Coherence and Reduction

2.1 An Example of a GNS Reduction

TD is a branch of physics which describes those phenomena observable in macroscopic systems (e.g. solids, liquids, gases, plasma) and their relation to energy, radiation, and properties of matter. The behaviour of these entities can be expressed by the four laws of thermodynamics which make use of macroscopic properties (e.g. pressure, temperature). Yet, such behaviour can also be explained in terms of their microscopic constituents by SM. In fact, based on statistical methods, probability theory and microscopic physical laws, SM explains the behaviour of macroscopic systems in terms of those dynamical laws governing its microscopic quantities (e.g. molecules, particles). The laws of TD can therefore be expressed in terms of the laws of SM. This hints at our first definition of Nagelian reduction, according to which reducing a theory T _A to another theory T _B is possible if and only if the laws of T _A are derivable from T _B with the help of bridge laws, which are empirical facts linking concepts of the reduced theory to terms of the reducing theory. This, however, is not sufficient to properly describe a successful reduction. Consider an example of reduction between TD and SM, namely the Boyle–Charles law (Dizadji-Bahmani, Frigg, and Hartmann 2010, pp. 395–396),^[1] to better understand the role of bridge laws. The Boyle–Charles law states that the temperature T of a gas is directly proportional to the product of the values of its pressure p and the volume V over which it evenly distributed:

where k is a constant. This law together with some specific conditions (i.e. gas in thermodynamic equilibrium with the surrounding environment and relatively low pressure) forms the core of the thermal theory of the ideal gas. In SM, there is a corresponding theory for ideal gases: the kinetic theory of the ideal gas. This theory describes the motion of n particles with mass m of a gas spread over the volume of, for example, a vessel according to Newtonian mechanics. The theory includes two assumptions:

1. the gas should be ideal to the extent that its molecules, which collide elastically, are point particles;

2. the three components of the velocity v → (v _x , v _y , v _z ) should be evenly distributed (i.e. there is no favoured direction).

Following the definition of pressure in Newtonian physics and the first assumption, the gas hitting a wall of the vessel exerts a pressure:

where 〈 v z 2 〉 is the average of the square of v _z , a particle’s velocity in z-direction with respect to the x–y plane of the wall. After few other calculations^[2] and following the second assumption, the left-hand term of the equation in the Boyle–Charles law can be expressed as:

where n 〈 E kin 〉 is the average kinetic energy of the gas. T can therefore be seen as:

This process shows how to derive the Boyle–Charles law from the laws of Newtonian physics. In fact, first, a particular theory, the thermal theory of the ideal gas, (here, eq. (3)), was derived by combining Newtonian physics with the two assumptions of the kinetic theory of the ideal gas. Second, eq. (4), that would stand as a bridge law in the GNS, has connected relevant terms, such as T and 〈 E kin 〉 , and it has yielded a version of the Boyle–Charles law bounded to some conditions. Finally, it has been shown that this particular version of the Boyle–Charles law bounded by particular conditions is strongly analogous (or even coincide) to the standard version of the Boyle–Charles law. Nowadays, scientists consider the reduction of TD to SM successful.^[3] The reduction of TD to SM is considered to be a synchronic intertheoretic reduction, namely a reductive relation between two coexisting theories which deal with different levels of a largely overlapping domain. In this reduction, the concepts and the laws of one theory can respectively be expressed as the concepts and be derived from the laws of the more fundamental theory. Accordingly, a correct reduction of TD to SM involves the derivations of the laws of TD from the laws governing the microconstituents of macroscopic systems together with probabilistic assumptions. For this reason, DFH suggest that this reductive relation resembles the GNS, which applies to synchronic intertheoretic reductions.^[4]

In the reductive relation, TD is the reduced theory T P and SM is the reducing one T F .^[5] According to the GNS, T P corresponds to the set of empirical propositions { T P ( 1 ) , … , T P ( n ) } associated with T P . Sp, T F is the set of empirical propositions { T F ( 1 ) , … , T F ( n ) } associated with T F . Here, the empirical propositions of T P and T F are the various laws of the theories.^[6] As shown in the case of the Boyle–Charles law, the reduction of T P to T F would then follow three steps.

1. Use auxiliary assumptions to help deriving a restricted version of each element T F ( i ) of T F .^[7] Let T F ∗ ≐ { T F ∗ ( 1 ) … T F ∗ ( n F ) } be the set of the restricted versions.

2. Adopt bridge laws in order to connect the relevant terms which are not share by the vocabularies of the theories involved.^[8] Substituting the terms in T F * with the terms from T F shows that the bridge laws yield the set T P ∗ ≐ { T P ∗ ( 1 ) … T P ∗ ( n P ) } .

3. Show that each element of T P * is strongly analogous to the corresponding element of T P .

If these conditions are satisfied, it is believed that T P is reduced to T F with respect to the GNS.

2.2 The Bayesian Analysis of DFH

Amongst the desiderata of reductions in science, coherence and confirmation are the main ones (Dizadji-Bahmani, Frigg, and Hartmann 2010, 2011).^[9] Nagel (1961, p. 341)^[10] himself sensed that reduction should reconcile two self-consistent and well-confirmed theories whose domains of application (largely) overlap whenever the two sketch a contradictory view of the world. The example of the reduction of TD to SM perfectly fits into this picture. In fact, TD (here, T P ) and SM (here, T F ) should be consistent with each other and evidence confirming TD should support SM, and vice versa. Obviously, both criteria must be met after the reduction occurs. DFH (Dizadji-Bahmani, Frigg, and Hartmann 2011) use Bayesian networks to represent the relation between reduction and confirmation.^[11] In fact, Bayesian networks, as illustrated in Nagel (1974), exploit (un)conditional independences and dependencies in order to represent large instances in little space by graphical structures and to perform probabilistic inferences in little time. The type of statistical calculation involved in Bayesian networks is called Bayesian inference, that is, an inference in which Bayes’ theorem is one of the main rules used to update the probability for a hypothesis as more evidence and information become available.^[12] Bayesian networks are a type of probabilistic graphical model that uses Bayesian inference for probability computations. Confirming an hypothesis H with a piece of evidence E in Bayesian terms means having a conditional probability P ( H | E ) larger than the prior probability P ( H ) . In other words, an hypothesis is confirmed by E if:

More precisely, a Bayesian Network is a directed acyclical graph (DAG),^[13] which satisfies the Markov condition^[14] and whose nodes represent discrete propositional variables, and edges capture their conditional independences and dependencies.^[15] To frame GNS in Bayesian terms, DFH introduce few simplifications which I will use throughout the article.

1. To simplify the calculations, DFH assume that T F and T P have only one element, namely T _F and T _P respectively. Their corresponding propositional variables will be T _F and T _P respectively.

2. The propositional variables represented by the nodes of a Bayesian network can take two values, i.e. T _F and ¬ T F . While the latter means that the proposition T _F is false, the former asserts that it is the case that T _F is true.

3. The probability of every node can lie in the open interval (0, 1). I set j ‾ = 1 − j for all parameters j, unless a parameter is a logical consequences of another variable: in this case, its conditional probability on such variable is 1.

Furthermore, three different pieces of evidence supporting the theories in the reduction relation are gained from experimental tests. They are defined with the propositional variables E, E _F , and E _P by DFH (Dizadji-Bahmani, Frigg, and Hartmann 2011, p. 324). The same simplifications introduced for T F and T P are applied to E , E F , and E P . These three respectively support both theories, only the fundamental theory, and only the reduced theory.

1. Evidence supporting only TD, e.g. the Joule–Thomson process.^[16]

2. Evidence supporting only SM, e.g. the dependence of a metal’s electrical conductivity on temperature.^[17]

3. Evidence supporting TD and SM simultaneously, e.g. the second law in TD.^[18]

According to DFH, the situation before the reduction would then look like the network in Figure 1. Let P ₁ be the probability distribution over the variables in such network. The relevant probabilities specifying the network are:

Figure 1:

The Bayesian Network representing the situation before T _P is reduced to T _F .

Before the reduction occurs, T _F and T _P are probabilistically independent because they do not share the same vocabulary and they are not supported by the same evidence. In fact, E _F is independent of T _P given T _F and, vice versa, E _P is independent of T _F given T _P . Formally:

The independences in (8) hold because, in the aforementioned Bayesian network, the paths E F − T F − E − T P and E P − T P − E − T F are respectively blocked at T _F and T _P by { T F } and { T P } . So, E _F and T _P are d-separated,^[19] and E _P and T _F as well. Therefore, the conjunction of the prior probabilities of the root nodes T _F and T _P looks like the following:

Already before the reduction, one notices that in the Bayesian network there is a connection between T _P and T _F , namely evidence E. Such link has lead scientists to investigate the intimate relation among those two theories. DFH present then the network for the situation after the reduction (see Figure 2). To reduce one theory to the other, DFH complete three steps: derive T F ∗ from T _F together with some auxiliary assumptions; introduce bridge laws which, together with T F ∗ , yield T P ∗ ; show that T P ∗ is strongly analogous to together with T _P . They make important remarks which help defining the values of the conditional probabilities of the three nodes which follow from the only remaining root node T _F . The derivation of T F ∗ from T _F and the interpretation of strong analogy between T P ∗ and T _P depend on the judgment of the scientists and on the specific context in which the reduction occurs (Dizadji-Bahmani, Frigg, and Hartmann 2011, p. 328). Regarding bridge laws, which are not factual claims in a rigorous sense, T P ∗ is a logical consequence of T F ∗ , according to DFH. Let then P ₂ be the probability distribution over the propositions one has after reducing T _P to T _F . The same simplifications introduced for T F and T P are applied to T F * and T P * . The relevant probabilities specifying the second network in Figure 2 are:

Figure 2:

The Bayesian Network representing the situation after the reduction of T _P to T _F , according to DFH.

Following such network, DFH show that, after the reduction, evidence confirming one theory confirms the other and vice versa. In fact:

The two theorems maintain that in order to have a confirmation flow from E _F to T _P and from E _P to T _F : i) E _F should confirm T _F and E _P should confirm T _P ; ii) T _F should confirm T F ∗ and T P ∗ should confirm T _P . The two conditions are satisfied because:

1. one of the original assumptions of the network is the fact that two evidence support their own respective theory;

2. T _F likely confirms T F ∗ and T R ∗ likely confirms T _P , because T F ∗ was derived by T _F , and T R ∗ is strongly analogous to T _P . Again, such construction of the GNS model is justified by the example in 2.1.

Once one constructs a Bayesian network like Figure 2 and assumes the presence of a confirmatory flow from T _F to T _P via T F ∗ and T P ∗ , she can prove that, after reducing T _P to T _F , E _F confirms T _P , and E _P confirms T _F .

2.3 The Bayesian Analysis of Tešic

The Bayesian analysis offered by DFH helps representing the reductive relation among T _P and T _F . However, Tešic (2019) points out to two difficulties faced by their analysis. I omit the details of the less relevant difficulty,^[20] because it does not undercut the main project laid out by DFH.^[21] I focus, instead, on the main critique Tešic presents, which regards the value of the probabilities P 2 ( T P ∗ | T F ∗ ) and P 2 ( T P ∗ | ¬ T F ∗ ) . As showed in (10), while P 2 ( T P ∗ | T F ∗ ) = 1 , P 2 ( T P ∗ | ¬ T F ∗ ) = 0 . Tešic claims that thinking the propositional variables T P * and T F * as interchangeable with each other (Dizadji-Bahmani, Frigg, and Hartmann 2011, pp. 329–330) without explicitly stating that bridge laws are assumed is misleading. In fact, DFH’s representation of the bridge law in their Bayesian network suffers from three problems (Tešic 2019, pp. 1108–1111):

1. Recall eq. (4), that is, the bridge law in the example of the Boyle–Charles law. According to the Bayesian network in Figure 2, one should then have not only P 2 ( T P ∗ | T F ∗ ) = 1 , but also P 2 ( T P | T F ∗ ) = 1 , because p V = 2 n 3 〈 E kin 〉 . Instead of the incorrect entailment p V = 2 n 3 〈 E kin 〉 | = p V = k T , Tešic notices that such entailment is only possible by supposing the bridge law B:

The problem arising in Figure 2 is that it does not incorporate B in the background, according to Tešic. B therefore needs to be included in the probability function P ₂ for the entailment to hold.

1. The fact that from eq. (4) P 2 ( T P ∗ | T F ∗ ) = P 2 ( T P ∗ | T F ∗ ) = 1 and P 2 ( T P ∗ | ¬ T F ∗ ) = P 2 ( T P ∗ | ¬ T F ∗ ) = 0 follow (Tešic 2019, p. 1118) makes the reduction symmetric. The Bayesian network in Figure 2 seems to imply that the Boyle–Charles law is reduced to the kinetic theory of gases, and vice versa. This clearly goes against the main idea behind every kind of scientific reduction: reduction is anti-symmetric. Furthermore, the interchangeability between T P ∗ and T F ∗ would prevent partial reductions, which are still important in science, according to DFH (Dizadji-Bahmani, Frigg, and Hartmann 2010, p. 399). In fact, scientists are not always able to connect every term of T P ∗ to T F and deduce every law of T P ∗ from T F plus bridge laws.

2. From eq. (4) it also follows that the marginal probabilities P 2 ( T P ) , P 2 ( T P ∗ ) and P 2 ( T F ∗ ) are equal (Tešic 2019, p. 1118). It is hard to conceive of them as equal given that the equation p V = 2 n 3 〈 E kin 〉 is deduced from the kinetic theory of gases together with some auxiliary assumptions, and that the equation p V = k T is then deduce from the equation p V = 2 n 3 〈 E kin 〉 and the bridge laws. This suggests that P 2 ( T P ∗ ) and P 2 ( T F ∗ ) should be left open.

Because of these three problems, Tešic presents an alternative Bayesian network to Figure 2. The main idea behind his network is to explicitly include the propositional variable B representing the bridge law as a root node. Let P ₃ be a probability distribution over the variables in Figure 3. The same simplification applied to T F and T P applies to B . Thus, assume that the only element of B is B, and that the two values assignable to the propositional variable B are: B and ¬ B , Then:

Figure 3:

The Bayesian Network representing the situation after reducing T _P to T _F , according to Tešic.

Two reasons motivate this explicit specification of the bridge laws (Tešic 2019, p. 1121): a) different scientists (often) have different credence about a particular bridge law (e.g. scientists share in fact different degree of belief on eq. (4)); b) the flow of confirmation depends on the value assigned to the probability of the bridge law. Thus, Tešic assigns the following values to the probabilities P 3 ( T P ∗ | T F ∗ , B ) , P 3 ( T P ∗ | T F ∗ , ¬ B ) , P 3 ( T P ∗ | ¬ T F ∗ , B ) , and P 3 ( T P ∗ | ¬ T F ∗ , ¬ B ) :

where a ∈ ( 0 , 1 ) . Accordingly, the new probability assignments do not face the three problems noticed by Tešic. In fact, the first problem is avoided because now one has P 3 ( T P ∗ | T F ∗ , B ) = 1 and, thus, T F ∗ , B | = T P ∗ , instead of simply having T F ∗ | = T P ∗ . The second problem is evaded by showing that the values in (13) entail 0 < P 3 ( T F ∗ | T P ∗ , B ) < 1 : this means the reduction represented by Tešic’s network is not symmetric. Finally, the third problem is successfully addressed because it is proved that the prior probabilities P 3 ( T P ∗ ) and P 3 ( T F ∗ ) can be either different or equal: in fact, it depends on the particular values one assigns to the relevant probabilities. According to this analysis, Theorems 2.1 and 2.2, which have already appeared in DFH’s network, follow.

For all the reasons mentioned in this section, I will compare the coherence pre- and post-reduction between the reducing theory and the reduced one by following the probability assignments specified in the Bayesian network in Figure 3.

3.1 Schupbach’s Measure

To introduce Schupbach’s measure, consider a finite and non-empty information set S , that is a set of ordered pairs:

where R _i is a source reporting that A _i , the content of the report R _i , is true. The content of the information system S is the ordered set of report contents 〈 A 1 ∧ … ∧ A n 〉 . Let P be the probability distribution over the propositions 〈 A 1 ∧ … ∧ A n 〉 , where P ( A i ) gives the degree of confidence of a rational agent in A _i . A coherence measure is a function C which maps every finite and non-empty set S of propositions with positive probability to a single real-valued outcome C ( S ) . In other words, the degree of coherence C ( S ) consists precisely in the degree of coherence of its content 〈 A 1 ∧ … ∧ A n 〉 (Bovens and Hartmann 2003). Then, let S be the set of all finite and non-empty information sets S of propositions with positive probability. Shogenji (1999) proposes to define the coherence of a set S ∈ S as:

Shogenji conceives of the coherence of S as a mutual support between the propositions in S . The measure is sensitive to the number n of sources in cases of logically equivalent propositions. Given that the join probability of the propositions of S equals to 1 when all sources reports the same proposition, as n, the number of agreeing reports, tends to infinity, so does the degree of coherence: indeed, ”the more coherent beliefs are, the more likely they are together” (Shogenji 1999, p. 338). Instead, if the propositions of a set are independent, the variables representing them will be probabilistically independent and the coherence measure will equal to 1: in this case, the set S is neither coherent nor incoherent. Finally, if the joint probability of the propositions is lower than the probability of their products, the set will be incoherent. A series of criticisms^[25] has been offered to the way this measure tries to capture the meaning of coherence (Schupbach 2011, pp. 3–5). For this reason, Schupbach suggests to generalise Shogenji’s measure by calculating the degree of coherence as a weighted average of the degree of coherence of all subsets of S :

Schupbach then proposes several definitions which are needed to avoid the problems arising with Shogenji’s original measure and, therefore, to measure the degree of coherence in a more precise manner. As I will show in the next section, I am only interested in information sets with cardinality two. If sets have cardinality two, the subset-sensitive generalisation of Shogenji’s measure offered by Schupbach can be simply represented by the Log-normalised version of the Shogenji’s measure (Schupbach 2011, p. 9), namely eq. (16).^[26] In fact, sets with cardinality two avoid all the problems faced by sets with cardinality strictly larger than two. So, if n = 2 and S = { A 1 , A 2 } , then:

If one measures the degree of coherence of a set containing probabilistically independent propositions as members, 0 will be the outcome since the joint probability of probabilistically independent variables is their product and therefore the logarithm of the ratio of two same values is 0. The outcome of S ( S ) will tend to negative infinity if the set S is not coherent. Vice versa, it will tend towards positive infinity if S is coherent.

3.2 Olsson’s Measure

Olsson (1999) understands coherence as a total agreement between the propositions in an information set S ∈ S . In the case of a set S = { A 1 , A 2 } , Olsson then proposes the following coherence measure:

which would look like the following for a set S = { A 1 , … , A n } :^[27]

This measure ranges over the closed interval [0, 1]. O indeed assigns 0 to cases of minimal agreement, that is, when the propositions involved are logically inconsistent. Regarding eq. (18), A ₁ and A ₂ would therefore not overlap in cases of minimal agreement. Whilst, in cases of maximal agreement between the propositions reported, the measure will give 1 as outcome. This means that the propositions in the set are logically equivalent. This measure also presents counter-intuitive results (Dietrich and Moretti 2005, p. 407), such as the possibility of reporting the degree of coherence of two positively dependent propositions as lower than that of two negatively dependent propositions. This remark about O is not relevant for the aim of this article: in next section, I will compare the degree of coherence before and after reduction by exploiting (and assuming), respectively, the independence and the positive dependence amongst the two theories.

3.3 Bovens and Hartmann’s Measure

Bovens and Hartmann (2003)^[28] change the course of previous coherence measures. Instead of trying to make the intuition one can have on the notion of coherence precise (i.e. mutual support, total agreement, relative overlap), they focus on the role that coherence, as a property of information sets, plays: boosting our confidence that the content of an information set is true ceteris paribus once the information is received from independent and partially reliable source. The model Bovens and Hartmann construct aims to measure the degree of confidence in the joint truth of an information set. Such degree is determined by the combination of conditions such as results’ expectation, tests’ reliability, coherence of the information. Each of these conditions has a specific measure. Respectively, one has i) an expectance measure, which is about the degree of prior expectance of the joint truth of an information set, ii) a reliability measure, which computes the degree of reliability of the sources, and iii) the coherence measure, which is needed to assess to degree of coherence of the set.

Consider again n partially reliable sources i which report proposition A _i , for i = 1 , … , n , so that the information set is { A 1 , … , A n } . The propositional variable A _i is defined for the proposition A _i and it can take on two values: A _i and ¬ A i . Similarly, the propositional variable R _i can also take on two values: R _i and ¬ R i . If the report mentions that A _i is the case after consulting the proper source, then R _i ; otherwise, ¬ R i . Then, let P be a probability distribution over the variables A 1 , … , A n , R 1 , … , R n , which satisfy the constraints of having independent and partially reliable sources. One needs sources to be:

1. independent, or else, sources would present information either by looking at other reports and bring additional information or by conveying what they think a coherent report is. For coherence to play a role as confidence boosting, the sources should rather gather information through and only through their own observations which they will report without biasedly inferring what they think a coherent report would look like;

2. partially reliable, or else, sources would either be truth-tellers or randomisers. The latter would report the information needed in entirely random manner and assessing the degree of coherence of information reported without any degree of reliability is useless. The former would report only true information making the property of coherence redundant, because it would no longer be important whether the report is coherent or not as it is already true given that its information comes from a fully reliable source.

These two points can be formally translated.

1. Having independent sources means that R _i should only report that A _i is the case given the she has (likely) observed A _i without her observations being affected by additional facts. Probabilistically speaking, A _i screens off R _i from all other variables A _j and R _j . Thus, there is a conditional independence between R _i and A 1 , R 1 , … , A i − 1 , R i − 1 , A i + 1 , R i + 1 , … , A n , R n , given A _i , for i = 1 , … , n :

1. Partial reliability can be specified with two parameters, namely the true positive rate P ( R i | A i ) = p and the negative positive rate P ( R i | ¬ A i ) = q . Bovens and Hartmann then assume that all sources are equally reliable, that is, all sources have the same p and the same q. The assumptions is introduced because knowing how much one trusts a source is not relevant to assessing the degree of coherence of an information set. For the reasons stated above, all sources in this model are deemed epistemically imperfect, which means that they are more reliable than randomisers, but less reliable than truth-tellers. Thus, the following constraint is imposed on P:

Bovens and Hartmann then define the degree of confidence in the information set as equal to the posterior joint probability of the propositions in the set after all reports have been collected:

Then, they apply the Bayes rule on eq. (22) and simplify it with respect to the independence constraint (eq. (20)):

While the numerator can be seen as:

where p n is the positive rate P ( R i | A i ) = p to the nth degree, and ξ 0 is the prior probability of the conjunction of n − 0 positive values and 0 negative values of the variables A 1 , … , A n . The denominator looks like:

In the denominator, Bovens and Hartmann gather all terms in which the variables A 1 , … A n take, first, n positive values and 0 negative values, then n − 1 positive values and 1 negative value and so on, until the term in which those variables take 0 positive values and n negative values is reached. This means that, for instance, ξ 1 is a prior probability where one proposition is false. Finally, if both numerator and denominator are divided by p n , the posterior probability P * ( A 1 ∧ … ∧ A n ) would be:

where x = q p , that is, the likelihood ratio, and ∑ i = 0 n ξ i = 1 . ξ i is therefore the conjunction of all the joint probabilities of the instances of n − i positive values and i negative values of the variables A 1 , … , A n .

According to the information collected so far, the three measures of Bovens and Hartmann can be finally presented. The expectance measure is defined by the prior joint probability of the propositions in the information set, i.e. the probability before any report was received:

The more ξ 0 increases, the more the degree of coherence of the set increases. The degree of coherence of the information set, i.e. eq. (22), is a monotonically increasing function of r, that is, the reliability measure:

where x is the likelihood ratio. This measure ranges over the open interval (0, 1), because the sources are neither fully reliable ( r ≠ 1 ), nor entirely unreliable ( r ≠ 0 ). The last relevant measure is the coherence measure. In order to evaluate the coherence of an information set, they measure the proportion of the confidence boost b, defined by the ratio P * ( A 1 ∧ … ∧ A n ) P ( A 1 ∧ … ∧ A n ) , relative to the confidence boost b max : a confidence boost which would have been received if the same information had been received in the form of maximally coherent information. A maximally coherent information set would contain only logically equivalent propositions and it has a specific distribution of ξ:

After calculating the posterior joint probability of a maximally coherent information set and its confidence boost, Bovens and Hartmann compute the coherence measure of an information set S = { A 1 , … , A n } . This measure is functionally dependent on the expectance and the reliability measure (cf. Bovens and Hartmann 2003, p. 612):

According to Meijs (2005), the maximal requirement is what makes Bovens and Hartmann’s measure produce counter-intuitive results, because B may, in some cases, consider the degree of coherence of a set containing independent proposition as higher of the degree of coherence of a set whose members are positively dependent. This feature of B can be threatening for the aim of the article: comparing the degree of coherence pre- and post-reduction among the two theories. Therefore, it will be necessary to check whether or not the results which come from the comparisons of the degree of coherence pre- and post-reduction, and which are obtained through different coherence measures, are similar. In case the results will contrast with each other, further remarks regarding the choice of the proper coherence measure will have to be made. It would, in fact, become a contextual question of which coherence measure one should prioritise with respect to the epistemic conception of coherence (e.g. mutual support, boosting confidence) they represent.

4.1 Schupbach’s Measure

Recall Schupbach’s coherence measure. Before the reduction, T _P and T _P are probabilistically independent. Hence, according to Schupbach, the value of coherence will be 0:

The situation changes after T _P is reduced to T _P as the two become probabilistically dependent:

Therefore, according to Schupbach, the two theories will be consistent with each other after the reduction if and only if what is inside the logarithm is strictly higher than 1, that is, if and only if the joint probability of T _F and T _P is higher than the product of their prior probabilities. Thus, one can have the following theorem (see Appendix A):

The first part of theorem means that if T _P and T P ∗ are independent or T _F and T F ∗ are independent, therefore T _P and T _F remain independent after the reduction and their coherence does not improve. The second part of the theorem, instead, means that coherence between T _P and T _F is gained after the reduction if and only if: i) the probability of the conjunction of T _P and T P ∗ is higher than the conditional probability of T _P on ¬ T P ∗ ; and ii) the probability of the conjunction of T _F and T F ∗ is higher than the conditional probability of T _F on ¬ T F ∗ .

4.2 Olsson’s Measure

The same theorem will appear with the comparison made with Olsson’s measure. Before T _P is reduced to T _F , one have to maintain the independence between their variables. Thus, the coherence measure formulated by Olsson would look like the following:

Once the reduction happens, O for S ′ is:

Here, one would need to fix the prior probability P 1 ( T P ) , i.e. P 1 ( T P ) = P 3 ( T P ) , to meaningfully compare the two measures. Then, the sufficient and necessary condition for S ′ ≻ ‾ S to hold is that the denominator of O ( S ′ ) is less than or equal to the denominator of O ( S ) , because the numerator of O ( S ′ ) is greater than the numerator of O ( S ) .^[30] Once the values are substituted in the coherence measure O ,^[31] the following theorem is obtained:

This is the same condition one finds with Schupbach’s measure. The higher the difference P 3 ( T F ∗ | T F ) − P 3 ( T F ∗ | ¬ T F ) and the difference P 3 ( T P | T P ∗ ) − P 3 ( T P | ¬ T P ∗ ) , the higher the degree of coherence.

4.3 Bovens and Hartmann’s Measure

To evaluate the degree of coherence pre- and post-reduction in light of the analysis of Bovens and Hartmann, the reliability measure they formulate should not play any role.^[32] In fact, B construct a quasi-ordering relation over the set { S , S ′ } and this binary relation is formally independent of the reliability measure. Reconsider the two sets S and S ′ which have the same size, i.e. 2. Recall that while P ₁ is the joint probability distribution for the pre-reduction propositions T _F and T _P , P ₃ is the joint probability distribution over the post-reduction propositions T _F and T _P . By calculating the weight vectors < ξ 0 , ξ 1 > for P ₁ and < ξ 0 ′ , ξ 1 ′ > for P ₃, the following difference function can be constructed:

The relation which induces a quasi-ordering over the set of the two information sets S and S ′ is then defined as:

Finally, to determine whether f r ≥ 0 , one needs to assess the conditions under which the sign of f r is actually positive for all values of r ∈ ( 0 , 1 ) . For this reason, Bovens and Hartmann calculate the following condition:

1. ξ 0 ≤ ξ 0 ′ ∧ ξ 1 ≥ ξ 1 ′ , or

2. ξ 0 ≥ ξ 0 ′ ∧ ξ 1 ξ 1 ′ ≥ ξ 0 ξ 0 ′ .

This condition is necessary and sufficient for S ′ ≥ S to hold. First, consider the first condition. ξ 0 ≤ ξ 0 ′ resembles what has been showed above: P 1 ( T F ) × P 1 ( T P ) ≤ P 3 ( T F ∧ T P ) . This is our usual condition, which reports:

Then, ξ 1 ≥ ξ 1 ′ means that:

Once P 1 ( T P ) is fixed and made equal to P 3 ( T P ) (for details, see Appendix C), a similar theorem to the ones seen above follows:

These are the same conditions one has with Olsson and Schupbach’s measures. The second part of the condition mentioned by Bovens and Hartmann is ruled out because it has been showed earlier that the probability of the conjunction of the two propositions T _F and T _P is higher after the reduction rather than before. Thus, the second part of Bovens and Hartmann’s condition does not hold.