this sense, the same formalism may have distinct applications, in the sense of accounting for distinct target phenomena. What we have claimed so far is that the formalism of quasi-truth cannot account for the situations it was intended to account, but that it may be understood as modeling a distinct kind of phenomenon.
• 12
  Later, Brown has improved his ideas in several papers, until the 2004 paper with Priest ⁵ introducing the “chunk and permeate” technique which, roughly speaking, consists in separating logically consistent “chunks” in a theory and study the kinds of information that can “permeate” from one chunk to another without leading to trivialization. For a general explanation and references, see M. Friend and M. del R. Martínez-Ordaz ¹⁶ and, of course, ⁵.
• 13
  Later, these consistent fragments of the theory were termed “chunks”.
• 14
  This is precisely what we meant above in mentioning complementarity. There is a “rationale” for such a procedure by means of paraclassical logic; these are non-adjunctive logics where we can have two propositions α (say, A) and β (which in particular could be ¬α) but not their conjunction α ∧ β. Furthermore, due to its concept of deduction, from two contradictory theses we cannot derive any proposition, that is, the logic is not trivial; see da Costa and Krause ^{(11), (12).}
• 15
  The discussion here could go to deeper levels. In ¹³, da Costa, Bueno and French discussed “the logic of pragmatic truth”, arguing that is consist of a kind of modal non-adjunctive logic, called Jas´kowski’s discussive logic, which is grounded on S5. But our sketch takes the minimal points. Another similar account of inconsistency in a classical modal context may be found in Jean-Yves Béziau’s paper ³.

• Article Info: CDD: 121

Publication Dates

• Publication in this collection
  21 July 2023
• Date of issue
  Apr-Jun 2023

History

• Received
  18 Nov 2022
• Reviewed
  28 Apr 2023
• Accepted
  04 May 2023