A Structural Analysis of the Phlogiston Case

ORI GIN AL ARTICLE

A Structural Analysis of the Phlogiston Case

Maria Caamano

Received: 30 November 2006 / Accepted: 19 November 2008 / Published online: 14 February 2009

� Springer Science+Business Media B.V. 2009

Abstract The incommensurability thesis, as introduced by T.S. Kuhn and P.K.

Feyerabend, states that incommensurable theories are conceptually incompatible

theories which share a common domain of application. Such claim has often been

regarded as incoherent, since it has been understood that the determination of a

common domain of application at least requires a certain degree of conceptual

compatibility between the theories. The purpose of this work is to contribute to the

defense of the notion of local or gradual incommensurability, as proposed by late

Kuhn. The application of this notion would allow to render the incommensurability

thesis coherent. To support this view, a typical example of incommensurability will

be formally analyzed by applying the structuralist metatheory developed, among

others by W. Balzer, C.U. Moulines and J.D. Sneed. The structural reconstruction of

the relation between the phlogiston theory and the oxygen theory offered here will

reveal that they are locally incommensurable, and will even make possible to

determine the ontological reduction relation that they also exemplify.

1 Preliminary Remarks

1.1 Focus, Goals and Structure of the Work

This work deals with the problem of the incommensurability of scientific theories,

focussing on a specific historical episode proposed by Thomas S. Kuhn as a

paradigmatic case of incommensurability. It should be remembered that the notion

of incommensurability was introduced simultaneously and independently by Kuhn

and Feyerabend in (1962), with the aim of characterizing relations between rival

M. Caamano (&)

Departamento de Filosofıa, Facultad de Filosofıa y Letras, Plaza del Campus s/n,

47005 Valladolid, Spain

e-mail: [email protected]

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Erkenn (2009) 70:331–364

DOI 10.1007/s10670-008-9141-y

scientific theories.1 The relationship is characterized by each author in a partially

divergent manner, although both coincide in presenting it as a radical form of

conceptual disparity between theories that are applied to the same field of research.

The radicality of conceptual disparity was due not only to the fact that theories

would be made up of different concepts, but also to the fact that the conditions for

the formation of these concepts would be incompatible. In accordance with both

authors, this becomes apparent in the incompatibility between the rules of use for

the terms that transmit these concepts. Incommensurability between rival scientific

theories would be patent not only in the impossibility of mutually translating the

languages the express these theories, but also in the impossibility of expanding one

of these languages in order for it to incorporate the other.

The defence of the theory of incommensurability, by Kuhn and Feyerabend, has

run parallel both with regard to their rejection of the verificationalist and

falsificationalist models of the relation between successive theories, and in their

claims for a radical version of the theory of the theoretical load of observation. The

theoretical dependence of all observation, understood as the dependence of all

observation with respect to the theory for which it serves as observation, implies a

relativization of all observation to the theoretical context in which it is taken into

account. This entails three closely connected consequences: (1) the rejection of the

difference between theoretical and observational enunciations; (2) the acceptance of

a radical semantic holism, in accordance with which the meaning of a term depends

on the theoretical context in which it occurs; and (3) a radical semantic

incommensurability, deriving from the global scope that radical semantic holism

gives to the variation of meaning that is characteristic of passing from one theory to

another, incommensurable one.

The research that is presented below is carried out as response to the demand,

made openly by Kuhn, Feyerabend and the proponents of the structuralist trend, for

the need to structurally analyse historical cases of incommensurable theories in

order to be able to formally specify and empirically support the notion of

incommensurability.2 Another general motive behind the work is that of contrib-

uting to a more coherent characterization of the notion of incommensurability,

developing Kuhn’s final proposal of presenting this as a local or partial relation

between theories,3 which would make it possible to overcome the paradoxes

provoked by other more extreme notions. In particular, the aim is to overcome the

paradox that consists of affirming that globally incommensurable theories can,

nevertheless, be applied to common domain.

With regard to the approach taken in the present study, its specificity lies in

combining a formal-type structuralist approach with one of an informal historicist

nature. The novelty of this approach lies in the application of a formal treatment to a

specific case of incommensurability, taking into account the historical context of the

1 This initial appeal to the notion of incommensurability is found in Kuhn (1962/1970) and Feyerabend

(1962/1981, pp. 44–96).2 Cf. Kuhn (1976), as well as Feyerabend (1977).3 Cf. Kuhn (1983).

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knowledge and beliefs in which this case arises.4 This very methodological choice

contrasts with that underlying Andreas Kamlah’s structural reconstruction of the

phlogiston case. His account, however serious and helpful, is not only explicitly

anachronistic,5 but also limited by its reliance on the observational/theoretical

distinction, which latter structuralists avoided and replaced with one between T-

theoretical and T-non-theoretical concepts.

Using the structuralist, history-sensitive approach, I will attempt to defend the

plausibility of the theory of local incommensurability that was introduced in Kuhn’s

later work. Thus, I will uphold that the structuralist approach makes it possible to

establish a restrictive criterion for the scope of incommensurability, and this, in turn,

an interesting comparability between incommensurable theories. This comparability

will require the analysis of conceptual coincidences or equivalences, as well as

conceptual divergences and incompatibilities. That is, it will be attempted to

determine not only the incommensurable components, but also the commensurable

components of the theories of phlogiston and of oxygen.

With regard to structure, the work shall be laid out as follows: firstly, an

explanation of the type of theoretical entity that would arise from the theories of

phlogiston and oxygen; secondly the presentation of the corresponding structural

reconstructions; and finally, the analysis of the intertheoretical relations between the

theories of phlogiston and of oxygen, showing how the manner in which they are

related satisfies the conditions established in the definitions of weak ontological

reduction and of incommensurability. Given the constraints on space available to

4 Besides the structuralist approach, there are other highly sophisticated and interesting approaches for

tacking the problem of incommensurability. Among these, three are worthy of special mention: the

cognitivist computational treatment of Thagard (1992); the treatment of Laudan, the inaugurator of the

so-called ‘problem-solving approach’, which is characterized in (1977); and the proposal of Niiniluoto in

the setting of his critical scientific realism, which he expounds in (1980).5 ‘‘I shall use a quite modern formulation for my purpose, and I shall disregard completely the fact that

chemists in the eighteen century were not able to express their ideas in the same way. I still think that I do

not miss the point by my anachronistic account. Furthermore I do not try to do justice to the historical

facts. I just want to formulate a phlogiston theory which is the best I can think of and which explains as

many chemical facts as possible. After all the problem at issue is not only an historical one but also a

problem of logical analysis’’, Kamlah (1984, p. 223). Kamlah’s anachronistic treatment of the relation

between the phlogiston theory and the oxygen theory essentially relies on concepts like those of

molecular or atomic weights, that were not available until John Dalton developed his atomic theory at the

beginning of the nineteenth century. Described in these terms, the relation between the above rival

theories may seem structurally closer than it actually was, since more information than that available at

the time is being employed in redefining the theoretical concepts of one theory in those of the other. But

let us not forget that, in order to understand the reasons why the transition from the phlogiston theory to

the oxygen theory finally happened, it is crucial to provide a historically faithful description of both rival

theories. In making use of anachronistic notions, Kamlah implicitly introduces an experimental

background clearly beyond the experimental possibilities available for identifying substances during

Pristley’s and Lavoisier’s historical period. In particular, the chemical distinction between atoms and

molecules was not drawn until Amedeo Avogadro, by experimentally determining the density of gases

under different temperature and pressure conditions, extensively developed the Daltonian atomic theory.

Furthermore, Kamlah’s conclusion that the phlogiston theory is reducible to a restricted oxygen theory,

even despite their discrepancies relative to ontological postulates, only holds if we accept as the standard

version of the former theory that according to which phlogiston is ascribed a negative atomic weight (cf.

ibid., p. 227). This, however, does not seem right, for ascribing a negative weight to phlogiston is

commonly understood as an ad hoc hypothesis that emerged only once the phlogiston theory was already

in crisis.

A Structural Analysis of the Phlogiston Case 333

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expound the different matters, both a certain degree of familiarity with the historical

episode referred to and a basic knowledge of structuralist methodology shall be

assumed.

1.2 Characterization of the Type of Theoretical Entity that is Constituted

by the Theories of Phlogiston and Oxygen

Taking into account that the aim of the model-theoretical reconstructions that are

offered below is the structural comparison of two incommensurable theories, just

which of the different structuralist notions of theory is to be applied in this case

should be clarified and justified beforehand.

From a synchronic structural approach two principal possibilities are contem-

plated: conceiving theories as theoretical elements or as theoretical networks.6

Although this is mentioned as a preliminary matter, the type of theoretical entity

that is represented by the theory of phlogiston and that of oxygen is an empirical

matter that can only be elucidated a posteriori. The empirical task of construction

was started with the aim of proving the simplest hypothesis: that both theories are

theoretical elements. Once this hypothesis had been confirmed by the reconstruction

that are offered in the sections below, the aim was to evaluate the more complex

hypothesis that the two combustion theories analysed belonged to one type of

theory, which was also more complex, and represent able in the structuralism as a

theoretical network.7 Specifically, it was attempted to characterize the theories of

phlogiston and oxygen as specializations of the theoretical networks of chemical

principles, composition and reactions. It was not possible to corroborate any of these

hypotheses, and one possible reason for this, which is of a general nature, is the fact

that the theories of phlogiston and oxygen belong to an incipient stage of Chemistry,

during which both the application settings and the conceptual apparatus of the

6 Cf. Balzer et al. (1987, pp. 167–177).7 In structuralist literature, the notion of a theory-net is defined by establishing two conditions: (1) that

there should exist a finite non-empty set of theory-elements TE and a specialization relation r; (2) that the

specialization relation be restricted to the set TE. By adding new conditions to the two aforementioned

one, it is possible to define a more restricted theory-net. This is a theoretical tree network or theoretical

tree, the specific characteristics of which are as follows: (1) the connection between the theory-elements

that belong to TE, such that for any pair of different theory-elements belonging to TE it is held that either

both are specialization of some other common theory-element, or that both have some specialization in

common; (2) the existence, within this set, of a single basic theory-element, i.e. of a single theory-element

that is not a specialization of any other element of the aforementioned set. Given that the definition of a

theory-net presupposes that of specialisation, the principle defining features of this should be mentioned,

even if in passing. They are as follows: (a) equality between the classes of potential models and partially

potential models of the respective related theory-elements; and, (b) the inclusion of the current class of

models, the class of constraints, the class of links and that of intentional applications of the theory that it

specializes, respectively, in the class of actual models, the class of constraints, the class of links, and that

of the intentional applications of the theory that is specialized. Expressed in other terms, two theory-

elements that are related by means of a specialization relation will share their conceptual apparatus, while

they will diverge with regard to the scope of their laws and, consequently, with regard to the extension of

their classes of intentional applications, since the theory-element that specializes restricts the laws and the

empirical scope of the specialized theory-element.

334 M. Caamano

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theories frequently overlapped, without the theoretical network having developed

yet.8

Another, final, possibility was evaluated, namely, that of characterizing both

theories as alternative theoretical elements belonging to successive theory-holons.

Within structuralism, the term ‘theory-holon’ is reserved for those theoretical

‘totalities’ or scientific macro-units that are generated from the interconnection of a

group of theories by means of intertheoretical links or bridging laws.9 These links,

which are represented by means of partial functions, will have a specific

directionality (they will not be reversible), they will link all the theoretical

elements in the domain to at least one other, and will be transitive. The related

theories may form part of one discipline, or even of several. This, precisely, seems

to be the case with the two theories under study; both are linked not only to

chemical theories, such as Cameralism, but also the physical theories, such as

Newtonian mechanics, and even to others converging with medicine, such as occurs

with the theory of phlogiston and iatrochemistry. The general link that is

characteristic of theory-holons may take the form of any type of intertheoretical

relation (specialization, theorization, reduction, equivalence),10 due to which, in

each case, it will be necessary to specify its particular mode of concretion. Hence, a

mixed criterion of the identification of theoretical totalities comes into play. By

applying the aforementioned criterion to the case under study in this work, it can be

verified that the dependence of the theories of combustion with regard to others

complies with each of the first ones constituting different theorizations of the second

ones. In spite of not having undertaken the reconstruction of these, the last

hypothesis mentions would seem to be sufficiently confirmed on the basis of crucial

data obtained in the reconstructions of the rival theories of combustion.

Without denying the importance that a thorough examination of this type of macro-

theoretical unit would have for the understanding of the global structure of scientific

disciplines, such as chemistry, it should be borne in mind that such a task is beyond

the scope of the current analysis, namely, the clarification of the type of relation of

incommensurability that arises between the theory of phlogiston and the theory of

oxygen.11 On one hand, the restriction on the scope of the reconstruction explains

8 As will be clear from the reconstructions, both theory-elements diverge, particularly with that

concerning its dependence (PHLO) or not (OXG) on the theory of principles, and consequently, on a

theory of the qualitative chemical composition. Neither of the theories studies constitutes a theory-net as it

has not been necessary to establish more specific laws to account for the different types of specific cases of

combustion, nor is it the case that both theories specialize the same theory. Nor do they constitute part of a

theory-net given they do not satisfy any of the conditions that must be fulfilled for all specialization; i.e.

neither do the laws nor either of the two theory-elements restrict any fundamental law established in the

other, nor do its corresponding classes of potential and partially potential models coincide with any other

theory-element. As will be seen in the reconstructions, P and Q, for example determine Mp of PHLO and

of OXG, respectively, but not Mp of the Cameralist theory of elements (on which both depend). In the

same manner, the relation of equal or lesser respirability, R, does not determine Mpp of either the theory of

principles, or of composition or reactions, while it does determine Mpp of the theories of combustion.9 Cf. ibid., pp. 386–423, and Dıez and Moulines (1999, p. 365).10 Cf. Balzer et al. (1987, p. 390).11 Below, the theory of phlogiston will be represented by the abbreviation ‘PHLO’, and the theory of

oxygen by ‘OXG’.


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why concepts that were central to the chemistry of that age have not been taken into

account, concepts such as those of principle and element, which are characteristic of

other theories related to the theories of combustion. The exclusion of such concepts

from the reconstructions consequently involves the omission of all reference to the

change undergone by them, which is habitually highlighted by theorists of

incommensurability when dealing with the phlogiston case.

2 Reconstruction of the Phlogiston Theory

By means of PHLO the aim is to explain the combustion of other phenomena

related with it, such as calcinations, reduction and the composition of chemical

substances.12 This set of phenomena of different types would thus constitute the

class of intentional applications of PHLO. In order to give an account of these, we

shall postulate the existence of a chemical principle, phlogiston, establishing it as

the fundamental law of the emission of this substance in combustion (by the

chemical substances subjected to this process).

In order to simplify the meta-theoretical analysis, only examples of complete

combustion shall be taken into account, i.e. instances of combustion in which the

phlogiston of the body that is subjected to combustion is exhausted. However, all

chemical reactions that are considered shall be instances of combustion, it coming

about that each model of PHLO is an instance of combustion. Before undertaking

the formal definitions of the different types of models, it should be stressed that the

present study aims to remain faithful to the historical context in which the theories

arose; thus, in order to avoid falling into anachronisms, certain quantitative terms

will not be used (e.g. ‘density’, ‘volume’, etc.).

2.1 Potential Models of PHLO

Mp(PHLO): x is a potential model of the theory of phlogiston (x [ Mp(PHLO)) iff

there exist S, T, C, A, F, g, w, R, 8 such that13

(1) If x = hS, T, C, A, F, R?, g, w, R, 8i [ Mp(PHLO) then

(2) S is a finite, non-empty set

(3) T = {t1, t2}

(4) C , S and C = [

(5) A , S and A = [, likewise A \ C = [

(6) F , S and F = [

(7) g: (C 9 A 9 {t1}) ? (S 9 A 9 {t2})

(8) w: S ? R?

(9) R � A 9 A, and R determines a weak order

(10) 8: S 9 S ? S

12 The principal historiographic sources used in the reconstructions of the theories of phlogiston and

oxygen are those mentioned in Sect. 3.1.13 Adhering to the method of presentation used in An Architectonic for Science, I shall include the set of

numbers among the constituents of the models.

336 M. Caamano

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Basic intended interpretation: S is a set of portions of specific chemical

substances (not a class of types of substances), T is a set whose only elements are

two temporal instants: one prior to the combustion (t1) and another subsequent to it

(t2). C is a subset of S which excludes A, and is interpreted as being the set of

portions of specific combustible substances. A is a set of chemical substances in the

air, i.e. of the substance that surrounds the bodies subjected to combustion. Finally,

with regard to the domains, F is a set of portions of an element called ‘phlogiston’.

The function g determines the combustion or chemical reactions, by assigning to

each portion of the combustible substance c and to each portion of air a in t1 a

portion of substance s and a portion of air a0 in t2. w is the function that determines

the weight, assigning to each chemical substance s its weight w(s) as an element of

R?. R lists different specific portions of air in terms of its equal or lesser degree of

respirability. It should be noted that R does not have a functional character. It may

come about that the same air may be as breathable or less breathable than more than

one air; i.e., there may be several airs that are more or less breathable than it. The

combination of substances that form aggregates is represented by means of the

function 8. Each pair of substances has assigned to it another substance, an

aggregate, which arises from the combination of the initial two.

We now go on to define certain notions derived from the primitive concepts of

PHLO.

If x = hS, T, C, A, F, R?, g, w, R, 8i [ Mp(PHLO) then

(1) Let \c, a, t1[ and \s, a0, t2[ be such that g(\c, a, t1[) = \s, a0, t2[ then

g1(c) = defs and g2(a) = defa0 (i.e., the image of a substance c under the

combustion function, g1(c), is the residue of s obtained on the basis of c, and

the image of a substance a under the combustion function, g2(a) is the

substance a0 obtained on the basis of a)

In (1) the concepts of the residue of the combustible substance obtained after the

combustion and of air obtained after the combustion on the basis of the air prior to

the combustion are defined. These concepts are defined by mean of the restriction of

c with regard to the domains C and A.

(2) Let s, s0, s00 be such that 8(s, s0) = s00 then s00 = defs 8 s0 (s00 should be

understood as the aggregation of s and s0)

In (2) the concept of aggregate is defined by means of applying the function o as

the value assigned to the arguments of this function, i.e. as that substance which

results from the combination of the other two.

(3) s contains s0 iffdef: there is some s00 s = s0 8 s00 or s = s00 8 s0

In (3) the mereological relation of containing is defined, also on the basis of the

function 8, as the relation between the value assigned to the arguments of this

function and one or other of these; i.e. as the relation between that substance

resulting from the combination of two other and one or other of these two.

(4) fs is the total (or maximum) quantity of phlogiston contained in s iffdef: scontains fs and for every f0 if s contains f0 then fs contains f0


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On the basis of the aggregation function and of the mereological containment

relation (derived from it), it is possible to define in (4) the concept of total or

maximum value of the phlogiston contained in a substance as that portion of

phlogiston that constitutes one of the arguments of the function 8 giving a value s,

or, expressed in other terms, as that portion of phlogiston that is combined with

some other substance to produce the aggregate s.

(5) P � S 9 S and sPs0 iffdef: w(fs) B w(fs0) (understood as—applied to combus-

tion—s0 is depleted of phlogiston giving rise to s iff the total quantity of

phlogiston s weighs less than or the same as the weight of the total phlogiston

in s0)

In (5) P is defined on the basis of the primitive concept of weight, and on the

derived concept of the total quantity of phlogiston in a substance, as a relation

between different specific portions of substances according to their equal or lesser

content of phlogiston. The relation P determines a weak order.

2.2 Actual Models of PHLO

M(PHLO): x is an actual model of the theory of Phlogiston (x [ M(PHLO)) iff

there exist S, T, C, A, F, g, w, R, 8 such that

(1) If x = hS, T, C, A, F, R?, g, w, R, 8i [ Mp(PHLO) then

(2) x [ Mp(PHLO)(3) for every \c, a, t1[ [ Dom(g):

(3i) c contains g1(c)

(3ii) there exists some f [ F such that s contains f in t1 and g1(c) does not contain fin t2 and g2(a) contains f in t2 and a does not contain f in t1

(3iii) for each f [ F: if a contains f in t1 then g2(a) will contain f in t2

(4) for each a [ A: g2(a)Ra iff w (fg2

(a)) C w(fa)

Condition (3) is the requirement that there should be a type of entity acting as a

principle in combustion, being emitted by the substance subjected to this process.

(3i) requires that the substance subjected to combustion should contain the residue

remaining after combustion. (3ii) establishes that the substance that is responsible

for the combustion should pass from the body to the air. (3iii) requires that none of

this substance should pass from the air to the body. Lastly, in (4) the decrease in

respirability for the air substance that, on having absorbed phlogiston, now contains

a greater quantity of that substance is established.

The demonstration of the following theorem, which is implied by axioms (3) and

(4), allows the conclusion to be reached that that air has less respirability after the

combustion.

Theorem 1 for every \c, a, t1[ [ Dom(g): g2(a)Ra and it is not the case thataRg2(a)

Proof Supposing that \c, a, t1[ [ Dom(g):

\g1(c), g2(a), t2[ = g(\c, a, t1[) by definition of g

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for all f [ F: if a contains f in t1 then g2(a) contains f in t2, by (3iii)

g2(a)Rathere exists f [ F such that g2(a) contains f in t2 and a does not contain f in t1not for all f [ F: if g2(a) contains f then a contains fit does not hold that aRg2(a), by (4) h

On the basis of Axiom (4) and Definition (5) a new theorem follows in which the

lower ‘phlogistication’ of the more breathable air and the greater ‘phlogistication’ of

the less breathable air are established. That is, the theorem expresses a correlation

between the lower respirability of air that takes part in the reaction and its greater

‘phlogistication’, presenting the restriction of P over the domain A as the inverse

relation of R, and, thus, as implying a lower quantity of phlogiston in the more

breathable air.

Theorem 2 for every \c, a, t1 [ [ Dom(g): aPg2(a) iff g2(a)Ra

Proof Let that \c, a, t1 [ [ Dom(g)

\g1(c), g2(a), t2 [ = g(\c, a, t1[) by definition of gfor all a, a0 [ A: g2(a)Ra iff w(fg

2(a)) C w(fa), by (4)

cPs iff w(fc) B w(fs), by def. (5)

aPg2(a) iff w(fa) B w(fg2

(a)), by the restriction of P over the domain Aw(fa) B w(fg

2(a)) iff w(fg

2(a)) C w(fa), by the inverse of C/B

aPg2(a) iff g2(a)Ra, by substitution on the basis of def. (5) and axiom (4) h

2.3 Partial Potential Models of PHLO

Mpp(PHLO): y is a partial potential model of the theory of phlogiston (y [Mpp(PHLO)) iff there exists x such that x = \ S, T, C, A, F, R?, g, w, R, 8[ [Mp(PHLO) and y = \ S, T, C, A, R?, g, w, R, 8[

The matter of theoricity concerns S, T, C, A, F, g, w, R, 8. The first two basic sets,

along with the two following derived ones, the (non-functional) relation R and the

functions, are non PHLO-theoretical, given that they may all be determined without

resorting to the notion of phlogiston, and ignoring the fundamental laws of PHLO.

Whereas F and the non-primitive, derived relation P must be considered as PHLO-

theoretical given their dependence of these laws. In particular, both the determi-

nation of F and that of P require the assumption of the law when affirming the

existence of an element, referred to as ‘phlogiston’, which acts as a principle in

combustion. This law appears in the formal definition of M(PHLO) as Axiom (3).

In spite of ‘phlogiston’ and related terms such as ‘phlogisticate, ‘dephlogisticate’,

etc., being the only PHLO- theoretical terms, the importance of the concepts that

are related to them are evident in the fact that they necessarily intervene in the

determination of one of the sets derived from the theory. This importance stems

from the key conceptual role that they play in articulating the specific conceptual

schema of PHLO.

Thus, I shall now go on to examine the criteria for determining the functions and/

or notions expressed by non PHLO-theoretical terms. The phenomena or entities to

which these terms refer will be recognized, at least in part, by taking these criteria


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into consideration. This is especially clear in the cases of S, A, g and R, when they

are determined (during the historical period in which PHLO prevailed) by

principally applying notions that were independent of all scientific theory,

belonging to the sphere of ordinary knowledge.

Certain terms from ordinary language are introduced into scientific theories as

primitive terms, i.e. which are not defined in these theories. This occurs in PHLOwith the notion of combustion. This must be understood as a primitive notion that

can only be determined on the basis of the disjunction of descriptions or predicates

that express different properties: increase in temperature, reduction of size, emission

of gases, reduced respirability of the surrounding air,14 etc. On the basis of the

aforementioned reasons, it can be inferred that the criteria for determining g will be

identified with those that may be used for determining Mpp. Thus, as combustion is

the principal phenomenon dealt with by PHLO, and given that the notion of

combustion must be characterized within the non PHLO-theoretical language (more

specifically, in ordinary language, i.e. at the most basic level of conceptualization),

it is deduced that in effect the determination of g is governed by the same criteria as

that of Mpp. g is the non PHLO-theoretical phenomenon that it is aimed to explain.

I shall postpone the analysis of the non PHLO-theoretical concepts until Sect.

2.5, where we shall specify the essential intertheoretical links of PHLO, i.e., those

links that are necessary to define or characterize (in the case of primitive terms)

some of the non PHLO-theoretical concepts. Nevertheless, I shall deal with some of

the different theories of conceptualizations on which the determination of these

concepts depends. In order to determine the function w we must resort to classical

particle mechanics. The determination of T presupposes some type of chronometric

theory. The cameralist theory of composition, along with the physical theories of

Boyle and Newton will be essential in the determination of the aggregation function

8. As has already been pointed out above, S, A, g and R will be determined using

essentially ordinary language, although the characterization of the notions of air,

combustion and respirability will similarly require taking into account the

iatrochemical theory of composition, as well as Boyles physical and chemical

ones. T will not be dealt with in 1.5, since given that it is a very common notion in

scientific theories and not specific to chemistry, the analysis of its intertheoretical

links is of no real interest for the present study.

Before moving on to the following section, one last matter should be clarified;

this is the problem that seems to arise from the inclusion of the PHLO-theoretical

sub-domain F in the non PHLO-theoretical domain S. It is possible to think that this

inclusion assumes the necessary PHLO-theoretical determination of S, and thus, its

PHLO- theoretical character. Nevertheless, this is only an apparent problem, as the

domain S is not necessarily determined by recourse to its different sub-domains,

rather on the basis of certain non PHLO-theoretical properties that each element of

14 The reduced respirability of the surrounding air is not explicable by any law of PHLO, thus it is

possible to identify the combustion on the basis of other properties.

340 M. Caamano

123

the domain in question must have (have weight, certain size, etc.).15 In the case of

the sub-domain F, which historically was discovered as empty, its inclusion in the

domain of substances would be explained by the acceptance that the substances to

which chemists attempt to refer will not be empirically determined in all cases, thus

allowing their existence to be inferred on the basis of certain determinable, non

PHLO-theoretical properties (such as, for example, the greater or lesser respira-

bility of the air surrounding the substance that is subjected to combustion).

2.4 Constraints for PHLO

As we shall see, the different constrictions that are identifiable in PHLO, in the

same way as those identifiable in OXG, are not strictly from these theoretical

elements, rather they are inherited or imported from other theories (in particular

Newtonian mechanics) with which they are connected by intertheoretical links.

There are two conditions for natural links: identity of weight and extensivity of

weight. The constrictions C1 and C2 are the same ones that are presented in

Classical Collision Mechanics relative to mass. In C1 the weight of any substance is

required to be independent from the system in which that substance appears. C2

requires that relative to a given aggregation operation, represented by the basic

notion 8, the weight should behave as an additive under this relation. To each pair of

chemical substances, 8 assigns another chemical substance whose weight results

from the addition of both substances.

C1(PHLO): for all X: X [ C1(PHLO) iff [ = X � Mp(PHLO) and for all x, y[ X and for all s: if s [ Sx \ Sy then wx (s) = wy (s)

C2(PHLO): for all X: X [ C2(PHLO) iff [ = X � Mp(PHLO) and for all x, y,

z [ X:

(a) 8: S 9 S ? S where S : = [ {Sx/x [ Mp(PHLO)}

(b) for all s [ Sx, s0 [ Sy (if s 8 s0 [ Sz then wz(s 8 s0) = wx(s) ? wy(s0))

The global constraint of PHLO would be:

GC PHLOð Þ ¼ C1 PHLOð Þ \ C2 PHLOð ÞThe theorem that is demonstrated below, which is implied by Axiom (3) and by

C2(FLO), establishes the reduction in weight of every body subjected to

combustion.

Theorem 3 for all \c, a, t1[ [ Dom(g): w(g1(c)) \ w(c)

Proof Let that \c, a, t1 [ [ Dom(g)

\g1(c), g2(a), t2 [ = g(\c, a, t1[), by definition gc contains g1(c), by (3i)

w(s0 8 g1(c)) = w(s0) ? w(g1(c)), by C2(FLO)

w(c) = w(s0) ? w(g1(c)), by substitution on the basis of (3i)

w(g1(c)) \ w(c) h

15 For the same reasons, the inclusion of the theoretical-OXG O sub-domain in the non-theoretical OXGdomain S does not imply the necessary theoretical-OXG determination of S.


123

2.5 Intertheoretical Links of PHLO

Below I shall examine the different links that should be considered as essential

constituents of PHLO. The first of them is the one that provides PHLO with the

values of the function w. These are determined using a variety of instruments,

namely, analytical balances, beam balances and dynamometers. The use of such

instruments requires the assumption of some model of rigid body mechanics

(RBM). Bearing in mind that this theory is reducible to classical particle mechanics

(CPM), it can be stated that a link L1 is needed between M(RBM) (or M(CPM))

and Mp(FLO).

L1(FLO) � M(RBM) 9 p(RBM, 1, 2, 5) 9 Mp(PHLO) 9 p(PHLO, 1, 2, 9)

which is determined by the following condition:16

For each x [ M(RBM) and x0 [ Mp(PHLO):

\x, \Px, Tx, mx[, x0, \Sx0, Tx0, wx0� [ L1(PHLO) iff there exists s [ S y t = Tx \Tx0 such that

(a) Sx0 = Px

(b) wx0(s) = mx(s)

P is interpreted as a set of particles, T as a set of temporal instants and m as the

function for determining mass.

Besides w, there are other non PHLO-theoretical functions, relations and

domains (basic sets) that are determined by specific essential links with other

theories. Nevertheless, as here we are dealing with theories that are pre-scientific

and/or non-axiomatized, neither the structural reconstruction nor the formal

definition of these intertheoretical links will be dealt with here. What constitutes

the intertheoretical links that determine S, C, A, g, R and 8 will simply be informally

indicated,17 by pointing out the manner in which the non PHLO-theoretical notions

that intervene in this determination depend on certain theories. As has already been

mentioned, the characterization of some of these notions principally involves a type

of conceptualization that is expressed in ordinary language, and not presented in the

form of a theory. Only a certain aspect of the characterization will depend directly

on pre-scientific or obsolete scientific theories. This is the case with the notions

represented by C, g and R, which will require an analysis of both the strictly

intertheoretical links and the connection with conceptualization from the sphere of

ordinary knowledge. Needless to say, this ordinary knowledge, transmitted via

ordinary language, will have to be relativized to the historical period being

considered, in the same way as theories. I shall now go on to examine the terms

‘substance’, ‘air’, ‘respirability’, ‘combustion’, ‘combustible substance’ and

‘aggregation’, in that order.

In the determination of S, the notion of matter that is characteristic of the physical

or corpuscular theory of matter is used. At the end of the seventeenth century and

16 The axioms given in p(RBM, 1, 2, 5) are in keeping with those established in Balzer et al. (1987, p.

269), where the class of potential models for rigid body mechanics is defined.17 The informal analysis that follows is based on the historical study of chemistry by Brock (1992, pp.

43–86).

342 M. Caamano

123

the beginning of the eighteenth century the mechanistic theories of Boyle and

Newton (precursors of Dalton’s atomic theory) consolidated a conception of

substances in terms that were eminently physical, with regard to conglomerates of

particles or bodies. In order to define the term ‘matter’ use is made of geometric

qualities and the mechanisms of affinity and repulsion, whilst the use of secondary

qualities (temperature, colour) and forms from Aristotelian physics are rejected.18

Undeniably, weight has to be considered as an essential property of matter. In

reality, it was only with the development of Newtonian mechanics that matter and

weight started to be distinguished. It seems obvious that this association leads

necessarily to the dependence of the theory of phlogiston on classical particle

mechanics, as the methods for determining weight presuppose the use of CPM or

some other theory that is reducible to it, such as RBM.19 At the beginning of this

epigraph, the connection between the models of RBM and the potential models of

PHLO in terms of weight is pointed out.

The determination of the basic set A is carried out in accordance with the

manners of determining air stipulated in the iatrochemical theory presented by Joan-

Baptista van Helmont.20 Here, air is conceived as the setting for the chemical

reactions, and is defined as an elastic fluid that transports particles, substances and

chemical elements. The results of chemical studies by Boyle in this respect reaffirm

those put forward by Helmont.21

The relation of equal or reduced respirability of air is derived from the property

of air of being breathable, this being determined by means of ordinary knowledge,

e.g. on the basis of the actions of inhaling and exhaling. Nevertheless, in order to

define the concept of respirability, which is indissociable from the concept of air, it

is necessary to once again turn to the iatrochemical definition of the term ‘air’. Some

of the models from the theory of iatrochemistry necessarily coincide with some of

the potential models of PHLO, due also to the fact that the conception of respiration

as the absorption and emission of gases, along with the corresponding character-

ization of the notion of gas, constitutes another of Helmont’s contributions to the

chemistry of his time. The subset of substances formed by gases would thus be

determined in accordance with the iatrochemical analysis of the latter, and by means

of which it is established that the gases are made up of air and earth of water.22

With regard to notion of combustion, in spite of there being certain links which

are constitutive with other theories that are essential for the determination of g, these

links do not contribute to forming a criterion for determination, i.e. a set of

conditions that are necessary and sufficient to do so. As modes of determining g, the

verification of the increase in temperature, of the reduction in size, of the emission

of gases, of the reduced respirability of the surrounding air, etc., must also be taken

into consideration. All of this principally involves different sections of ordinary

knowledge. Nevertheless, insofar as the concept of gas plays a role in this

18 Cf. ibid., pp. 66, 75.19 Balzer et al. (1987, pp. 53–54).20 Cf. ibid., p. 49.21 Cf. ibid., p. 59.22 Cf. ibid., p. 51.


123

determination, and this is considered scientific, there will be at least one essential

link derived from this intervention. As has already been mentioned when analyzing

the notion of respiration, in accordance with the theory of iatrochemistry and the

chemistry of Boyle, gases are considered as compounds and/or mixtures of either

earth and air, or water and air.23 Consequently, and given that the determination of

the function g requires both the verification that gases are emitted in the combustion

process, and the reduced respirability of the air surrounding the reaction, the

essential link of g with the aforementioned theories will be generated in a dual

manner: firstly, directly, due to its dependence on the notion of gas; and secondly,

more indirectly, due to its dependence on the notion of respiration, which in turn is

directly dependent on that of gas.

The concept of aggregation (characterized by means of the function 8) specifies

two different cases within the cameralist theory: that of compounds, and that of

mixtures. Its origin is physical, as it is based on the physical theories of Boyle and

Newton.24 In accordance with these, matter is made up of particles that are

hierarchically organized into groups or clusters to form mixtures or compounds. The

former are characterized by having been constituted by means of a quantitatively

unstable combination of elements. On the contrary, the latter are defined on the

basis of a quantitatively stable combination of elements.

Nevertheless, certain significant differences between the physical and chemical

concepts of composition should be pointed out. The former refers to the

concatenation relation between particles, the result of which would be the

concatenation or grouping together of these particles. The second, however,

explains the aggregation relation between chemical substances, which would lead to

the formation of an aggregate. The essential difference lies in the fact that an

aggregate does not consist of a mere grouping of substances, rather of a new

substance that is obtained on the basis of other different ones. This distinguishing

aspect is not a case of using specific concepts from any scientific theory, rather of

notions that are characteristic of ordinary language. Thus, in the description of this

distinguishing aspect only ordinary terms (e.g. ‘substance’, ‘formation’, etc.) are

involved. Finally, the need to take into consideration both the physical and the

strictly chemical meaning of the notion of aggregation must to be stressed, as, even

though part of the meaning of the term ‘aggregation’ depends on notions from the

sphere of ordinary language, another fundamental part requires notions that are

characteristic of physics for its determination. More specifically, the physical

concepts of ‘mass’ and ‘conservation of mass’ will come into play. Thus, it would

seem reasonable to uphold the existence of an essential intertheoretical link of 8with the corpuscular theories of Boyle and Newton, so that the notion of

composition of these theories would be essential for setting the meaning of

‘aggregation’. Nevertheless, as has already been alluded to, the specification of this

link will be necessary but not sufficient to lend meaning to this term, it also being

necessary to take into account the aforementioned ordinary knowledge.

23 Cf. ibid., pp. 51, 71–72.24 Cf. ibid., pp. 55, 75.

344 M. Caamano

123

If we concede that the physical theories of Boyle and Newton are essential for the

determination of w, 8 and S, with the iatrochemical theory and Boylean chemistry

being required to determine A, g and R, then the global link of PHLO could be

defined as the intersection of all the intertheoretical links of PHLO.

2.6 The Theory-Element of PHLO and Its Empirical Claim

T(PHLO) := \K(PHLO), I(PHLO)[ where

K(PHLO) := \Mp(PHLO), M(PHLO), Mpp(FLO), GC(PHLO), GL(PHLO)[and I(PHLO) � Mpp(PHLO) is such that

(1) I0 � I(PHLO) where I0 is {the combustion of non-metallic inorganic

substances (wood, coal), the calcination of mercury, the reduction and

composition of mercury}

(2) all the members of I(PHLO) are sufficiently similar to those of I0.

Formulation of the empirical claim of PHLO: The intentional applications may

be extended to a set of models that are connected among themselves by GC(PHLO)

which satisfy the laws and the links.

More specifically, the express assertion that the members of I(PHLO) that are

made up of a number of substances S, such as air A and combustible substances C,

may be extended to structure containing the substance phlogiston P. The structures

have to comply with the law and the bonding conditions. Members of I(PHLO) will

consist of a structure of temporal instants T, with a series of functions g for

determining the reactions of combustion, functions w for weight, and 8 for

aggregation. The relations R and P also intervene by giving account of the members

of I(PHLO), as they establish connections respectively between different

substances, on the basis of the property of being equal or less breathable, and in

accordance with their equal or reduced phlogistication. In each individual reaction

or process of combustion determined by g, T and S (and consequently their subsets

C, A, P) along with the other functions and relations that have been mention, the

laws of PHLO must hold. However, each substance in S will be applicable in at

least one member of I(PHLO). The string of functions w must assign to each

substance the same weight throughout the chemical reaction in I(PHLO) in which it

intervenes.

3 Reconstruction of the Oxygen Theory

In the same way as with PHLO, with OXG the aim is to give account of the

phenomenon of combustion as well as other phenomena associated to it; principally,

the calcination, reduction (analysis) and composition (synthesis) of chemical

substances. The set comprising the (types of) phenomena mentioned make up the

class of intentional applications of OXG. In order to give an account of these, we

shall postulate the existence of a new chemical element, oxygen, with the absorption

of this substance in combustion (by the chemical substances subjected to this

process) being established as a fundamental law.


123

3.1 Potential Models of OXG

Mp(OXG): x is a potential model of the theory of oxygen (x [ Mp(OXG)) iff there

exist S, T, C, O, g, w, R, 8 such that

(1) x = hS, T, C, A, O, R?, g, w, R, 8i(2) S is a finite, non-empty set

(3) T = {t1, t2}

(4) C , S and C = [

(5) A , S and A = [, likewise A \ C = [, it also being the case that A \O = [

(6) O , S and O = [

(7) g: (C 9 A 9 {t1}) ? (S 9 A 9 {t2})

(8) w: S ? R?

(9) R � A 9 A, and R determines a weak order

(10) 8: S 9 S ? S

Basic intended interpretation: S is a set of portions of specific chemical

substances, T is a set whose only elements are two temporal instants: one prior to the

combustion (t1) and another subsequent to it (t2). C is a subset of S which includes

elements of A, and is interpreted as being the set of portions of specific combustible

substances. A is a set of chemical portions of air, i.e. of the substance made up of

portions of elements, some of which intervene in the process of combustion. Finally,

with regard to domains, O is a set of portions of an element called ‘oxygen’.

The function g determines the combustion or chemical reactions, by assigning to

each portion of the combustible substance c and to each portion of air a in t1 a

portion of substance s and a portion of air a0 in t2. w is the function that determines

the weight, assigning to each chemical substance s its weight w(s) as an element of

R?. R lists different specific portions of air in terms of their equal or lesser degree of

respirability. It should be borne in mind that, unlike the rest of the relations that

appear in the definition of Mp(OXG), R does not have a functional character. It may

come about that the same air may be as breathable, or less breathable than more than

one air; i.e., there may be several airs that are more or less breathable than it. The

combination of substances that form aggregates is represented by means of the

function 8. Each pair of substances has assigned to it another substance, an

aggregate, which arises from the combination of the former two.

We now go on to define certain notions derived from the primitive concepts of

OXG, in a similar manner to that done for the reconstruction of PHLO.

If x = hS, T, C, A, O, R?, g, w, R, 8i [ Mp(OXG) then

(1) Let \c, a, t1[ and \s, a0, t2[ of which g(\c, a, t1[) = \s, a0, t2[ then

g1(c) = defs and g2(a) = defa0 (i.e. the image of a substance c under the

combustion function, g1(c), is the residue of s obtained on the basis of c, and

the image of a substance a under the combustion function, g2(a) is the

substance a0 obtained on the basis of a)

In (1) the concepts of the residue of the combustible substance obtained after

combustion and of air obtained after combustion on the basis of the air prior to the

346 M. Caamano

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combustion are defined. These concepts are defined by means of the restrictions of gwith regard to the domains S and A.

(2) Let s, s0, s00 be such that 8(s, s0) = s00 then s00 = defs 8 s0 (s00 should be

understood as the aggregation of s and s0)

In (2) the concept of aggregate, by means of applying the function 8, as the value

assigned to the arguments of this function; i.e. as that substance which results from

the combination of the other two.

(3) s contains s0 iffdef: there is some s00 s = s0 8 s00 or s = s00 8 s0

In (3) the mereological containment relation of containing is defined, also on the

basis of the function 8, as the relation between the value assigned to the arguments

of this function and one or other of these; i.e. as the relation between that substance

resulting from the combination of two others and one or other of these two.

(4) os is the total (or maximum) quantity of oxygen contained in s iffdef: s contains

os, and for every o0 if s contains o0 then os then o0

On the basis of the aggregation function and of the mereological containment

relation (derived from it), in (4) it is possible to define the concept of total or

maximum value of the oxygen contained in a substance as that portion of oxygen

that constitutes one of the arguments of the function o giving as a value s, or,

expressed in other terms, as that portion of oxygen that is combined with some other

substance to produce the aggregate s.

(5) Q � S 9 S and sQs0 iffdef: w(os) B w(os0) (when applied to combustion,

understood as s0 deoxygenated (loses oxygen) giving rise to s iff the total

quantity of oxygen s weighs less than or the same as the weight of the total

oxygen in s0)

In (5) Q is defined on the basis of the primitive concept of weight, and on the

derived concept of the total quantity of oxygen in a substance, as a relation between

different specific portions of substances according to their equal or lesser content of

oxygen. The relation Q determines a weak order.

3.2 Actual Models of OXG

M(OXG): x is an actual model of the theory of oxygen (x [ M(OXG)) iff there exist

S, T, C, A, O, g, w, R, 8 such that

(1) x = hS, T, C, A, O, R?, g, w, R, 8i(2) x [ Mp(OXG)(3) for all \c, a, t1[ [ Dom(g):

(3i) a contains g2(a)

(3ii) there exists o [ O such that a contains o in t1 and g2(a) does not contain o in t2and g1(c) contains o in t2 and c does not contain o in t1

(3iii) for all o [ O: if c contains o in t1 then g1(c) contains o in t2(4) for all a [ A: g2(a)Ra iff w(og

2(a)) B w(oa)


123

Condition (3) is the requirement that there should be a type of entity that is

absorbed in combustion by the substances subjected to this process. (3i) requires

that the substance air with participates in the combustion should contain the residue

remaining after combustion. (3ii) establishes that the substance that is responsible

for the combustion should pass from the air to the body. (3iii) requires that none of

this substance should pass from the body to the air. The decrease in respirability for

that substance air with a lower quantity of oxygen is established in (4).

The demonstration of the following theorem, which is implied by axioms (3) and

(4), allows us to conclude that that air has less respirability after the combustion.

Theorem 1 for all\c, a, t1[ [ Dom(g): g2(a)Ra and it is not the case that aRg2(a)

Proof Let \c, a, t1[ [ Dom(g)

\g1(c), g2(a), t2[ = g(\c, a, t1[) by definition of ga contains o in t1 and g2(a) does not contain o in t2, by 3 (ii)

g2(a)Ra, by (4) and by def. (5)

Not for all o [ O is not the case that a contains o in t1 and g2(a) does not contain oin t2

Not for all o [ O is the case that if a contains o in t1 then g2(a) contains o in t2it is not the case that aRg2(a), by (4) h

On the basis of Axiom (4) and Definition (5) a new theorem is followed, in which

the lower ‘phlogistication’ of the less breathable air and the greater ‘oxygenation’ of

the more breathable air are established. That is, the theorem expresses a correlation

between the lower respirability of air that intervenes in the reaction and its reduced

‘oxygenation, presenting the restriction of Q over the domain A as an isomorphic

relation of R, and, thus, as implying a lower quantity of oxygen in the less

breathable air.

Theorem 2 for all \c, a, t1[ [ Dom(g): g2(a)Qa iff g2(a)Ra

Proof Let \c, a, t1 [ [ Dom(g)

\g1(c), g2(a), t2[ = g(\c, a, t1[), by definition of gfor all a, g2(a) [ A, for all o [ O: g2(a)Ra iff w(og

2(a)) B w(oa), by (4)

s0Qs iff w(os0) B w(os), by def. (5)

g2(a)Qa iff w(og2

(a)) B w(oa), by restriction on Q with respect to A

g2(a)Qa iff g2(a)Ra, by substitution on the basis of definition (5) and

axiom (4) h

3.3 Partial Potential Models of OXG

Mpp(OXG): y is a partial potential model of the theory of oxygen (y [ Mpp(OXG))iff there exists an x such that x = \S, T, C, A, O, R?, g, w, R, 8[ [ Mp(OXG) and

y = \S, T, C, A, R?, g, w, R, 8[The matter of theoricity affects S, T, C, A, O, g, w, R, 8. ‘Oxygen’, ‘oxidation and

derived terms such as ‘deoxigenation’, are the only OXG-theoretical terms, as only

these depend on the laws of OXG for the determination of their meaning. The

determination of O and of the non-primitive relation Q specifically require the

348 M. Caamano

123

assumption of Axiom (3) from the formal definition of M(OXG), where one of the

fundamental laws of OXG is expressed. In this law, the existence of an element,

oxygen, which is absorbed by all bodies subjected to a process of combustion is

stated. The role of the concepts of oxygen and oxidation in the organization of the

specific conceptual schema of OXG is fundamental, given their involvement in the

constitution of the fundamental laws of this theory. The concepts of oxygen and

oxidation intervene in the conceptual schema supplied by Mp(OXG), which in turn

is proposed in the one offered by M(OXG).The rest of the sets, the basic ones (S and T) and those that are characterized on

the basis of them (C and A), along with the (non-functional) relation R and the

functions (g, w, and 8), may be considered non OXG-theoretical, as they can be

determined without resorting to the OXG-theoretical concept of oxygen, or to the

fundamental laws of OXG. The fact that oxygen appears as a component of air does

not imply that ‘air’ must be conceived as an OXG-theoretical term, since the

identification of the substance air is possible by means that are independent of the

laws supplied by the theory of iatrochemistry. According to that established in this

theory, air is defined as an elastic fluid that is a transporter of particles that acts as an

environment for combustion. It can be seen that PHLO and OXG share their non T-

theoretical terms. The Kuhnian notion that theoretical terms of a theory could infect

the non-theoretical ones is not borne out in the light of the results of the above

analysis.25 In the case of the transition from the theory of phlogiston to that of

oxygen, in the determination of the non PHLO-theoretical concepts inherited by

OXG there is no need for the fundamental laws of OXG, due to which they are kept

as non-theoretical concepts in the latter theory.

Taking into account that the partial potential models are determined by means of

non T-theoretical models, and that, according to what has already been said, PHLOand OXG share all their non-theoretical concepts, it can be concluded that

Mpp(PHLO) = Mpp(OXG). The coincidence that both theories supply an

explanation for the non-theoretical phenomenon of combustions which they

exemplify in the same manner, along with the assumption that the classes of

intentional applications are subsets of the classes of partial potential models, makes

it possible to infer that I(OXG) \ I(PHLO) = [. This is the same as saying that

the classes of intentional applications of OXG and PHLO are not exclusive, rather

they overlap, either totally or partially.

The above analysis enables a study to be carried out on the discrepancies and

coincidences between the successful and unsuccessful intentional applications of

both theories. Such a study should give account of the success of both theories in

explaining the reduced respirability of the air surrounding combustion. Similarly,

it would have to explain the success of OXG, as opposed to PHLO, in resolving

the anomaly of the increase in weight of calcined metals, and its failure, also in

contrast to PHLO, of leaving unsolved the problem of the apparent decrease in

weight of non-metallic substances that are subjected to combustion(as will be

proved below).

25 Kuhn expresses this concern in (1983, p. 673).


123

At this point no explanation will be given of how the different notions expressed by

non OXG-theoretical terms are determined, given that as these notions are shared with

PHLO, their determination criterion has already been examined in Sect. 2.5.

3.4 Constraints for OXG

OXG shares with PHLO the ‘‘inherited’’ (not its own) constraints C1 and C2 for the

identity and extensivity of the weight. It should be remembered that in C1 the

weight of any substance is required to be independent of the systems in which the

said substance appears. C2 requires that relative to a given aggregation operation

(represented by the basic notion 8) the weight should behave as an additive under

this relation. To each pair of chemical substances, 8 assigns another chemical

substance whose weight results from the addition of the weight of both substances.

Nevertheless, there is one crucial discrepancy between PHLO and OXG that

should not be overlooked. The point of divergence is related with Theorem 3 of the

theory of phlogiston, implied by (3) and C2(PHLO), and therein is established the

decrease in weight of every substance after having been subjected to the process of

combustion. As has already been pointed out, this theorem is deduced on the basis

of the constriction that stipulates the additivity of the aggregation and of one of the

fundamental laws of PHLO according to which every substance emits phlogiston

on being burnt. Given that OXG shares C2 with PHLO, but this is not the case with

the aforementioned fundamental law, the deduction of the same theorem on the

basis of the axioms of OXG is impossible. One important consequence emerges

from this, namely that from OXG no account is given of the apparent decrease in

weight that certain bodies undergo when subjected to combustion. It only supplies

an explanation of the increase in weight of metals and certain minerals (such as

sulphur) when burnt. By combining C2(OXG) and Axiom (3), included in the

formal definition of M(OXG), where the absorption of oxygen by the some of the

bodies that react in combustion is established, it is possible to obtain a theorem in

which the increase in weight undergone by combustible substances when burnt is

expressed. It should be noted that this theorem contradicts the second theorem of

PHLO, which is derived on the basis of C2(PHLO) and Axiom (3) from the formal

definition of M(PHLO). This serves to highlight how the increase in weight of

metals is the principal anomaly that the defenders of the theory of phlogiston

tackled, unsuccessfully, and how the solution of this anomaly from the theory of

oxygen requires the theory of phlogiston to be rejected, by denying certain

consequences that follow the fundamental law of that theory.

We now go on to give the definitions of the constraints for OXG that, as has

already been mentioned, coincide with those of PHLO.

C1(OXG): for all X: X [ C1(OXG) iff [ = X � Mp(OXG) and for all x, y [ Xand for all s: if s [ Sx \ Sy then wx (s) = wy (s)

C2(OXG): for all X: X [ C2(OXG) iff [ = X � Mp(OXG) and for all x, y,

z [ X:

(a) 8: S 9 S ? S where S := [ {Sx/x [ Mp(OXG)}

(b) for all s [ Sx, s0 [ Sy (if s 8 s0 [ Sz then wz(s 8 s0) = wx(s) ? wy(s0))

350 M. Caamano

123

The global constraint for OXG would be:

GC(OXG) = C1(OXG) \ C2(OXG)

The theorem that is demonstrated below, which is implied by Axiom (3) and by

C2(OXG), establishes the increase in weight of every body subjected to combustion.

Theorem 3 for all \c, a, t1[ [ Dom(g): w(g1(c)) [ w(c)

Proof Let \c, a, t1[ [ Dom(g)

\g1(c), g2(a), t2 [ = g(\c, a, t1[), by definition of gg1(c) contains o in t2 and c does not contain o in t1, by (3ii)

w(g1(c)) = w(c) ? w(o), by C2(OXG)

w(g1(c)) [ w(c) h

3.5 Intertheoretical Links of OXG

Six essential links can be identified for OXG. Of them, those relative to S, C, w and

8 are shared with PHLO. As here we are dealing with links that have already been

examined, only their origin is given, and no analysis is made of them. As was

already pointed out in the previous section, when determining the values of the

function w some model from classical particle mechanics (CPM) needs to be

assumed: i.e. a link L1 is required between M(CPM) and Mp(OXG). In the same

manner, with regard to the determination of S, the need to resort to the notion of

material as it is defined in the mechanistic theories of Boyle and Newton, as well as

the characterization of w offered in CPM, became evident. Lastly, 8 also conserves

its links, which connect OXG with classical mechanics, with the cameralist theory

and with Boyle’s corpuscular theory. The dependence on ordinary knowledge,

already described in the previous section, should not be forgotten.

The concepts of air, respiration, combustion and combustible substance lose

some of their links with the iatrochemical theory as with the physics of Boyle.

Nevertheless, their determination, which is principally dependent on ordinary

knowledge, does not vary, so that the changes in beliefs that justify the

disappearance of these links do not, however, justify the consideration that this

variation generates a conceptual change.

I shall now touch on a pair of historical aspects relative to alterations in

intertheoretical links. The first of these is the evolution of the concept of air,26 and

the second, the role of Boyle as a driving force behind the scientific revolution, and

precursor of Dalton’s chemical atomism. Both topics are partially linked, as air not

only ceases to be considered as an environment to go on to be considered as an

aggregate made up of a variety of elements that are active in combustion, but it is

also completely disassociated from the state in which it appears. Air ceases to be a

sui generis substance, which is merely the carrier of particles and passive in the

combustion process, to go on to be considered as a substance composed of different

types of elements, among which appears oxygen, and which is active in the process

of combustion. To the extent in which they depend on the concept of air, some of

the intertheoretical links of the concepts of respiration, combustion and combustible

26 Cf. Brock, op. cit., pp. 124–127.


123

substance with the iatrochemical theory and the physics of Boyle will also be

modified. For the determination of the three concepts, with everything, the use of

ordinary knowledge will still be fundamental. In general, states are dissociated from

elements, the former being explained on the basis of changes in temperatures or

affinities with other substances, and not in terms of chemical principles or intrinsic

properties of elements. As a result of this, in the chemistry of Lavoisier, air will be

identified with gases (aggregates of elements) without the substances thus named

having necessarily to remain in a gaseous state. According to the above, the

elements are dissociated from both states and principles. Consequently, chemical

reactions (including combustion) can no longer be explained by resorting to

principles but to proportional relations between substances. The effective

disappearance of the notion of principle in chemistry will not come about until

the emergence of Dalton’s atomic theory. Nevertheless, proponents of OXG will

make only nominal use of this notion, varying the meaning of ‘principle’ until they

make it coincide with that of ‘element that is systematically active in certain types

of reactions’.

If, as has been supposed in the previous section, it is acknowledged that the

physical theories of Boyle and Newton are essential in the determination of w, 8 and

S, then the global link of OXG could be defined as the intersection of all the

intertheoretical links of OXG.

3.6 The Theory-Element of OXG and its Empirical Claim

T(OXG) := \K(OXG), I(OXG)[ where

K(OXG) := \Mp(OXG), M(OXG), Mpp(OXG), GC(OXG), GL(OXG)[and I(OXG) � Mpp(OXG) is such that

(1) I0 � I(OXG) where I0 is {the combustion of non-metallic inorganic

substances (wood, coal), the calcination of mercury, the reduction and

composition of mercury}

(2) all the members of I(OXG) are sufficiently similar to those of I0.

Formulation of the empirical claim of OXG: The intentional applications may be

extended to a set of models that are connected among themselves by GC(OXG)

which satisfy the laws and the intertheoretical links.

The assertion states that the members of I(OXG), which consist of various

structures of substances S (such as air A and combustible substances C) can be

extended to the structures of the substance oxygen O. The structures must comply

with the laws and constraints. Members of I(OXG) will consist of a structure of

temporal instants T, with a series of functions c for determining the reactions of

combustion, functions w for weight, and 8 for aggregation. The relations R and Qalso intervene by giving account of the members of I(OXG), as they establish

connections between different substances, on the basis of the property of being

equal or less breathable, and in accordance with their equal or reduced oxidization,

respectively. In each individual reaction or process of combustion determined by g,

T and S (and consequently their subsets C, A, O) along with the other functions and

relations that have been mention, the laws of OXG must hold. However, each

352 M. Caamano

123

substance of S will be applicable in at least one member of I(OXG). The string of

functions w must assign to each substance the same weight throughout the chemical

reaction in I(OXG) in which it intervenes.

4 Analysis of the Incommensurability Relation Between PHLO and OXG

In this section, I shall argue in favour of the thesis, assumed by some structuralists,

that every coherent notion of incommensurability implies that of applicative

reduction,27 and even, that of ontological reduction, although I shall propose a

modified version of the latter, which does not presuppose that of theoretical

reduction.28 Note that, if incommensurability is presented as being incompatible

with any type of reduction relation, it must thus have other resources alternative to

reduction in order to lend meaning to the affirmation that incommensurable theories

that are not reducible in any way nevertheless give account of the same setting of

reality. However, defenders of the thesis of incommensurability have not supplied a

sufficiently developed proposal in this sense.

The episode of incommensurability that is reconstructed in the present work will

show the possibility of applying a weak applicative or ontological reduction

between incommensurable theories, even when the tout court exact, approximate or

ontological reductions may not be possible. The notion of weak or applicative

ontological reduction does not presuppose compliance with the conditions for a

strict or theoretical reduction, rather only compliance with the conditions for a

correspondence between [a certain especially relevant subset of] the intended

applications of the theories, the domains of which would be shared or correlation-

able with relations structures from the other theory. The proposal of this type of

reduction responds, in part, to methodological considerations (such as guaranteeing

a coherent notion of incommensurability) and, in part, to the acknowledgement of

the meta-theoretical evidence (derived from the study of incommensurable theories)

that is available.29

The stance that I shall take can be summarized in terms coined by Stegmuller

(and employed also by Moulines), by stating that the incommensurability studied

herein will be theoretical and not empirical.30 That is, the relation of incommen-

surability will affect the class of potential models of the theories involved, without

extending to the class of partial potential models (i.e. those structures that determine

27 Cf. Dıez and Moulines (1999) op. cit., p. 459.28 Dieter Mayr anticipated this notion of reduction by establishing the explanation of anomalies as a

criterion for reduction between incompatible successive theories (cf. Mayr 1976, pp. 275–294).29 The relevant sources to this respect are mentioned in Moulines (1984, pp. 69–70). The author refers

there to numerous pairs of theories related by ontological reduction, being plausible to assume that this

relation would imply some sort of translation between the partial potential models of these theories (at

least for the case of homogeneous ontological reduction, which will be discussed below). Among the pairs

of theories mentioned by Moulines are the following: Kepler’s Planetary Theory and Newtonian Particle

Mechanics, Geometric Optics and Ondulatory Optics, Simple Equilibrium Thermodynamics and Kinetic

Theory of Gases, Newtonian Particle Mechanics and Quantum Mechanics, Mendelian Genetics and

Molecular Biology (cf. pp. 61–62).30 Cf. Stegmuller (1976, § 11); Moulines, Exploraciones Metacientıficas, cit., p. 207.


123

the intentional applications of theories). The latter class will form a setting that is

susceptible to reduction.

4.1 Relation of Theoretical Incommensurability Between PHLO and OXG

Below, I shall reproduce the definition of incommensurability given by J.A. Dıez

and C.U. Moulines in a recent work in which they present certain new developments

in the structuralist programme.31

‘‘Let N, N* be two different theoretical networks. We say that N is supplantable

(incommensurably) by N* iff there is a relation q � Mp* 9 Mp and a non-empty

set Ia such that:

(1) q is not effectively calculable.

(2) is a function, is effectively calculable and Rec( ) = Mpp.

(3) There do not exist n non-empty sets M1*, …, Mn

* included in M0* such that q[M1

*

[ �� [ Mn*] �& M0.

(4) (i)Ia � I0 \ qe[I0*] y (ii) for each y [ Ia (y [ r[M0

*] ^ y 62 r[M0])’’.32

When explaining the above formal definition, both authors stress one aspect of

great interest. This is the distinction between two factors of incommensurability:

one which is conceptual (referred to in Condition (1), which affects potential

models, and thus, the conceptual apparatus of the respective theories), and another,

which is propositional (referred to in Condition (3), which affects the actual models,

and consequently, the laws of the respective theories). The former determines the

impossibility of establishing a systematic correlation between the models (and, thus,

between the basic concepts) of the two theories. The latter determines the

impossibility of deducing the fundamental laws of the supplanted theory on the

basis of the laws of the supplanting one. In any case, Conditions (2) and (4) give

account of the partial nature of incommensurability (or of the degree of

commensurability that it implies), considering, equally, the two aforementioned

factors. In (2) the possibility of systematically correlating the partial potential

models (and consequently, the basic non –theoretical concepts) of each theory is

determined. Finally, (4) determines that the set of anomalies from the supplanted

theory forms part of the set of intentional applications of both theories, and that

these anomalies should not be submissible under the laws of the supplanted theory,

but that they should be submissible under the laws of the supplanting theory.

31 The definition of theoretical suplantation with incommensurability is established in Dıez and Moulines

(1999, pp. 456–460). The reason why the definition of incommensurability given in Balzer et al. (1987,

pp. 306–320) is not given here has to do with its lesser interest for the analysis of this case in particular. In

the present study the case requires a definition of incommensurability that may be independent of the

theoretical reduction and, in that manner, it may comply with all possible cases of incommensurability,

among them, these that are not susceptible to theoretical reduction, such as the one dealt with herein.

There is a more recent and suited structuralist proposal for defining the relation of incommensurability. Its

character is more global, it allows the substitution of highly specific but not very illuminating conclusions

for the present case to be substituted by other more general and revealing ones. In particular, it makes it

possible to do without the prerequisite of the reduction relation, resorting to the T-non-theoretical relation

to establish a sufficiently significant relation between the theories.32 Dıez and Moulines (1999, p. 459).

354 M. Caamano

123

In order to demonstrate that the relation between PHLO and OXG fulfils

condition (1), it needs to be demonstrated that this relation does not satisfy the

definition of exact reduction. Let us first consider the six constituent conditions of

the definition of reduction that are established in the structuralist programme.33

Firstly, the condition of derivability (D) of the laws of the reduced theory T is

established on the basis of the reducing theory T0 and through the mediation of q.

Secondly, the reduction needs to be compatible (C) with the constraints for both

theories. A third condition is the compatibility (L) of the reduction with the links of

both theories. The fourth condition requires that each potential model of T be related

(‘‘translatable’’) (T) by means of reduction with some potential model of T0. In fifth

place, independence (I), with respect to the definition of q, from the derivation

expressed in (D) is required. Lastly, as a sixth condition a connection (IA) is

established between the intentional applications of the two theories.

I shall now reproduce the formal definition of reduction proposed in the

structuralist programme:

If T and T0 are idealised theoretical elements, then q directly reduces T to T0

(T0 q T) iff

(1) q � Mp0 9 Mp(2) Rng(q) = Mp (T)

(3) for all x0, x: if \x0, x[ [ q y x0 [ M0 then x [ M (D)

(4) for all X0 � Dom(q): if X0 [ GC0 then q(X0) [ GC (C)

(5) for all x0, x: if x0 [ GL0 y \ x0, x[ [ q then x [ GL (L)

(6) for all y [ I there exists y0 [ I0 such that \y0, y[ [ (IA)

Let us now consider the difficulties in applying the structuralist definition of

reduction to the present case, where the impossibility of calculating the reduction

relation between PHLO and OXG will be obvious. Firstly, the fulfilment of

condition (3) is incompatible with that of condition (4). The intervention of

phlogiston in combustion, which is necessary for the fulfilment of (3), involves

consequences that are incompatible with the role of oxygen in the same

phenomenon, since the constraint C2(OXG) of the extensivity of the weight, which

is required for the satisfaction of (4), means that in one case the combustible

substance decreases in weight, and in the other, the same substance increases in

weight. The conceptual framework in which oxygen is the substance that is

principally active in the process of combustion, recognisable due to it being

absorbed by the substances that are submitted to this process, and in which these,

due to C2(OXG), will have to increase in weight, is incompatible with the other one

in which the substance that is principally active in combustion is phlogiston,

recognisable due to having been emitted by the substances that are subjected to this

process, and in which these, C2(PHLO), will have to decrease in weight. Similar

inconsistencies arise when attempting to determine from OXG the derived concept

of dephlogistication, which forms part of PHLO. For the reasons explained,

conditions (3) and (4) cannot be satisfied jointly, and given that both are necessary

33 Cf. Balzer et al. (1987, pp. 275–279).


123

conditions for exact reduction, it can now be concluded that this will not come

about.

Proof Let (3) for all x0, x: if\x0, x[ [ q y x0 [ M(OXG) then x [ M(PHLO), y (4)

for all X0 � Dom(q): if X0 [ GC(OXG) then q(X0) [ GC(PHLO)(4) for all X0 � Dom(q): if X0 [ GC(OXG) then q(X0) [ GC(PHLO)Let X0 [ GC(OXG)for all X0 if X0 [ GC(OXG) then for all s [ Sx0, s0 [ Sy0 (si s 8 s0 [ Sz0 then

wz0(s 8 s0) = wx0(s) ? wy0(s0)), by definition of C2(OXG) and of GC(OXG)

for all \ c, a, t1[ [ Dom(g): w(g1(c)) [ w(c), Theorem 3 of OXGfor all X if X [ GC(FLO) then for all s [ Sx, s0 [ Sy (si s 8 s0 [ Sz then

wz(s 8 s0) = wx(s) ? wy(s0)), by definition of C2(PHLO) and of GC(PHLO)

for all c\, a, t1[ [ Dom(g): w(g1(c)) \ w(c), Theorem 3 of PHLOfor all\c, a, t1[ [ Dom(g): w(g1(c)) \ w(c) and for all\c, a, t1[ [ Dom(g): not

w(g1(c)) \ w(c)

it is not the case that q(X0) [ GC(PHLO), by ECQIf X0 [ GC(OXG) then it is not the case that q(X0) [ GC(PHLO)for all X0 � Dom(q): if X0 [ GC(OXG) then q(X0) [ GC(PHLO) and if X0 [

GC(OXG) then it is not the case that q(X0) [ GC(PHLO)It is not the case that (3) for all x0, x: If \x0, x [ [ q y x0 [ M(OXG) then x [

M(PHLO), and (4) for all X0 � Dom(q): if X0 [ GC(OXG) then q(X0) [GC(PHLO), by reduction ad absurdum.

Condition (6) is not satisfied either, given that there are successful intentional

application of PHLO that are lost in the transition to OXG, although they are

subsequently recovered in the development of chemistry. In spite of the condition

that establishes the connection (IA) between the intentional applications of the two

theories not being satisfied in a literal interpretation, it is satisfied in a more

restricted version in which I and I0 are substituted, where appropriate, by Ia, (set of

anomalies contained in the theory of phlogiston), I0, (set of paradigmatic

applications of the theory of phlogiston) and I00 (set of paradigmatic applications

of the theory of oxygen). In effect, although not for each successful application of

PHLO is there a corresponding successful application in OXG (remember, for

example, the combustion of hydrogen), it is true that there is a set made up of the

aimed applications of both theories such that these applications are not submissible

under the laws of PHLO but are so under those of OXG. Expressed formally, the

new version of the condition (IA) would be as follows:34

Ia � I0 \ [I00] and for all y [ Ia (y [ r[M0

0] ^ y 62 r[M0]);

where r represents the ‘‘cut off’’ of T-theoretical concepts, assigning partial

potential model to actual or potential models that are formed by adding T-theoretical

concepts to the said partial potential models.

After revising which defining conditions of exact reduction satisfy the relation

between PHLO and OXG, and which do not, it is verified that the only condition

that is satisfied is the final one (IA), that which principally involves the intentional

34 As shall be reiterated at a later point, this condition appears in Dıez and Moulines (1999, p. 459), as

one of the defining conditions of the relation of incommensurability.

356 M. Caamano

123

applications and, thus the non T-theoretical conceptual framework (Mpp) that is

common to both theories in which such intentional applications are characterised. In

conclusion, the relation between FLO and OXG does not fit in with an exact

reduction relations, given that it does not satisfy five of the six conditions necessary

for this, but, as it does satisfy the condition (IA), it does fit in with a new type of

ontological reduction, of a weak or applicative nature (which shall be characterized

below), and it satisfies one of the two general conditions of exact reduction, namely,

the resolution of anomalies in the theory that it is aimed to reduce. Consequently,

the irreducibility is due to the non-satisfaction of the other general condition of

exact reduction, the derivability of the laws of theory that it is aimed to reduce on

the basis of the laws of the theory that is supposed to be the reducing one.

Condition (2) of the definiens of the relation of incommensurability is thus

fulfilled, as is deduced from the definitions of Mpp(PHLO) and of Mpp(OXG), to

each partial model of OXG (which intervenes in the reduction relation),

corresponds a partial model of PHLO until the domain of the partial models of

PHLO is used up. What is more, as we have seen, each partial model of OXG is

identified with a partial model of PHLO.

Thus, condition (3) of the definiens of the relation of incommensurability is

satisfied, as correlation between the actual models is impossible in all cases, due to

the application of the laws and the satisfaction of the constraint being incompatible.

That is, the application of the laws and the satisfaction of the constraints in one

theory are incompatible with the application of the laws and the satisfaction of the

constraints in the other theory. The deducibility of the laws of the supplanted theory

on the basis of the laws of the supplanting theory is impossible in all cases given the

general scope of this incompatibility.

Finally, condition (4) of the definiens of the incommensurability relation is also

satisfied, as the (interesting) set of anomalies from the supplanted theory contains

anomalies that are submissible under the laws of the supplanting theory (and not

under those of the supplanted theory). Any example of the calcination of metal is a

case in which the aforementioned is satisfied.

4.2 Empirical Commensurability and Weak Ontological Reduction Between

PHLO and OXG

Going back to the question of empirical commensurability between theoretically

incommensurable theories, it is possible to ask about the types of global inter-

theoretical relations in which this come about in the present case.

Firstly, the empirical convergence between both theories is clear in their

convergence in the pre-theoretical approach or in the theoretical presuppositions. As

is established in their definition, the inter-theoretical relation of theorization is

characterised by the presence of a special type of determining like, such as

interpretive links.35 The existence of determining links between the theorized theory

and that which theorizes constitutes a presupposed condition that must be satisfied

in order to guarantee the presence of interpretive links. On the other hand, these

35 Cf. Balzer et al. (1987, pp. 250–251, 278–279).


123

links must satisfy the requirements that determining links should always culminate

in actual models of the previous (or underlying) theory, and that they should not be

‘‘reversible’’, i.e. that they should not imply another determining link in the opposite

direction. The semantic determination of the PHLO-non-theoretical and OXG-non-

theoretical terms, be they either referential of definitional, presupposes the

acceptance of the laws of theories that underlie both theoretical elements. Given

that PHLO and OXG share all their non-theoretical concepts, the theories

presupposed in their determination or interpretation will be the same. The theories

of phlogiston and oxygen thus coincide in theorizing (or, at least, presupposing) the

same theories of matter, respiration and composition.

Secondly, both theories are reducible, either applicatively or by means of a weak

ontological reduction, as shall be seen below. Firstly, it should be borne in mind

that, as Moulines explicitly states, the structure formed on the basis of the non-T-

theoretical terms constitutes the ‘‘outside world’’ of T, while the structure

corresponding to the T-theoretical terms can be conceived as being the specific

apparatus of T for ‘‘seeing the world’’.36 The first type of substructure shall be

considered as a partial potential model of T, and will play decisive role in the

empirical interpretation and application of the potential model in which it is

included. With a more general character, the class of partial potential models is

identified with the basis of empirical contrasting or the empirical application of T,

the relevant subset of them being known as the ‘‘intentional applications’’ of T. With

regard to PHLO and OXG, and how this differs with that previously established,

both would be theorizations of Boyle and Newton’s physics (which are necessary in

order to determine the domain of the substances along with the aggregation and

weight functions), at the same time as of iatrochemistry and Boylean chemistry

(which are essential in the interpretation of the domain of the air substances, the

combustion function and the relation of respirability). Similarly, in accordance with

the above, the substructures of PHLO and OXG shaped on the basis of the shared

non-theoretical terms ‘substance’, ‘combustible substance’, ‘air’, ‘aggregation’,

‘weight’, ‘combustion’ and ‘respirability’ make up the terminology in which the

partial potential models that are common to both theories would be characterised,

and thus would necessarily intervene in the characterisation of a shared empirical

basis for theory testing.

The notion of ontological reduction was introduced in the structuralist meta-

theoretical context by C. U. Moulines with the aim of enriching the schema of exact

reduction, reproduced in the previous section. In such a schema the ontological

aspect that is inherent to our intuition about reduction would, in his opinion, be

avoided. Moulines explains his proposal in ‘‘Ontological Reduction in the Natural

Sciences’’,37 as well as in Sect. III.3.6. of Pluralidad y recursion.38 It also comes

into play, although in an implicit form, in the definition of incommensurability

given the structuralist programme.39 The interest in the notion of ontological

36 Cf. ibid., p. 277.37 This work is included in the compilation Reduction in Science, cit., pp. 51–70.38 ‘‘Conexiones ontologicas en la reduccion de teorıas’’, cit., pp. 364–374.39 Cf. Balzer et al. (1987, pp. 317–320).

358 M. Caamano

123

reduction lies in the fact that it makes it possible to identify the relation between the

domains in the respective theories, thus assuring that the theoretical reduction

relation that is established between them gives us information on the world, on the

empirical.

The notion of ontological reduction is highly relevant for the treatment of the

problem of incommensurability, since it allows the conception of a new type of

applicative ontological reduction, which clarifies certain aspects of this problem.

We should start by pointing out that the reconstruction of the correspondence

between domains, which is characteristic of ontological reduction, can be used not

only to verify the empirical adaptation of inter-theoretical reduction, but also to

determine the empirical connection between theories that have an inter-theoretical

relation other than theoretical reduction. For example it could be used to examine

the type and extent of empirical connection between two incommensurable theories

on a theoretical level. This, in fact, will be the aim of the analysis that is offered

below, in which it will be shown how the relation between PHLO and OXG falls

into the definition of a new type of ontological reduction, and, consequently, how

theoretical incommensurability (which is associated to theoretical irreducibility)

entails empirical commensurability (which is associated to ontological reduction).

Let us now examine the formalisation of the relation between basic domains

carried out by C. U. Moulines. Given that the structural relation between the basic

sets of different theories may reach a high level of complexity and that, moreover, it

involves direct consequences for the structural relation between the respective

relations and functions that are typified over these basic sets, it is worthwhile

analysing the form of these typifications first. These will indicate which class of

arguments the functions take, or to which class of accounts the relations are applied;

they will do so by establishing a procedure for constructing the sets corresponding

to these functions or relations on the basis of previously given sets, over which the

set-theoretical operations of power set and Cartesian product are repeatedly

applied.40

Let T and T0 be two theories related by means of reduction, each one of their

respective potential models will have the following form:

x ¼\D1; . . .;Dn;A1; . . .;Am; r1; . . .; rp[ and

x0 ¼\D01; . . .;D0v0 ;A01; . . .;A0w0 ; r

01; . . .; r0z0[

Di (or D0i) are the basic sets, Ai (or A0i) are the basic auxiliary sets, and ri (or r0i) the

functions or relations that constitute the potential model. For each ri there will be a

typification si making it possible to establish that:

ri 2 si Di1; . . .;Dih;Aj1; . . .;Ajk

� �

where Di1; . . .;Dihf g � D1; . . .;Dnf g and

Aj1; . . .;Ajk

� �� A1; . . .;Amf g

(and analogically for each r0i in T0).

40 Ibid., pp. 6–7.


123

It will be said that q � M0p 9 Mp is an ontological reductive link between T0

and T if, besides satisfying the formal definition of reduction, ‘‘it is formed at least

in part by a connection between some of the Di and some D0i and perhaps also some

A0j’’,41 the last of which will not be taken into account in the formal definition due to

the fact that they belong to the setting of mathematics and are not specific to

empirical theories. Based on an informal distinction by M. Spector between domain-

preserving reduction and domain-eliminating reduction,42 Moulines establishes a

formal distinction between homogeneous ontological reduction and heterogeneous

ontological reduction. Both types correspond to two general forms in which

different reductively linked theory domains can be related. If the basic sets of the

reduced theory are related by means of a total or partial identity (proper inclusion)

with those of the reducing theory, the theories will be considered to be connected by

means of a homogeneous ontological reduction relation. Whilst, if one of the basic

set of the reduced theory is related with one or more in the reducing theory, in such

a way that does not imply the identification of elements, rather a bi-univocal

correspondence between domains, the theories will be considered to be related by

means of a heterogeneous ontological reduction. The theories connected by means

of both types of ontological reduction links would maintain a relation of mixed

ontological reduction. Let us now see the definition of the first type of reduction, the

one that is relevant for the present case.

If T is reducible to T0 by means of q, then: q It is a homogeneous (ontological)reductive link of T to T0 iff:

For all x [ Mp, x0 [ Mp0, si q(x0) = x,

Then there exist i, j [ N such that

Di(x) is a basic set of x, Dj(x0) is a basic set of x0,

and Di(x) � Dj(x0)

As the theories of phlogiston and oxygen are non-reducible, it is clear that the

condition that is presupposed in the two definitions above, the reducibility between

the theories will not be fulfilled. Nevertheless, leaving this point to one side, it still

makes sense to ask about the satisfaction or not of the remaining definitional

conditions—those that are directly expressed in the conditions. As was stressed

above, on demonstrating that between PHLO and OXG there is no strict reduction

relation q; nevertheless, between both theories there is a special type of reduction

relation (which can be represented by ‘ a’) to the level of their respective classes of

paradigmatic intentional applications. If in the definition of the ontological

reduction the precondition that there must be a relation q is substituted by another

that established that there must be a relation a, it would be possible to establish a

new type of ontological reduction relation. Thus, in order to define this new type of

relation first we need to define the relation a, on which the novel character the

former depends.

41 Moulines (1991, p. 266).42 Spector (1978).

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123

If T and T0 are two idealised theoretical elements, then a weakly or applicatively

reduces T to T0 (T0 aT) iff there exists a relation � Mpp0 9 Mpp and a non-empty

set Ia such that:

(1) is a function, is effectively calculable and for all y, y0: if\y0, y [ [ then

y0 = y(2) (i) Ia � I0 \ [I0

0] and (ii) for all y [ Ia (y [ r[M0

0] ^ y 62 r[M0])’’.

The relation between PHLO and OXG fulfils condition (1) by virtue of the

definitions of classes of partial potential models in each theory. The satisfaction of

condition (2) by this relation has already been commented on in the previous

section.

We shall now verify that a is a homogeneous weak (ontological) reductive linkof PHLO to OXG.

We know that PHLO is reducible to OXG by means of a, by the definitions of

Mpp(PHLO) and Mpp(OXG) as well as by the analysis of the relation between the

classes of intentional applications of both theories that was carried out above.

and that for all y [ Mpp(PHLO), y0 [ Mpp(OXG), if a(y0) = y,

Then there exist S, S0 such that

S(y) is a basic set of y, S0(y0) is a basic set of y0,and S(y) = S0(y0), by the definitions of Mpp(PHLO) and Mpp(OXG)

Similarly, we know that PHLO is reducible to OXG by means of a

and that for all y [ Mpp(PHLO), y0 [ Mpp(OXG), if a(y0) = y,

Then there exist T, T0 such that

T(y) is a basic set of y, T0(y0) is a basic set of y0,and T(y) = T0(y0), by the definitions of Mpp(PHLO) and Mpp(OXG)

Thus, for OXG aPHLO, all the condition of the definiens of the homogeneous

ontological reduction are satisfied. All the basic sets of PHLO, namely, S y T, are

homogeneously linked with the basic sets of OXG, S0 y T0.The fact that the situation is different for any of the derived sets or domains is due

to some of them depending on T-theoretical concepts, for which there does not, nor

can exist an equivalent in the other theory. Specifically, I refer the domains F of

PHLO and O of OXG. Not only does it not occur that F is not identifiable with any

set or subset derived from OXG nor correlationable with any typification of OXG,

but also that any similar attempt at identification or correlation by means of any

modification in OXG would lead to incompatibilities within the theory itself. The

ontological irreducibility of the derived set F and O, both dependent on T-

theoretical concepts with incompatible implications, demonstrates one of the

defining features of the incommensurability between the theory of phlogiston and

that of oxygen. With regard to the rest of the derived domains, A and C are

homogeneously linked with A0 and C0.

4.3 Resolution of Anomalies and the Scientific Progress from PHLO to OXG

With regard to the comparative or gradual relations, which is commensurable with

respect to the substructures of the potential models (Mpp, I), incommensurability


123

makes it possible to compare the success of both theories on the basis of the shared

intentional applications that constitute anomalies in PHLO (consisting of the

increase in weight of certain substances when burnt) and not in OXG. Thus there is

an objective (inter-theoretical) procedure for comparing the greater or lesser success

of the theory of oxygen in relation to the theory of phlogiston. The explanation of

scientific progress by virtue of the relation between the intentional applications of

the rival theories is expounded in detail by C. U. Moulines, in his article ‘‘Is There

Genuinely Scientific Progress?’’.43 The author argues that the thesis of incommen-

surability, as well as that of the theoretical baggage accompanying it, is compatible

with epistemological comparability at the level of intentional applications, and at

the same time with scientific progress, which can be evaluated on the basis of this

comparability. Nevertheless, this would no longer be understood as additional

knowledge on the same things, which would presuppose a conceptually linear and

veritatively accumulative conception of scientific development in which theories

would be presented as sets of enunciations referring to things, but as additional

knowledge on certain identical or homomorphous sub-domains of paradigmatic

intentional applications of special epistemological or pragmatic relevance. This

conception of scientific progress allows conceptual discontinuities or partial

changes in the universe of discourse of competing theories to be produced, but it

also requires a minimal degree of continuity between some of them. In the same

way, conceptual discontinuity does not allow us to consider an accumulation of

truths linked to the transition between theories, but it permit the consideration of a

build-up of solved problems (or of especially interesting solved problems). From the

viewpoint of structuralism the existence is confirmed of an inter-theoretical

epistemic-pragmatic criterion for prioritising certain intentional applications, whose

possible extension to models is of special interest in this sense. This corresponds

with Condition (4) (i) of the aforementioned second definition of incommensura-

bility. Having more knowledge on the same things is substituted by knowing better

how to solve the same problems. The highly pragmatic nature of intentional

applications reinforces this idea. The possibility of extending more intentional

applications, or simply more interesting applications, to the models of a theory is

equivalent to being able to resolve more (epistemic-pragmatic) problems or more

interesting problems. Moulines formalises this last, more quantitative, aspect of

scientific progress in the following way:44

S ¼: I \ r Mð Þ;

where ‘S’ is the success of a theory, I its class of intentional applications and r(M)

the class of partial potential models that make up part of its models. Thus, the

success of a theory is defined by virtue of those intentional applications that can

effectively be extended to models of the theory to which they belong.

43 This article appears in Poznan Studies in the Philosophy of the Sciences and the Humanities (Moulines

2000).44 Moulines provides a formalization relative to theory-nets (ibid., p. 190), the one offered here, relative

to theory-elements, amounts to a simplified version of the former.

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123

In order to compare the success of two rival theories by applying the

aforementioned definition, these will have to share at least part of their intended

applications and, consequently, part of their partial potential models; since,

otherwise, neither theory could be compared with respect to the same property, i.e.

its success in the application of the theory to the same phenomena. By sharing I and

Mpp, PHLO and OXG share the type of property that is required to examine the

greater or lesser success of one of them with respect to the other. In spite of the fact

that a number of decades will be needed for the experimental demonstration of the

greater success of the theory of oxygen with respect to that of the theory of

phlogiston, the fact is that in the former there are many more paradigmatic

applications that are extendible to models that in the latter. Examples of this are all

those cases of calcination of metals. Expressed formally:

S½PHLO� � S½OXG�:From what has been said up until this point, it can be inferred that the decision

between scientific theories is governed by a rational criterion of success (resolution

of certain cognoscitive and/or paradigmatic problems); i.e. that this inter-theoretical

criterion permits the determination and comparison of scientific progress between

incommensurable (non-T-theoretically commensurable) theories. In this way, the

notion that the thesis of incommensurability is linked to the defence of irrationalism,

as well as to an absolute theoretical relativism, is refuted.

5 Conclusions

I will now highlight the three principal results that the study of the case that has

been presented has made it possible to reach:

(1) The contribution of a pro-example supporting the thesis that there exist locally

incommensurable theories, more particularly, those which are theoretically

incommensurable but empirically commensurable.

(2) The demonstration of strict irreducibility between each theory given the

incompatibility between the satisfaction of the condition of ‘‘deducibility’’ and

the satisfaction of the condition of maintaining the constraints.

(3) The possibility of an applicative and weak homogeneous ontological reduction

between the two theories and, consequently, the determination of the key

factor for explaining the objectivity of the progress that the transition between

incommensurable theories supposes.

Acknowledgments I am indebted to Jose Antonio Diez Calzada for his extremely insightful revisions to

many previous versions of this paper. I am also thankful to Ulises Moulines for providing me clear

guidelines on different intricate matters related to the structuralist treatment of the phlogiston case. This

work was financially supported by the Spanish Ministry of Education.


123

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