Carnapian Structuralism

Holger Andreas

Munich Center for Mathematical Philosophy

(penultimate draft; to appear in Erkenntnis)

Abstract

This paper aims to set forth Carnapian structuralism, i.e., a syntactic

view of the structuralist approach which is deeply inspired by Carnap’s

dual level conception of scientific theories. At its core is the axiomati-

sation of a metatheoretical concept AE(T) which characterises those

extensions of an intended application that are admissible in the sense

of being models of the theory-element T and that satisfy all links, con-

straints and specialisations. The union of axiom systems of AE(T)

(where T is an element of the theory-net N) will allow us to present

scientific theories in an axiomatic fashion so that deductive reasoning

in science can be formalised. Thereupon defeasible and paraconsistent

means of reasoning will be introduced. The logical study of scientific

reasoning is the key motivation of Carnapian structuralism. A fur-

ther motivation is to help overcome the syntactic-semantic split in the

philosophy of science.

1 Introduction: micrologic vs. macrologic

The relationship between Sneedian structuralism and Carnapian Wissen-

schaftslogik is ambivalent. On the one hand, the structuralist framework

aims to provide formal means of rationally reconstructing scientific theo-

ries and as such continues the Carnapian enterprise. On the other hand,

structuralism breaks with the syntactic view of scientific theories which is

essential to Carnap’s logic of science. As is well known, the structuralist no-

tion of a scientific theory is explicated in terms of interrelated sets of models

as opposed to interrelated sets of axioms.

1

In order to explain and to highlight the distinctive traits of the Sneed formal-

ism, Wolfgang Stegmuller [16, p. 12] came to speak about the difference be-

tween micrologic and macrologic. The former concerns inferential relations

between sentences and formulas, whereas the latter explicates such metathe-

oretical concepts as an intended application, a model of a theory-element,

a global link etc. Needless to say, Stegmuller considered the macrological

perspective of scientific theories to be more advanced and superior to the

micrological one. It seems to be an implication, moreover, of his account

of structuralism that the two perspectives exclude one another. Holding

macro- and micrologic to be incompatible is a telling case of the general

split into semantic and syntactic views in the philosophy of science.

As a consequence of introducing structuralism as a particular species of

the semantic view of scientific theories, research on the structuralist frame-

work has become largely disconnected from the analysis of scientific reason-

ing. The label non-statement view, which was introduced by Stegmuller,

emphasises this dissociation from the logician’s enterprise of studying the

norms and the laws of human reasoning. This characterisation was sup-

ported by the doctrine of the indivisibility of a theory-elememt’s and a

theory-net’s global empirical claim into single axioms. Not all structural-

ists, however, were happy about the non-statement characterisation and the

split into micro- and macrologic, let alone philosophers of science outside the

structuralist community. Nonetheless, the issue of scientific reasoning in the

structuralist framework has rarely been addressed in the literature. Hence,

the doctrine of the indivisibility of global empirical claims has remained

unrefuted.

The objective of the present paper is to reestablish the micrological per-

spective in the macrologic of structuralism. This will be brought about by

introducing a novel metatheoretical relation AE(T) into the structuralist

framework by means of Carnapian postulates. More precisely, I will set

forth axioms that characterise those theoretical ex tensions of an intended

application that are admissible in the sense of being members of a system

of extensions in which all links, constraints and specialisations are satisfied

and where all extensions are models of the respective theory-element. The

resulting system qualifies as a scientific theory in the sense of Carnap’s good

old-fashioned dual level conception. Carnapian structuralism appears a fit-

ting label for logical and philosophical investigations being guided by these

axioms.

2

Carnapian structuralism allows for the logical analysis of deductive and de-

feasible reasoning in science. It retains the expressive power of the struc-

turalist framework without losing the expressive resources of predicate logic.

This is its chief motivation. Carnapian structuralism, thus, aims to con-

tribute to the understanding of how formal mathematical logic is related

to actual scientific reasoning. It should be of interest, therefore, to any lo-

gician who thinks that formal logical systems are explanatory of scientific

reasoning.

The present paper refines and summarises results obtained in [2, 3, 4]. The

major refinements concern, first, the formulation of links and constraints,

which is unified in the present account. This deviation from classical struc-

turalism achieves a significant simplification of structuralist theory presen-

tation. Second, the inference system of preferred subtheories by Brewka

[6] now introduces defeasible forms of reasoning into structuralism. This

system is simpler and behaves logically more nicely than prioritised default

logic, which was used in [3].

2 The formal framework of Carnapian Structural-

ism

Structuralism makes essential use of set-theoretic predicates in the sense of

Suppes [17]. Such predicates have a semantic and a syntactic side such that

a statement view on set-theoretic predicates and their explicit definitions

seems to remain a viable option. In fact, Suppes [18, p. 30] himself considers

the analysis of scientific theories in terms of set-theoretic predicates as a

proper way of axiomatising such theories:

Although a standard formalization of most empirically signifi-

cant scientific theories is not a feasible undertaking for the rea-

sons set forth in the preceding section, there is an approach to an

axiomatic formalization of such theories that is quite precise and

satisfies all standards of rigor of modern mathematics. From a

formal standpoint the essence of this approach is to add axioms

of set theory to the framework of elementary logic, and then to

axiomatize scientific theories within this set-theoretical frame-

work. From the standpoint of the topics in the remainder of

3

this book, it is not important what particular variant of set the-

ory we pick for an axiomatic foundation. From an operational

standpoint what we shall be doing could best be described as

operating in naive set theory.

Both standpoints described by Suppes, the formal and the operational one,

are in play in the present investigation. As in classical structuralism, I will

proceed in a semiformal fashion using naive set-theory. However, the format

of structuralist theory presentation will be an axiomatic one as a theory-net

is given by a set of postulates and definitions. Moreover, the descriptive

vocabulary of these postulates and definitions is divided into theoretical

and observational terms as in Carnap’s dual level conception. Unlike the

latter conception, the key theoretical and observational terms of Carnapian

structuralism are set-theoretic predicates.

As has been pointed out by Suppes [18, Ch. 2], the semiformal style of defin-

ing set-theoretic predicates is precise enough to indicate how a complete

formalisation using axiomatic set theory can be obtained. Such a full for-

malisation of the definitions of set-theoretic predicates allows for a full for-

malisation of the axiomatic system of a theory-net to be developed in the

present investigation. This view upon the axioms of a theory-net amounts

to the formal standpoint.

The formal standpoint requires an axiomatisation of set theory in first order

logic.1 The first order system that comprises some variant of axiomatised

set theory is assumed to satisfy the following conditions:

(1) There is one and only one domain of interpretation, which comprises

empirical objects and sets.

(2) The axiomatisation of set theory must make room for Urelements in

order to distinguish empirical objects from one another.

Several systems satisfying these two conditions have been developed in the

literature. For an extension of ZF that accounts for distinguishable Ure-

lements see Loewe [14]. (This system is favoured by Gerhard Schurz (this

volume) in his axiomatic approach to structuralism.) Another promising

1Using higher order logic without set theory would be an alternative, though. This

strategy is not pursued here.

4

strategy is to use the formal axiomatisation of naive set theory in a para-

consistent framework, such as that of adaptive logics (see Peter Verdee [19]).

Throughout the paper I assume the reader to be familiar with the fundamen-

tal metatheoretical concepts of the structuralist framework. So, for example,

the symbols M(T), Mp(T) and r(T) (which stand for the set of models,

potential models and the restriction function) are understood in the stan-

dard way. Any structuralist reconstruction of a scientific theory comprises,

for any T ∈ N, explicit semi-formal definitions of these symbols. Carnapian

structuralism builds upon such definitions.

The notion of a theory-net is understood here in the wider sense of capturing

a set of axioms coming from whatever scientific theories. This understand-

ing conforms to the formal definition of a theory-net [5, p. 172] but does

not require such a net to have a tree-like structure. Theory-nets thus un-

derstood resemble theory-holons as introduced in [5, Ch. VIII]. Unlike the

latter, however, they do not capture relations of theoretisation, reduction

and equivalence between scientific theories. Two theory-elements T, T′ ∈ N

may well differ in regard to the type of their potential models.

3 Postulates for structuralism

In the original exposition of the structuralist framework by Sneed [15], the

global empirical claim of a tree-like theory-net was described in terms of T-

theoretical extensions of sets of intended applications. More precisely, this

claim says that, for all theory-elements T in the net N, the set of intended

applications of T has a set E(T) of structures such that (i) any intended

application of T has a T-theoretical extension in E(T), (ii) the members of

E(T) are models of T, (iii) any member of E(T) is a model of T′ whenever

T is a specialisation of T′ and (iv) all constraints are satisfied among the

members of E(T). This formulation of the global empirical claim came

out as a refinement of the Ramsey sentence, wherefore it was labelled the

Ramsey-Sneed sentence by Stegmuller.

In An Architectonic for Science by Balzer et al. [5], the by now classical

exposition of structuralism, the global empirical claim of a theory-net is

captured by explicit set-theoretic constructions and complemented by the

consideration of links among theory-elements. The global empirical claim

of a theory-element T, for example, is the proposition that there is a set Y

5

of models of T such that (i) Y satisfies all constraints, (ii) the members of

Y are correctly linked to potential models of other theory-elements T′ and

(iii) the reduction of the members of Y to partial potential models yields

the set I(T) of intended applications of T. For the global empirical claim

of a theory-net N, specialisation relations among theory-elements are to be

considered in addition to these conditions.

Implicit in all variants of global empirical claims is the notion of an admissi-

ble T-theoretical extension of an intended application. For theory-nets, this

relation may be explained as follows:

Explanation 1. Admissible T-extension

We say that a structure x is an admissible T-theoretical extension, or T-

extension, of a structure y and write (x, y) ∈ AE(T) if and only if (i) x is

a T-theoretical extension of y, (ii) x is a model of T, (iii) y is an intended

application of T, (iv) x satisfies all links and constraints to the admissible

T′-extensions of other intended applications, (v) all specialisation relations

are satisfied among the intended applications of theory-elements in N, and

(vi) any intended application of T has an admissible T-extension.

The core idea of Carnapian structuralism is to formally introduce the re-

lation AE(T) as a theoretical concept by means of axioms that qualify as

postulates in the sense of Carnap’s [9] dual level conception of scientific lan-

guage. To this end, we need to axiomatise the above explanation of AE(T).

Let us start with the first condition. An admissible T-extension x is required

to be a T-extension of a structure y:

P1(T) ∀x∀y((x, y) ∈ AE(T)→ y = r(T)(x))

where r(T)(x) is the restriction function that “cuts off” the theoretical re-

lations as standardly defined in structuralism.

The second condition of the above explanation is that a theoretical structure

must be a model of T in order to be an admissible T-extension of some other

structure:

P2(T) ∀x∀y((x, y) ∈ AE(T)→ x ∈M(T))

Third, it is required that the range of AE(T) consists only of intended

applications of T:

P3(T) ∀x∀y((x, y) ∈ AE(T)→ y ∈ I(T))

6

I(T), of course, denotes the set of intended applications.

To obtain a concise axiomatic formulation of links and constraints, I suggest

a unification of these two notions along the following lines. Recall that links

are relations among potential models of different theory-elements, whereas

constraints are relations among potential models of one and the same theory-

element. Both links and constraints constrain the theoretical extensions of

intended applications that are admissible. If a tuple of potential models

of T violates a constraint, the members of this tuple will not together be

members of a set Y of potential models that verifies the global empirical

claim of T. Links and constraints will always be expressible by some formula

φ(x1, . . . , xn), where x1, . . . , xn are potential models of possibly different

theory-elements.

The simplest types of links and constraints are equality links and constraints,

which require the interpretations encoded by two structures x and x′ to be

consistent with one another. For example, if x and x′ contain an inter-

pretation of the mass function, then these two interpretations of the mass

function must agree for arguments that are in the domains of both x and

x′.2

Why do we need links and constraints other than equality ones? For a

number of theory-elements, there are constraints that express properties of

compound objects. For example, if c is a particle composed of particles a

and b, then it must hold that m(c) = m(a) + m(b), where m designates

the mass function. The corresponding constraint expresses that mass is an

extensive quantity. Further kinds of constraints for compound objects are

formulated in equilibrium thermodynamics. All these properties, however,

can be equally well expressed by separate theory-elements. There is nothing

formally wrong with establishing a theory-element whose substantial claim

is the extensivity constraint of the mass function. The set of models of such

a theory-element may be defined as follows:

Definition 1. Models of EXTm

x is a model of the extensivity law of the mass function (x ∈M(EXTm)) if

and only if there exist P,C,m such that

(1) x = 〈P,C,m〉2Admittedly, it is not obvious that the authors of [5] intended to have equality links.

The formal notion of a link, however, is expounded there in such a manner that it allows

for equality links.

7

(2) x ∈Mp(EXTm)

(3) ∀u∀v∀w(C(u, v, w)→ m(u) = m(v) +m(w)).

C(u, v, w) is a relation with the intended interpretation that u is composed

of the objects v and w but not of further objects. This relation is considered

to be non-theoretical with respect to classical particle mechanics. By the

definition of Mp(EXTm) it is specified that the cardinality of P is three.

The primary motivation for the introduction of (intertheoretical) links is to

account for the transfer of data between two theories. Such links encode

a correlation among relations of two theory-elements T and T′. Again, if

this correlation is not of the equality type, it can be expressed by a separate

theory-element. Let us therefore confine the consideration of links and con-

straints to equality ones and express all non-equality links and constraints

by separate theory-elements. This will lead to a more concise formulation

of the structuralist representation scheme.

One further consideration is necessary to achieve a unified axiomatic for-

mulation of links and constraints in terms of admissible T-extensions. A

typification of a relation R in structuralism has the form R ∈ σ(D1, . . . , Dk),

where σ(D1, . . . , Dk) is an echelon set over D1, . . . , Dk, which are base sets

of a structure x. Alternatively, we may typify a relation R by a proposi-

tion of the form R ⊆ σ(D1, . . . , Dk). In this formulation, σ(D1, . . . , Dk) is

the total relation corresponding to R in a structure x whose base sets are

D1, . . . , Dk. So, let σR(x) designate the total relation that corresponds to

R in a structure x.

This being said, the axiomatic formulation of equality links and constraints

for a relation R is almost straightforward:

P4(T,T′, R) ∀x∀y∀x′∀y′((x, y) ∈ AE(T) ∧ (x′, y′) ∈ AE(T′)→∀z(z ∈ σR(x) ∩ σR(x′)→ (z ∈ (x)R ↔ z ∈ (x′)R)))

where T and T′ need not be distinct. In less formal terms, this postulate says

that, wherever two structures x and x′ overlap in terms of their empirical

domains and in terms of a relation concept R, the interpretation of R by x

must agree with that of x′. (x)R denotes the extension of the relation R in

the structure x, as is standard in the structuralist notation.

Let us now deal with specialisations. T′ being a specialisation of T (in

symbols: σ(T′,T)) means that, first, all intended applications of T′ are also

8

ones of T and, second, that the admissible T′-extensions of y must also be

admissible T-extensions of y. The first of these two conditions can explicitly

be captured by the following postulate schema:

P5(T,T′) ∀x(σ(T′,T)→ (x ∈ I(T′)→ x ∈ I(T)))

The second condition is implied by this postulate schema and P1(T)–P4(T)

and P6(T).

So far, the postulates do not express any empirical claim concerning the

intended applications of T. It might just be the case that no intended ap-

plication of T is successful in the sense of having an admissible T-extension.

In this case, the extension of AE(T) would be the empty set. Carnapian

postulates, however, have the potential to also express some empirical con-

tent, in addition to specifying the meaning of theoretical concepts. We shall

therefore advance a postulate that specifies a condition of success, i. e., the

condition that T successfully applies to any of its intended applications. Be-

ing successful as an intended application of T means that it can be extended

to a T-theoretical structure that is a model of T and, moreover, satisfies all

the links, constraints and specialisations. Hence, any intended application

of T must have an admissible T-extension:

P6(T) ∀y(y ∈ I(T)→ ∃x((x, y) ∈ AE(T)))

The postulates P1(T)−P6(T) do capture the intended meaning of AE(T)

completely. Henceforth, the set of these postulates is designated by P (T)

and is referred to as the system of postulates for the theory-element T. D(T)

designates the set of definitions of M(T), Mp(T) and r(T). In the case of

a theory-net N, P (N) designates the set that contains P1(T) −P6(T) for

all T ∈ N. Likewise, D(N) designates the set that contains precisely the

definitions of M(T), Mp(T) and r(T) for all theory-elements T in N.

4 Semantics of theoretical terms

A formal study of scientific reasoning must have expressive resources for

the formulation of local empirical claims, i.e., claims about the particular

properties of particular empirical objects. To give a very simple example,

asserting the proposition that some object has a mass of such and such a

value is a local empirical claim. To prepare the formulation of local empirical

9

claims in the present system, it is necessary to spell out the semantics of

theoretical terms and postulates that underlies the postulates P (N). This

semantics will also be used for the formulation of N’s global empirical claim.

Following Carnap [9], we shall understand the notions of a theoretical term

and of a postulate such that a theoretical term is a symbol whose meaning

is determined through the axioms of a scientific theory, wherefore such ax-

ioms are also called postulates. What is it for the meaning of a term to be

determined by axioms? Carnap’s Foundations of Logic and Mathematics [8]

contains the seeds of a precise semantics of theoretical terms. There he de-

velops the idea that abstract, or theoretical, terms are indirectly interpreted

through axioms of a scientific theory. That means that the interpretation

is given by one or several sentences of the object language as opposed to

an assignment of an extension or intension in the metalanguage. Sentences

indirectly interpreting symbols are to be adopted as axioms in the calcu-

lus that is associated with the formal language and the theory in question.

This adoption does not rest on a prior understanding or assumption of truth

concerning the axioms. Only observation terms are directly interpreted to

the effect that only propositions of the observation language have a direct

factual content.

Carnap’s notion of indirect interpretation in the Foundations of Logic and

Mathematics [8] has a strong formalist flavour. In his Beobachtungssprache

und theoretische Sprache [9] we can recognise the elements of a model-

theoretic notion of indirect explanation. There he proposes to divide a

scientific theory TC into the Ramsey sentence TCR and the following con-

ditional:

TCR → TC

This conditional became labelled later on the Carnap sentence of a scientific

theory. Recall that TC designates the conjunction of T- and C-postulates.

T-postulates are those axioms of the theory that contain only theoretical

terms, whereas C-postulates have occurrences of both theoretical and ob-

servation terms, thereby establishing connections between the theoretical

and the observational vocabulary of the respective scientific theory. Let Vodesignate the set of observation terms and Vt the set of theoretical terms.

L(Vo) and L(Vo, VT ) designate the respective languages. PTC denotes the

set of T- and C-postulates.

10

The Ramsey sentence of a theory TC in the language L(Vo, Vt) is obtained by

the following two transformations of the conjunction of T- and C-postulates.

First, replace all theoretical symbols in this conjunction by higher-order

variables of appropriate type. Then, bind these variables by higher-order

existential quantifiers. As result one obtains a higher-order sentence of the

following form:

∃X1 . . . ∃XnTC(n1, . . . , nk, X1, . . . , Xn) (TCR)

where X1, . . . , Xn are higher-order variables.

Now, Carnap instructs us to understand the Carnap sentence as follows: if

there is an interpretation of the theoretical terms that satisfies TC (in the

context of the given interpretation of L(Vo)), the theoretical terms should

be understood as designating such an interpretation. Notably, the Carnap

sentence indirectly interprets the theoretical terms as their interpretation

is characterised by the condition that TCR → TC is always true. Carnap,

however, does not address the problem that TCR → TC usually does not

uniquely interpret the theoretical terms. Nor is the case considered where

the Ramsey sentence is false.3

Taking up this train of thought of Carnap, we can form the notion of an ad-

missible structure of the complete language L(Vo, Vt): an L(Vo, Vt) structure

is admissible if and only if it (i) extends the given L(Vo) interpretation to

interpret the Vt terms and (ii) satisfies PTC , provided there is such a struc-

ture. If there is no such structure, all extensions of L(Vo) that interpret the

Vt terms may be considered admissible. L(Vo, Vt) structures may interpret

Vt terms in a theoretical domain Dt that expands the observation domain

Do. Henceforth, let Ao designate the given or intended interpretation of

the observation language and EXT (Ao, Vt, Dt) the set of extensions of Ao

that interpret Vt, where Vt terms may be interpreted in Dt ∪Do. MOD(A)

designates, for a set A of sentences, the set of models of A.

To make these ideas about admissible structures precise [1]:

Definition 2. Admissible structures

Sa designates the set of L(Vo, Vt) structures that are admissible under an

3The former problem is addressed in Carnap [10], where he uses Hilbert’s ε-operator

for expressing that the Carnap sentence yields only an indefinite description of theoretical

terms.

11

interpretation of the Vt symbols by the postulates PTC . It is defined as

follows:

Sa :=

MOD(PTC) ∩ EXT (Ao, Vt, Dt)

if MOD(PTC) ∩ EXT (Ao, Vt, Dt) 6= ∅,EXT (Ao, Vt, Dt) if MOD(PTC) ∩ EXT (Ao, Vt, Dt) = ∅.

Understanding thus the notion of an admissible structure, the following

truth-rules are intuitive:

Definition 3. Truth rules for theoretical sentences

ν: L(Vo, Vt)S → {T, F, I}, where L(Vo, Vt)

S designates the set of L(Vo, Vt)

sentences:

(1) ν(φ) := T iff for every structure A ∈ Sa, A |= φ;

(2) ν(φ) := F iff for every structure A ∈ Sa, A |6= φ;

(3) ν(φ) := I (indeterminate) iff there are A1,A2 ∈ Sa such that A1 |= φ

and A2 |6= φ.

The idea lying behind these rules is rather simple. A theoretical sentence

is true if and only if it is true in every admissible structure. A theoretical

sentence is false iff it is false in every admissible structure. And a sentence

has no determinate truth-value iff it is true in, at least, one admissible

structure and false in, at least, another structure being also admissible.

The semantics of defined terms can be considered as a special case of the-

oretical terms. For a set Vd of defined terms and a set D of corresponding

definitions, MOD(D) ∩ EXT (Ao, Vt, Dt) will be a singleton. Hence defini-

tion 2 remains to be applicable to defined terms, where in such applications

a set Ad of definitions replaces the set PTC of postulates.

Things are slightly more complicated if the language of our theory TC con-

tains both defined and theoretical terms. Then, the definition of admissible

structures needs to be adjusted as follows:

Definition 4. Admissible structures of L(Vo, Vd, Vt)

Sa designates the set of L(Vo, Vd, Vt) structures that are admissible under an

interpretation of the Vt symbols by the postulates PTC and an interpretation

12

of the Vd symbols by the definitions DTC . It is defined as follows:

Sa :=

MOD(PTC ∪DTC) ∩ EXT (Ao, Vt ∪ Vd, Dt)

if MOD(PTC ∪DTC) ∩ EXT (Ao, Vt ∪ Vd, Dt) 6= ∅,EXT (Ao, Vt ∪ Vd, Dt) ∩MOD(DTC) otherwise.

The present truth-rules apply nicely to the postulates themselves. If

MOD(PTC)∩EXT (Ao, Vt, Dt) is not empty, all postulates will be assigned

to the value true. If the intersection of MOD(PTC) and EXT (Ao, Vt, Dt)

turns out to be empty, their truth-value will be indeterminate, apart from

some contrived cases.4 In any case, at least one postulate will not be as-

signed the value true if MOD(PTC) ∩ EXT (Ao, Vt, Dt) is empty. We can

say that the valuation of the postulates captures the empirical claim of the

corresponding scientific theory:

Explanation 2. Empirical claim of a theory

Let Θ be a scientific theory that consists of the postulates PTC . Then, Θ is

empirically true if and only if, for all φ ∈ PTC , ν(φ) = T .

This explanation rests on the following considerations. First, the Ramsey

sentence TCR is taken to represent the empirical content of an axiomatic

system PTC , following Carnap [9]. Second:

Proposition 1. Let TCR designate the sentence being obtained from TC

by Ramsification, where TC is interpreted in the domain 〈Do, Dt〉. Then,

TCR if and only if MOD(PTC) ∩ EXT (Ao, Vt, Dt) 6= ∅.

See [1] for proof.5 And third:

Proposition 2. For all φ ∈ PTC , ν(φ) = T if and only if MOD(PTC) ∩EXT (Ao, Vt, Dt) 6= ∅.

The forward direction of this proposition is trivial to prove. The other di-

rection can easily be proved by contradiction. In sum, explanation 2 follows

from propositions 1 and 2, and taking TCR to express the empirical content

of TC.4Here is an example of such a contrived postulate: α → β with α being an L(Vo)

sentence and being falsified by Ao and β being a theoretical sentence.5The specification of the domain of interpretation of TC is necessary for fixing the

range of the higher-order variables in the Ramsey sentence. If we were to allow these

variables to range over echelon sets of 〈Do, D′t〉 (Dt 6= D′

t) the equivalence of proposition

would not hold.

13

5 Semantics of AE(T)

Recall that the system of postulates P (N) in section 3 is intended to intro-

duce, for all T ∈ N, AE(T) as a theoretical term. Hence, in order to apply

the present semantics of theoretical terms to the above axiomatisation of

AE(T), we merely need to specify Vo, Vd, Vt, PTC , DTC and Dt in defini-

tion 4 for the language L(N) in which the postulates P (N) and definitions

D(N) are given. As for Vo, it seems obvious that

pI(T)q ∈ Vo(N) for all T ∈ N

since I(T) is assumed to be completely and directly interpreted in struc-

turalism. We do not need to use axioms of P (N) ∪D(N) to measure I(T),

which is the distinctive criteria of a T -non-theoretical term with T being

given by P (N) ∪D(N). Vo, furthermore, may or may not contain individ-

ual constants designating empirical objects. The observation language of

a theory-net N, however, does not contain symbols for T-non-theoretical

concepts since the interpretation of these concepts is specified by the respec-

tive components of the intended applications of T, which in turn are given

by the interpretation of pI(T)q. T-non-theoretical concepts are referred to

in the definitions of M(T) and Mp(T) as variables because their definientia

have the general form of saying that there are sets D1, . . . , Dk and relations

R1, . . . , Rn such that such and such conditions are satisfied.

Of Vt we can say that

pAE(T)q ∈ Vt(N) for all T ∈ N

pAE(T)q is a theoretical term whose meaning we have introduced through

the postulates P (N). Vt may or may not contain constants designating the-

oretical objects. Vt does not contain symbols for T-theoretical concepts for

the same reasons why Vo(N) does not contain symbols for T-non-theoretical

concepts.

As for PTC , we simply set

PTC := P (N)

where P (N) is the set of postulates for the relation AE(T), for all T ∈ N.

Besides theoretical terms, the language of the present axiom system contains

the defined terms pM(T)q, pMp(T)q and pr(T)q. These are set-theoretical

14

predicates which are introduced through explicit definitions, as is well known

from the standard format of structuralist theory presentation. Hence,

pM(T)q, pMp(T)q, pr(T)q ∈ Vd(N)

DTC := D(N).

As all domains of interpretation of T-theoretical concepts are given by the

intended applications of T, no separate theoretical domain of interpretation

needs to be specified. Hence, Dt(N) can be set to the empty set. All

placeholders of the present semantics of theoretical terms have thus been

specified for the axiomatic system P (N) ∪ D(N). We can thus define the

set of admissible structures of L(N):

Definition 5. Admissible structures of L(N)

Sa designates the set of L(N) structures that are admissible under an inter-

pretation of the Vt(N) symbols by the postulates P (N) and an interpretation

of the Vd(N) symbols by the definitions D(N).

Sa :=

MOD(P (N) ∪D(N)) ∩ EXT (Ao, Vt(N) ∪ Vd(N), Dt(N))

if MOD(P (N) ∪D(N)) ∩ EXT (Ao, Vt(N) ∪ Vd(N), Dt(N)) 6= ∅,EXT (Ao, Vt(N) ∪ Vd(N), Dt(N)) ∩MOD(D(N)) otherwise.

6 The global empirical claim of a theory-net

The formulation of the global empirical claim of a theory-net can be obtained

from explanation 2 and proposition 2 in a straightforward way (cf. [2]):

Explanation 3. Global empirical claim of a theory-net

The global empirical claim of a theory-net N is the proposition that

MOD(P (N) ∪D(N)) ∩ EXT (Ao(N), Vt(N), Dt(N)) 6= ∅.

In brief, the global empirical claim of N says that there is an interpretation

of L(N) that (i) satisfies the postulates P (N), (ii) satisfies the definitions of

D(N), and (iii) agrees with the given interpretation of I(T) for all T ∈ N.

This translates to the more intuitive proposition that there is a system, or

set, E(N) of structures such that (i) any intended application of T (T ∈ N)

has a corresponding T-extension in E(N) that is a model of T, (ii) all

members of E(N) satisfy all links and constraints among one another and

15

(iii) all specialisations among theory-elements are satisfied for any intended

application of any theory-element T ∈ N. This proposition strongly reminds

one of the Ramsey-Sneed sentence in Sneed [15]. In fact, the equivalence

between these two formulations of the empirical claim of a theory-net is easy

to show on condition that all non-equality constraints in the Ramsey-Sneed

formulation admit of a reformulation by theory-elements.

The comparison with the global empirical claim of a theory-net in Archi-

tectonics is less straightforward for one particular reason. The notion of

a link is introduced in classical structuralism such that links may relate a

theory element T to some theory-element T′ /∈ N. At least, such links are

neither explicitly nor intuitively excluded in classical structuralism. Note,

furthermore, that the specification of a global link from a theory-element

T to another theory-element T′ does not involve reference to T′’s intended

applications. The present system, by contrast, only considers links between

theory-elements in the net N and requires for all T ∈ N explicit considera-

tion of their intended applications. This makes it more puristic in the sense

of the original exposition of structuralism in Sneed [15].

7 Further empirical claims

Scientific reasoning involves propositions of a large variety of logical forms.

For a structuralist analysis of such reasoning we therefore need to be able to

translate propositions of whatever logical form into the structuralist frame-

work. (Such a translation is not available in classical structuralism). Let us

start with atomic propositions:

R(a1, . . . , an) (α)

Let R, in this notation, designate some “ordinary” scientific concept (i.e.,

a concept which is not part of the structuralist metatheory), such as the

concept of mass, force or space in classical particle mechanics. The particular

problem we encounter here is that α is inferentially inert in the context of

P (N) ∪D(N), i.e., we are not able to draw any interesting inferences from

α when we avail ourselves of the axioms of N. More precisely,

Cn(P (N) ∪D(N)) ∪ Cn({α}) = Cn(P (N) ∪D(N) ∪ {α})

where Cn denotes the consequence operation in classical logic. This is clearly

unacceptable since a major use of scientific theories is to help extend our

16

respective beliefs about empirical systems. If we know the value of the mass

function for a particular empirical object, classical collision mechanics allows

us to infer how this object will behave in collision experiments.

The inferential inertness of α is due to the existential quantification of the-

oretical terms in the definition of the models of T. To see this, recall that

such a definition has the general form of stating that x is a model of T if and

only if there are sets D1, . . . , Dk, R1, . . . , Rn such that (i) x = 〈R1, . . . , Rn,

D1, . . . , Dk〉 etc. Hence, the sentence R(a1, . . . , an) is inferentially inert in

the context of P (N)∪D(N) as this sentence is so also in the context of TCR

when R is a theoretical term of TC.

To bring the axioms of N to bear inferentially on α, I suggest the following

translation:

∃y∃z∃x(y ∈ I(z) ∧ (x, y) ∈ AE(z) ∧ (a1, . . . , an) ∈ (R)x) tr(α)

tr(α) says that there are structures x, y and a theory-element T such that

y is an intended application of T, x an admissible T-extension of y and x

verifies α. I(z) is a function that gives, for a theory-element z, the set of

intended applications.

Recall that for tr(α) to be true, it must be true in all systems of admissi-

ble T-extensions of the intended applications in N since AE(T) itself is a

theoretical term. In formal terms:

tr(α) iff for all A ∈ Sa(N),A |= tr(α) (1)

Having accomplished the translation of atomic formulas, let us move on to

the translation schema for formulas of arbitrary logical complexity. What

is called for is a function tf that maps formulas of L(Vd) onto formulas

of L(N) such that tf(α) qualifies as a translation of α, where L(Vd) is a

non-structuralist language such as is used in standard syntactic accounts of

scientific theories. To devise such a function, we need to consider atomic

formulas R(t1, . . . , tn) in place of atomic propositions R(a1, . . . , an), where

t1, . . . , tn designate individual terms. The rules for the translation of the

logical constants, then, can be given in the familiar recursive fashion:

Definition 6. Translation function tf

tf : L(Vd)F → L(N)F .

17

(1) tf(R(t1, . . . , tn)) := ∃y∃z∃x(y ∈ I(z) ∧ (x, y) ∈ AE(z)

∧ (t1, . . . , tn) ∈ (R)x)

(2) tf(¬α) := ¬ tf(α)

(3) tf(α1 ∧ α2) := tf(α1) ∧ tf(α2)

(4) tf(α1 ∨ α2) := tf(α1) ∨ tf(α2)

(5) tf(α1 → α2) := tf(α1)→ tf(α2)

(6) tf(∃uα) := ∃u tf(α)

(7) tf(∀uα) := ∀u tf(α)

8 Deductive reasoning

A primary motivation of Carnapian structuralism is to relate formal math-

ematical logic to actual scientific reasoning by means of the set-theoretic

notions of the structuralist framework. However, the present structuralist

formulation of atomic propositions and compounds thereof has little resem-

blance with the formulation of such propositions in science. The latter con-

tains constants for theoretical concepts, such as symbols for forces, masses,

temperatures etc., whereas in the structuralist formulation variables are used

to refer to theoretical concepts. This problem is familiar from the Ramsey

account of scientific theories, which is useful for certain metatheoretical in-

vestigations but not properly explanatory of scientific reasoning.

Let me indicate how theoretical constants and deductive reasoning therewith

can be recovered from the present formulation of propositions. Suppose some

system of empirical entities is an intended application of the theory-element

T. Let b designate the set-theoretic structure that represents that system.

By applying the substantial law of T to b, we aim to expand our beliefs about

the empirical entities of b. The logician - more precisely, the micrologician

- thinks of this expansion of beliefs in terms of inferences. Which types

of inferences are governing the expansion under consideration in our axiom

system?

Let us start from the proposition that b is an intended application of T:

b ∈ I(T) (2)

18

From this, we can infer using postulate P6(T), universal instantiation, and

modus ponens:

∃x((x, b) ∈ AE(T)), (3)

from which it follows, by existential elimination, universal instantiation ap-

plied to the postulates P1(T) and P3(T) and conjunction introduction,

that

(a, b) ∈ AE(T) ∧ b = r(T)(a) ∧ a ∈M(T). (4)

By this proposition we know (i) that a verifies all sentences that b verifies and

that (ii) a is a model of T. Since a is a constant, which designates a sequence

of sets, we can replace it with a symbol of the form 〈D1, . . . , Dk, R1, . . . , Rn〉.By (i) we know that the relations R1, . . . , Rn are such that all propositions

encoded by the structure b must hold. We can obtain these propositions

from (4) by reasoning in naive set-theory. (ii) lets us know that the relations

R1, . . . , Rn of a = 〈D1, . . . , Dk, R1, . . . , Rn, 〉 are such that the substantial

law of T must hold. In sum, we obtain from (4), first, propositions about

the empirical system that is an intended application of T and, second, a

proposition that expresses the substantial law of T itself, where all these

propositions are formulated in terms of T-theoretical and T-non-theoretical

constants. We have thus recovered a formulation of local empirical claims

and of the substantial law of T that conforms to the patterns of standard

syntactic accounts of deductive reasoning. For a fully worked out piece of

scientific reasoning in Carnapian structuralism see Andreas [4].

These considerations show how micrological deductive reasoning can be ex-

pressed in a structuralist setting. A proof that any deductive argument can

be represented in the structuralist framework would require consideration of

links and would have to rest on the assumption that any proposition of such

an argument may possibly be verified or falsified by a system of admissible

T-extensions of the intended applications in N.6 The surplus of Carnapian

structuralism – in comparison to both standard syntactic accounts of sci-

entific theories and classical structuralism – consists in the synthesis of the

micrological with the macrological perspective. This combination will be

further exploited when we come to introducing defeasible and paraconsis-

tent means of reasoning in the next section.

6To motivate the latter requirement, a deductive argument from a proposition that

concerns an object which is not a member of any base set of any intended application in

N cannot be expressed in a structuralist setting.

19

9 Defeasible and paraconsistent reasoning

Classical deductive reasoning proved insufficient to account for the variety

of putatively rational inferences that are being used by human minds. A

case in point is default reasoning, i.e., reasoning on the basis of axioms or

inference rules that lead to true conclusions most of the time but that do

have exceptions. Philosophers of science are familiar with this problem from

investigating ceteris paribus laws. The recognition of default reasoning gave

rise to the development of nonmonotonic logics. Nonmonotonicity means

that the consequnce operation C of the logic need not obey the following

property:

Definition 7. Monotonic consequence operation

A consequence operation C : P(LF ) → P(LF ) is monotonic if and only if

for all sets A, B ∈ LF it holds that C(A) ⊆ C(A ∪B).

In other words, nonmonotonic inferences are ones that may be invalidated

by an extension of the premise set. The addition of further premises may

invalidate previously drawn conclusions. Nonmonotonic reasoning has thus

also been described as defeasible reasoning.

Besides default reasoning, defeasible reasoning can be witnessed in the con-

text of abductive inferences being understood as inferences from phenomena

to explanations thereof. Novel explanations may override and thus defeat

older ones. Novel empirical findings may show a presently held explanation

to be untenable.

Another weakness of classical deductive reasoning is that it breaks down, as

it were, as soon as the premise set contains a minor or major inconsistency

since such premise sets entail simply any formula of the language. This

property has been described as explosion into triviality:

Definition 8. Explosive consequence operation

A logic with a consequence operation C : P(LF ) → P(LF ) is explosive if

and only if, for all formulas α, β ∈ LF , β ∈ C({α ∧ ¬α}).

Not being explosive is a minimal requirement for a logic or inference system

to qualify as paraconsistent.7 They aim to define an inference relation that

7The difference between an inference system and a logic be understood to consist in

that only the latter is required to have a fully fledged proof theory.

20

allows us to draw sensible inferences from a classically inconsistent set of

premises.

The need for paraconsistency in the logic of science arises from the fact that

there is a majority of scientific theories that exhibit minor inconsistencies

but remain in use for both prediction and theoretical understanding. This

point has been emphasised, perhaps with some exaggeration, by [11]. In our

system, the presumed universal validity of laws is expressed by the success

postulate P6(T). This requires any intended application of any theory-

element to be extensible to models of T in such a way that all links and

specialisations are satisfied. This is not possible in the case of classical me-

chanics, to give just one prominent example. Yet, classical mechanics has

remained in use and has not been rejected outright, as classical logic would

have it. Classical electrodynamics has been shown to have inconsistent ap-

plications - i.e., applications where the axioms lead to contradictions - to an

even higher degree than classical mechanics [12].

Nonmonotonic logics have hardly been used in the logical analysis of scien-

tific reasoning. Which paraconsistent inference system or logic works well

in the case of scientific reasoning is very much an open question. In what

follows, I shall explain how defeasible and paraconsistent forms of reasoning

can be introduced into the structuralist framework in a fairly intuitive way.

This introduction is based on the inference system of preferred subtheories

by Brewka [6, 7] whose essential concepts are as follows:

Definition 9. Default theory

A default theory is a sequence 〈T1, . . . , Tn〉, where each set Ti (1 ≤ i ≤ n) is

a set of first order formulas.

The different components of such a sequence are intended to represent differ-

ent degrees of reliability. Sentences of T1 are considered to have the highest

degree of reliability, whereas sentences of Tn are assumed to have the least

degree. If there are any hard facts and conceptual truths, these would have

to be members of the set T1.

The union of T1, . . . , Tn is usually not consistent. So, inconsistencies are

to be removed by the inference formalism. For this to achieve, maximal

consistent subsets are to be chosen for the definition of the inference relation.

An important refinement that comes with the preferred subtheories approach

consists in respecting the levels of reliability in the construction of maximal

21

consistent subsets. The intuitive idea is to exploit first the most reliable

propositions when drawing inferences and then move on to less reliable ones.

This idea is made precise by the notion of a preferred subtheory:

Definition 10. Preferred subtheory

A set S = S1 ∪ . . .∪Sn is a preferred subtheory of a default theory Θ if and

only if (i) S is classically consistent, (ii) for all i, 1 ≤ i ≤ n, Si ⊆ Ti and (iii)

there is no consistent set S′ of sentences with an i such that (S∩Ti) ⊂ (S′∩Ti)and for all j < i, (S ∩ Tj) = (S′ ∩ Tj).

The construction of a preferred subtheory starts with a maximal consistent

subset of T1, goes on to add as many formulas of T2 as can consistently

be added and repeats this process for T3, . . . , Tn. This construction, apart

from special cases, does not result in a uniquely determined preferred sub-

theory. Rather, there will usually be several sets of formulas that qualify as

a preferred subtheory of a default theory. One way to establish an inference

relation upon such a set of preferred subtheories is to accept just those for-

mulas as inferable which are a logical consequence of the intersection of all

preferred subtheories:8

Definition 11. Superskeptical inference relation

Let Θ be an default theory and P the set of preferred subtheories of Θ.

Then, φ is a superskeptical inference of Θ (in symbols: Θ |∼PS(su) φ) if and

only if φ ∈ Cn⋂

S∈PS.

This inference system of preferred subtheories is a means of formalising

both nonmonotonic and paraconsistent reasoning. It can be used for our

structuralist axiomatic system in a very intuitive way:

Definition 12. Structuralist default theory

Let N be a theory-net and H(N) a set of sentences of the form b ∈ I(T)

such that H(N) is a sentential representation of the intended applications

in N. Θ(N) is a structuralist default theory if and only if there are sets

T1, . . . , Tn of sentences such that

(1) Θ(N) = 〈T1, . . . , Tn〉;8Superskeptical inferences are intuitively more plausible in the case of our structuralist

system than standard (skeptical) inference relation in the preferred subtheories approach,

according to which a formula is inferable if and only if it is a classical consequence of any

preferred subtheory.

22

(2) T1 = P (N) ∪D(N);

(3) T2 ∪ . . . ∪ Tn = H(N);

(4) for all i, j(i 6= j), Ti ∩ Tj = ∅;

(5) if pb ∈ I(T)q ∈ Ti and β = pb′ ∈ I(T′)q ∈ Tj and i < j, then T must

have a higher reliability than T′.

This explanation, of course, takes the notion of the reliability of a theory-

element as primitive. For a discussion of this notion see [3]. Note that

the definitions and postulates of N have the highest reliability. L(N) has a

trivial interpretation that satisfies P (N)∪D(N), viz, the one where I(T) = ∅for all T ∈ N. Hence, any preferred subtheory of Θ(N) contains P (N) ∪D(N).

The construction of preferred subtheories gives us a system of successful

intended applications Hs(N) ⊆ H(N) in the sense that Hs(N) sententially

represents those intended applications of T (T ∈ N) that can in fact be

extended to models of T such that all links and specialisations are satis-

fied. More precisely, Hs(N) is the intersection of sets H ′(N)∪P (N)∪D(N)

(H ′(N) ⊆ H(N)) that are consistent and respect priorities among intended

applications. Hence, the inference formalism sorts out the unsuccessful ap-

plications of the theories under consideration. Once the sets of successful

intended applications are determined and sententially represented by a set

Hs(N), we can go on drawing inferences from Hs(N) ∪ P (N) ∪ D(N) by

means of classical deductive logic.

10 Conclusion

The present system is driven by the idea of combining the expressive power

of the structuralist framework with standard systems of deduction as well as

inference systems of defeasible reasoning. This combination opens up novel

strands of research for the structuralist approach:

(1) AGM-style belief revisions can be defined along the lines of Brewka [7].

The resulting framework will allow one, so it seems, to study belief changes

in science, including revolutionary ones, in greater detail than in standard

systems of belief revision and classical structuralism. Studies of this type

23

may include a diachronic account of the interpretation of theoretical con-

cepts. For a first attempt in that direction, see [3].

(2) On the basis of a structuralist definition of belief revisions, one can define

conditionals using the Ramsey test in the style of Gardenfors [13, Ch. 7].

This will allow one to incorporate counterfactual approaches to causation

and to explanation into structuralism. Such a counterfactual approach to

causation may be joined with a reconsideration of logical empiricist accounts

of explanation in the structuralist framework.

(4) The logical study of paraconsistent reasoning in science is almost straight-

forward as the inference system of preferred subtheories qualifies as a para-

consistent one. Unlike many other paraconsistent systems, preferred subthe-

ories allow for selective acceptance of the instances of universal axioms. This

amounts to selective acceptance of the intended applications of a theory-

element in Carnapian structuralism. In a less formal fashion, Cartwright [11]

has aimed to show that our theoretical understanding of nature is guided

by selectively using applications of universal axioms.

Pursuing these strands of research will connect the structuralist approach

with contemporary philosophy of science more strongly and help undo the

relative isolation that the structuralist community, unwillingly, has been

driving to in the last decades.

Acknowledgements

I am grateful to Gerhard Schurz and two unknown referees for very helpful

comments on an earlier draft of this paper.

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