Arguing Towards Truth: The Case of the Periodic Table

Abstract

Recently Erik Scerri has published an influential philosophical history of the development of the Periodic Table. Following Scerri’s account, I will explore the main thread of the arguments responsible for the remarkable advancement of scientific understanding that the Periodic Table represents. I will argue that the history of disputation at crucial junctures in the debate shows sensitivity to the aspects of truth that are captured by my model of truth in inquiry. The availability of a clear and explicit model of truth in inquiry is of crucial importance as a response to post-modernist and other relativistic accounts of inquiry. It shows that despite such apparent sociological constraints as acceptability a robust theory of truth is available as a foundation for evaluating argumentation.

Appendix

Part I:

1. A scientific structure, TT = ‹T,FF,RR› (physical chemistry is the paradigmatic example) where T is a set of sentences that constitute the linguistic statement of TT closed under some appropriate consequence relation and where FF is a set of functions F, such that for each F in FF, there is a map f in F, such that f(T) = m, for some model or near model of T. And where RR is a field of sets of representing functions, R, such that for all R in RR and every r in R, there is some theory T* and r represents T in T*, in respect of some subset of T.

A scientific structure is first of all, a set of nomic generalizations, the theoretic commitments of the members of the field in respect of a given body of inquiry. We then include distinguishable sets of possible models (or appropriately approximate models) and a set of reducing theories (or near reducers). What we will be interested in is a realization of TT, that is to say a triple ‹T,F,R› where F and R represent choices from FF and RR, respectively. What we look at is the history of realizations, that is an ordered n-tuple: ‹‹T,F1,R1›,…,‹T,Fn,Rn›› ordered in time. The claim is that the adequacy of TT as a scientific structure is a complex function of the set of realizations.

1.1. Let T′ be a subtheory of T in the sense that T′ is the restriction of the relational symbols of T to some sub-set of these. Let f′ be subset of some f in F, in some realization of TT. Let ‹T′1,…,T′n› be an ordered n-tuple such that for each i, j (i < j) T′i reflects a subset of T modeled under some f′ at some time earlier than T′j. We say the T is model progressive under f′ iff:

• (a) T′k is identical to T for all indices k, or

• (b) the ordered n-tuple ‹T′1,…,T′n› is well ordered in time by the subset relation. That is to say, for each T′i, T′j in ‹T′1,…T′n› (i < j ≥ 2), if T′i is earlier in time than T′j, T′i is a proper subset of T′j.

1.2. We define a model chain C, for theory, T, as an ordered n-tuple ‹m1,…,mn›, such that for each mi in the chain mi = ‹di,fi,› for some domain di, and assignment function fi, and where for each di and dj in any mi, di = dj; and where for each i and j (i < j), mi is an earlier realization (in time) of T then mj.

Let M be an intended model of T, making sure that f(T) = M for some f in F (for some realization ‹T,F,R›) and T is model progressive under f. We then say that C is a progressive model chain iff:

• (a) for every mi in C, mi is isomorphic to M, or

• (b) there is an ordering of models in C such that for most pairs mi, mj (j > i) in C, mj is a nearer isomorph to M than mi.

This last condition is an idealization, as are all similar conditions that follow. We cannot assume that all theoretic advances are progressive. Frequently, theories move backwards without being, thereby, rejected. We are looking for a preponderance of evidence or where possible, a statistic. Nor can we define this a priori. What counts as an advance is a judgment in respect of a particular enterprise over time best made pragmatically by members of the field.

1.3. Let ‹C1,…,Cn› be a well ordering of the progressive model chains of TT, such that for all i, j (i > j), Ci is a later model chain than Cj. TT is model chain progressive iff ‹C1,…,Cn› is well ordered in time by the subset relation. That is to say each later model includes and extends the models antecedent to it in time.

2. We now turn out attention to the members of some R in RR. The members of RR represent T in T* in respect of some subset of T, k(T). Let ‹k1(T),…,kn(T)› be an n-tuple of representations of T over time, that is if i > j, then ki(T) is a representation of T in T* at a time later that kj(T). We say that TT is reduction progressive iff,

• (a) k(T) is identical to Con(T) for all indices, or

• (b) the n-tuple is well ordered by the subset relation.

2.1. We call an n-tuple of theories RC = ‹T1,…,Tn› a reduction chain, and ‹T1,…,Tn› a deeper reduction chain than j-tuple ‹T′1,…,T′j›, iff n > j and for all i, j there is a ri in Ri such that ri represents Ti in Ti + 1 and similarly for T′i and further for all Tk (k ≤ j) Tk is identical in both chains Note, the index i must be different from the index j, since if i = j, there is no Ti + 1.

2.2. We call a theory reduction chain progressive iff T iff for an n-tuple of reduction chains ‹RC1,…, RCn› and for each RCi (i < 1), RCi + 1 is a deeper reduction chain than Rci.

2.3. T is a branching reducer iff there is a pair (at least) T′ and T* such that there is some r′ and r* in R′ and R*, respectively, such that r′ represents T′ in T and r* represents T* in T and neither T′ is represented in T* nor conversely.

2.3.1. B = ‹TT1,TT2,…,TTn› = ‹‹T1,F1,R1›, ‹T2,F2,R2›,…,‹Tn,Fn,Rn›› is a reduction branch of TT1 iff T1 is a branching reducer in respect of Ti, and Tj (i ≥ 2; j ≥ 3 for i, j ≤ n)

2.4. We say that a branching reducer, T is a progressively branching reducer iff the n-tuple of reduction branches ‹B1,…,Bn› is well ordered in time by the subset relation, that is, for each pair i, j (i > j) Bi is a later branch than Bj, that is, the number of branching reducers has been increasing in breadth as inquiry persists.

Part II:

The core construction is where a theory T is confronted with a counterexample, a specific model of a data set inconsistent with T. The interesting case is where T has prima facie credibility, that is, where T is at least model progressive, that is, is increasingly confirmed over time (Part I, 1).

A. The basic notion is that a model, cm, is a confirming model of theory T in TT, a model of data, of some experimental set-up or a set of systematic observations interpreted in light of the prevailing theory that warrants the data being used. And where

(1) cm. is either a model of T or

(2) cm is an approximation to a model of T and is the nth member of a sequence of models ordered in time and T is model progressive (1.1).

B. A model interpretable in T, but not a confirming model of T is an anomalous model.

The definitions of warrant strength from the previous section reflect a natural hierarchy of theoretic embeddedness: model progressive, (1.1), model chain progressive (1.3) reduction progressive (2), reduction chain progressive (2.2), branching reducers (2.3) and progressively branching reducers (2.4). A/O opposition varies with the strength of the theory. So, if T is merely model progressive, an anomalous model is type-1 anomalous, if in addition, model chain progressive, type-2 anomalous etc. up to type-6 anomalous for theories that are progressively branching reducers.

P1: The strength of the anomaly is inversely proportional to dialectical resistance, that is, counter-evidence afforded by an anomaly will be considered as a refutation of T as a function of strength of T in relation to TT. In terms of dialectical obligation, a claimant is dialectically responsible to account for type 1 anomalies or reject T and less so as the type of the anomalies increases.

P2: Strength of an anomaly is directly proportional to dialectical advantage, that is, the anomalous evidence will be considered as refuting as a function of the power of the explanatory structure within which it sits.

P*: The dialectical use of refutation is rational to the extent that it is an additive function of P1 and P2.