Assessing Theories: The Coherentist Approach

In this paper we show that the coherence measures of Olsson (J Philos 94:246–272, 2002), Shogenji (Log Anal 59:338–345, 1999), and Fitelson (Log Anal 63:194–199, 2003) satisfy the two most important adequacy requirements for the purpose of assessing theories. Following Hempel (Synthese 12:439–469, 1960), Levi (Gambling with truth, New York, A. A. Knopf, 1967), and recently Huber (Synthese 161:89–118, 2008) we require, as minimal or necessary conditions, that adequate assessment functions favor true theories over false theories and true and informative theories over true but uninformative theories. We then demonstrate that the coherence measures of Olsson, Shogenji, and Fitelson satisfy these minimal conditions if we confront the hypotheses with a separating sequence of observational statements. In the concluding remarks we set out the philosophical relevance, and limitations, of the formal results. Inter alia, we discuss the problematic implications of our precondition that competing hypotheses must be confronted with a separating sequence of observational statements, which also leads us to discuss theory assessment in the context of scientific antirealism.

1. The present paper’s main purpose is to discuss coherence measures. Its main purpose is not to discuss confirmation measures, nor is it to discuss in detail which confirmation measures are adequate for the purpose of comparing and evaluating (i.e., assessing) theories. Although it would be interesting to compare the various coherence measures and the measures of confirmation proposed in the literature with respect to the question of how they fare as measures for theory assessment, we leave this for another occasion.

2. For the Shogenji measure of coherence this question is already answered in Brössel (forthcoming).

3. There are so many coherence measures in the literature that one cannot discuss all of them in one single paper, at least not in the formal detail I am aiming at. For example Douven and Meijs (2007) discuss several coherence measures that could not be considered here. The Bovens and Hartmann (2003) quasi-ordering of coherence is not taken under consideration because it is not a measure and it is defined in the rigid framework of testimonial systems.

4. Popper (1954), Hempel (1960), Levi (1961, 1967), and Hintikka and Pietarinen (1966) are the most important early papers that try to embody this idea in formal measures of confirmation.

5. More formally: Let the true information provided by a theory T be the set \({\mathfrak{True}}{\text{-}}{\mathfrak{Inf}}(T)=\{A: A\; is\; a\; true\; statement\; that\; is\; implied\; by\; T\}\). Let the false information provided by a theory T is the set \({\mathfrak{False}}{\text{-}}{\mathfrak{Inf}}(T)=\{A: A\; is\; a\; false\; statement\; that\; is\; implied\; by\; T\}\). Suppose T ₁ implies T ₂. Hence, any consequence of T ₂ is also a consequence of T ₁. This implies that if \(A \in {\mathfrak{True}}{\text{-}}{\mathfrak{Inf}}(T_2)\) (i.e., if A is a true consequence of T ₂) then \(A \in {\mathfrak{True}}{\text{-}}{\mathfrak{Inf}}(T_1)\) (A is a true consequence of T ₁) and if \(A \in {\mathfrak{False}}{\text{-}}{\mathfrak{Inf}}(T_2)\) (i.e., if A is a false consequence of T ₂) then \(A \in {\mathfrak{False}} {\text{-}}{\mathfrak{Inf}}(T_1)\) (A is a false consequence of T ₁). What is called true respectively false information here is called truth respectively falsity content by Popper (1968). Miller (1974) and Tichý (1974) provide a detailed discussion of the true and false content or information provided by theories.

6. As already explained, i _p (T, E), respectively \(p(\neg T|\neg E)\), quantifies how much the theory informs us about the observational data. Since Hilpinen (1970) is interested in quantifying the degree of information provided by the observational data about the theory, he uses i _p (E, T), respectively \(p(\neg E|\neg T)\).

7. Hilpinen calls it the normalized common content measure because of its relation to the content measures cont _p . Formally the relation is the following: \(i_p(T,E)=\frac{cont_p(T\vee E)}{cont_p(E)}\). Hilpinen interprets cont _p (T ∨ E) as a measure of the common content of T and E. Hence, i _p (T, E) relates the common content of T and E to the overall content of E and quantifies what proportion of the content of E is also content of T. See Hilpinen (1970) for a detailed axiomatic motivation of this information measure.

8. This is obvious since \(\models (A_1\wedge \ldots \wedge A_n)\leftrightarrow (A_1\wedge \ldots \wedge A_n)\wedge (A_1\vee \ldots \vee A_n)\). This implies that \({{p(A_1\wedge\ldots \wedge A_n)}\over {p(A_1\vee \ldots \vee A_n)}}=p(A_1\wedge \ldots \wedge A_n|A_1\vee \ldots \vee A_n)\). It is remarkable and astonishing that our understanding of “hanging together” is the same as that of Shogenji (1999). He writes: “The crudest way of unpacking the idea that coherent beliefs ‘hang together’ is that they are either true together or false together. However, coherence comes in degrees; in other words we want to say that the more coherent beliefs are, the more likely they are true together” (Shogenji 1999, p. 338). But Shogenji does not agree with our formal interpretation of this intuitive notion of “hanging together,” since he concludes that “[t]he more coherent two beliefs are, the stronger is the positive impact of the truth of one on the truth of the other” (Shogenji 1999, p. 338).

9. Please note, p(T|E) > 0 implies not only that p(T|E) is defined, but also that p(E) > 0.

10. For a detailed argument in support of the Kemeny–Oppenheim measure of factual support, see Fitelson (2001), esp. Sect. 3.2.3.

11. For the coherence measures of Shogenji (1999) and Fitelson the proofs in the appendix actually show the following stronger result: all true theories cohere with the observational data after finitely many steps of observations and for every observation thereafter; false theories don’t . No such theorem is provable for Olsson’s (2002) measures of coherence since coherence is not defined for this measure. Nevertheless Olsson’s coherence measure satisfies the minimal conditions on good theory assessment functions put forward in Sect. 2.2. The reason for this is that the minimal conditions on good theory assessment functions only require that if one consider two theories, on of which is true and the other false, then the true theory cohere more strongly with the evidence than the false theory. This comparative requirement can be satisfied even if their is no definition of the qualitative notion of coherence available.

12. With the help of the content measure cont and theory T ^obs one can introduce a measure of the empirical content of a theory T instead of just the content of a theory. In particular, cont _p (T ^obs) can be said to measure the amount of empirical content of T.

13. It is important to note that this utilization of coherence measures to assess theories is inadequate from the perspective of an anti-realist of the second sort even if the coherence measures would favor true theories over false theories. The reason is that anti-realists of the second sort do not prefer true theories over false theories in general. In particular, if both theories are empirically adequate an anti-realist of the second sort would not necessarily favor the true theory over the false theory. In addition, anti-realists of the second sort would reject the consequence (Theorem 4.2) that all false theories should be treated on a par in the long run. After all, anti-realists of the second sort subscribe to the point of view that false but empirically adequate theories are to be preferred to false and empirically inadequate theories.

14. Van Fraassen (1980) propagates this second sort of anti-realist position most prominently. One might object that van Fraassen advocates a view on theories which stands in sharp contrast to the one presumed here. However, as noted by Hawthorne (2011, p. 333), van Fraassen’s semantic view of theories need not be in opposition to the assumption that scientific theories are expressible in a sentence or statement of some language. “Presumably, if scientists can express a theory well enough to agree about what it says about the world (or at least about its testable empirical content), it must be expressible in some bit of language.” Such a theory “should be subject to empirical evaluation [\(\ldots\) and] a theory of confirmation [or theory assessment] should apply to them.”

15. For Olsson’s coherence measure this is obvious since Olsson (2002) does not even define the qualitative notion of coherence.

16. Not all confirmation measures satisfy this minimal condition though. Huber (2005, 2008) argues that for example the log-likelihood confirmation measure championed by Fitelson (2001), Good (1960), and Kemeny and Oppenheim (1952) is not an adequate measures for the purpose of theory assessment. The reason is that the “log-likelihood ratio measure l neither distinguishes between informative and uninformative true nor between informative and uninformative false theories” (Huber 2005: 1158). This renders it even more interesting that Fitelson’s coherence measure distinguishes between informative and uninformative truth. After all, Fitelson’s coherence measure, when it is applied to pairwise coherence of theory and observational data, is the average of the two incremental confirmations (in the sense of Kemeny and Oppenheim 1952) between the theory and observational data.

17. Interestingly, however, the normalized log-ratio measure suggested by Shogenji (2012) and discussed in detail by Atkinson (2012) is motivated by similar considerations as the theory of theory assessment by Huber (2005, 2008). However, a detailed comparison of both approaches has yet to be undertaken.

Authors and Affiliations

1. Department of Philosophy II, Center for Mind, Brain, and Cognitive Evolution, Ruhr-University of Bochum, Bochum, Germany

   Peter Brössel

Corresponding author

Correspondence to Peter Brössel.

A precursor of this paper appears as “Theory Assessment and Coherence” in the conference proceedings of a graduate conference at the Ruhr-University of Bochum in 2008. The conference proceedings appeared as A. Newen and T. Schlicht, special issue “In Honor of Rudolf Carnap: Central Issues in Epistemology” of Abstracta. I am especially grateful to Franz Huber for many valuable discussions on the topic of this paper and Bayesian epistemology in general. In addition, his exceedingly helpful comments on various drafts of this paper helped tremendously in clarifying the scope and the line of argumentation of the paper. I am also very thankful to Ralf Busse, Anna-Maria A. Eder, and two anonymous referees of this journal for their helpful remarks and suggestions.

Appendix: Proof of Theorems

Proof of Theorem 3.3

\(\forall p \forall \epsilon >0\;\exists \delta_{\epsilon} >0: [p(T_2 \wedge E)> 0\;\&\;0<p(E)<1\;\&\;i_p(T_1,E)\geq i_p(T_2,E)+\epsilon\;\&\;p(T_1|E)\geq p(T_2|E)-\delta_{\epsilon}] \Rightarrow \mathcal{C}_{O, p}(T_1,E)> \mathcal{C}_{O, p}(T_2,E)\)

Proof

Let p be a probability function and let p(T ₂ ∧ E) > 0 and 0 < p(E) < 1. Let \(\epsilon\) be arbitrary with \(\epsilon >0\).

The proof is straightforward if \(\delta_\epsilon\) is chosen arbitrary with \(0< \delta_\epsilon \times p(E)< \frac{p(T_2 \wedge E)}{1-p(\neg T_2 \wedge \neg E)}\times [\epsilon \times p(\neg E)]. \) We know there is such an \(\delta_\epsilon\). because p(T ₂ ∧ E) > 0, 0 < p(E) < 1 and \(\epsilon >0\).

We have to prove that:

1. 1.

   \(i_p(T_1,E)\geq i_p(T_2,E)+\epsilon \Rightarrow [p(T_1| E)\geq p(T_2|E)-\delta_\epsilon \Rightarrow \mathcal{C}_{O,p}(T_1,E)> \mathcal{C}_{O,p}(T_2,E)]\)

2. 2.

   Suppose \(i_p(T_1,E)\geq i_p(T_2,E)+\epsilon, \) this means \(p(\neg T_1\wedge \neg E)\geq p(\neg T_2 \wedge \neg E) +\epsilon \times p(\neg E)\) which implies \(1-p(\neg T_1\wedge \neg E)\leq 1-[p(\neg T_2\wedge \neg E) +\epsilon \times p(\neg E)]\)

3. 3.

   Suppose \(p(T_1|E)\geq p(T_2|E)-\delta_\epsilon , \) i.e. \( p(T_1\wedge E)\geq p(T_2\wedge E)-\delta_\epsilon \times p(E)\)

4. 4.

   \(\frac{\delta_\epsilon \times p(E)}{p(T_2\wedge E)}< \frac{\epsilon \times p(\neg E)}{1-p(\neg T_2 \wedge \neg E)}\) (from 1)

5. 5.

   \([1-\frac{\delta_\epsilon \times p(E)}{p(T_2 \wedge E)}] > 1- \frac{\epsilon \times p(\neg E)}{1-p(\neg T_2 \wedge \neg E)}\) (from 4)

6. 6.

   \(\frac{p(T_2\wedge E)}{p(T_2\wedge E)}-\frac{\delta_\epsilon \times p(E)}{p(T_2\wedge E)}> 1-\frac{\epsilon \times p(\neg E)}{1-p(\neg T_2\wedge \neg E)}\) (from 5)

7. 7.

   \(\frac{p(T_2\wedge E)-\delta_\epsilon \times p( E)}{p(T_2\wedge E)}> 1-\frac{\epsilon \times p(\neg E)}{1-p(\neg T_2\wedge \neg E)}\) (from 6)

8. 8.

   \(\frac{p(T_1 \wedge E)}{p(T_2 \wedge E)}> 1-\frac{\epsilon \times p(\neg E)}{1-p(\neg T_2\wedge \neg E)}\) (from 3 and 7)

9. 9.

   \(\frac{p(T_1 \wedge E)}{p(T_2\wedge E)}> \frac{1-p(\neg T_2\wedge \neg E)}{1-p(\neg T_2\wedge \neg E)}-\frac{\epsilon\times p(\neg E)}{1-p(\neg T_2\wedge \neg E)}\) (from 8)

10. 10.

    \(\frac{p(T_1 \wedge E)}{p(T_2\wedge E)}> \frac{[1-p(\neg T_2\wedge \neg E)]-\epsilon \times p(\neg E)}{1-p(\neg T_2\wedge \neg E)}\) (from 9)

11. 11.

    \(\frac{p(T_1 \wedge E)}{p(T_2\wedge E)}> \frac{1-[p(\neg T_2\wedge \neg E)+\epsilon \times p(\neg E)]}{1-p(\neg T_2\wedge \neg E)}\) (from 10)

12. 12.

    \(\frac{p(T_1 \wedge E)}{p(T_2\wedge E)}> \frac{1-p(\neg T_1 \wedge \neg E)}{1-p(\neg T_2\wedge \neg E)}\) (from 2 and 11)

13. 13.

    \(\frac{p(T_1\wedge E)}{1-p(\neg T_1 \wedge \neg E)}> \frac{p(T_2\wedge E)}{1-p(\neg T_2\wedge \neg E)}\) (from 12)

14. 14.

    \(\frac{p(T_1\wedge E)}{p(T_1 \vee E)}> \frac{p(T_2\wedge E)}{p(T_2\vee E)}\) (from 13)

15. 15.

    \(\mathcal{C}_{O,p}(T_1,E)>\mathcal{C}_{O,p}(T_2,E)\) (from 14 and the definition of \(\mathcal{C}_{O,p}\))

Proof of Theorem 3.6

\(\forall p \forall \epsilon >0\) \(\exists \delta_{\epsilon}>0 : [cont_p(T_1)\geq cont_p(T_2)+\epsilon\;\&\;0\neq p(T_1|E)\geq p(T_2|E)-\delta_{\epsilon}] \Rightarrow \mathcal{C}_{S, p}(T_1,E)>\mathcal{C}_{S, p}(T_2,E)\)

Proof

Let p be a probability function with p(E ∧ T ₁) > 0 and let \(\epsilon>0\).

The proof is straightforward if \(\delta_{\epsilon}\) is chosen arbitrary satisfying \(0<\delta_\epsilon <\frac{p(T_1\wedge E)}{p(E)\times p(T_1)}\times \epsilon\). We know there is such an \(\delta_\epsilon \in \mathbb{R}\), because p(T ₁ ∧ E) > 0.

Proof of Theorem 3.10

\(\forall p \forall \epsilon >0 \exists \delta_{\epsilon}>0 : [p(T_1\wedge E)>0\;\&\;cont_p(T_1)\geq cont_p(T_2)+\epsilon\;\&\;p(T_1|E)\geq p(T_2|E)-\delta_{\epsilon}] \Rightarrow \mathcal{C}_{F, p}(T_1,E)> \mathcal{C}_{F, p}(T_2,E)\)

Proof

Let p be an arbitrary probability function with p(T ₁ ∧ E) > 0 and let \(\epsilon>0\).

The proof is straightforward if \(\delta_{\epsilon}\) chosen arbitrary satifying \(0< \delta_\epsilon \leq \frac{p(T_1\wedge E)}{p(E)\times p(T_1)}\times \epsilon\). We know that there is such an \(\delta_{\epsilon}\) because p(T ₁ ∧ E) > 0 and \(\epsilon>0\).

Proof of Theorem 4.1 for Olsson’s Coherence Measure

Let e ₁,…, e _n ,…be a sequence of sentences of \(\mathcal{L}\) which separates \(Mod_\mathcal{L}\), and let e ^w _i  = e _i , if \(w\,\vDash\, e_i\) and \(\neg e_i\) otherwise. Let p be a strict (or regular) probability function on \(\mathcal{L}\). Let p* be the unique determined probability function on the smallest σ-field \(\mathcal{A}\) containing the field \(\{Mod(A):A\in \mathcal{L}\}\) satisfying p*(Mod(A)) = p(A) for all \(A\in \mathcal{L}, \) where \(Mod(A)=\{w\in Mod_\mathcal{L}: w\,\vDash\, A\}\) and \(Mod_\mathcal{L}\) is the set of all maximal-consistent sets of sentences of \(\mathcal{L}\) including instances.

Then according to the Gaifman–Snir Theorem (Gaifman and Snir 1982) there is a \(X\subseteq Mod_\mathcal{L}\) with p*(X) = 1, such that the following holds for every w ∈ X and all theories T ₁ and T ₂ of \(\mathcal{L}\).

where \(\mathcal{I}(T_1,w)=1\), if \(w\,\vDash\, T_1\) and 0 otherwise.

1. 1.

   Suppose additionally that \(w^\prime \in X\) and \(w^\prime \,\vDash\, T_1\) and \(w^\prime \,\vDash\, \neg T_2\).

   By the Gaifman and Snir (1982) Theorem and the assumptions the following holds:

   $$ \mathop {lim}\limits_{{n \to \infty }}\left[p\left(T_1|E^{w^\prime}_n\right)\right]=1\,\hbox{and}\,\mathop {lim}\limits_{{n \to \infty }}\left[p\left(T_2 |E^{w^\prime}_n\right)\right]=0. $$

   And since the following holds true:

   $$ \mathop {lim}\limits_{{n \to \infty }}\left[p\left(T_1\vee E^{w^\prime}_n\right)\right] \in [p(T_1), 1]\,\hbox{and}\,\mathop {lim}\limits_{{n \to \infty }}\left[p\left(T_2 \vee E^{w^\prime}_n\right)\right]\in [p(T_2), 1] $$

   we can conclude that

   $$ \mathop {lim}\limits_{{n \to \infty }}\left[p\left(T_1| E^{w^\prime}_n\right)\right]>\mathop {lim}\limits_{{n \to \infty }}\left[p\left(T_2 |E^{w^\prime}_n\right)\times \frac{p\left(T_1\vee E^{w^\prime}_n\right)}{p\left(T_2 \vee E^{w^\prime}_n\right)}\right]. $$

   This implies \(\exists n \forall m\geq n: p(T_1|E^{w^\prime}_m)>p(T_2 |E^{w^\prime}_m)\times \frac{p(T_1\vee E^{w^\prime}_m)}{p(T_2 \vee E^{w^\prime}_m)}\).

   From which we can infer that: \(\exists n \forall m\geq n: \frac{p(T_1\wedge E^{w^\prime}_m)}{p(T_1\vee E^{w^\prime}_m)}>\frac{p(T_2 \wedge E^{w^\prime}_m)}{p(T_2 \vee E^{w^\prime}_m)}\).

   which implies that: ∃n∀m ≥ n:

   $$ {\mathcal{C}}_{O,p}(T_1,E^{w^\prime}_m)>{\mathcal{C}}_{O,p}(T_2 ,E^{w^\prime}_m) $$
2. 2.

   Additionally assume that \(w^\prime \in X\) and \(w^\prime \,\vDash\, T_1\wedge T_2\) and \(T_1\,\vDash\, T_2\) but \(T_2\nvdash T_1\). Since p is by assumption a strict probability function is must hold that cont _p (T ₁) > cont _p (T ₂) and hence that p(T ₂) > p(T ₁).

   By the Gaifman–Snir Theorem we can infer that:

   $$ {\mathop{lim}\limits_{m \rightarrow \infty}}p(T_1|E_m^{w^\prime} )={\mathop{lim}\limits_{m\rightarrow \infty}}p(T_2|E_m^{w^\prime})=1 $$

   with this we get that for all \(\epsilon\) there exists a \(n_\epsilon\) such that for all \(m\geq n_\epsilon\) holds that:

   $$ |p(T_2|E^{w^\prime}_m)-p(T_1|E^{w^\prime}_m)|<\epsilon $$

   Then define \(\epsilon =\frac{[p(T_2)-p(T_1)]}{2}\)

From this definition we get that for all

\(m\geq n_\epsilon\):

\(p(\neg T_1| \neg E^{w^\prime}_m)-p(\neg T_2 |\neg E^{w^\prime}_m)>\epsilon\) since:

1. 1.

   \(p(T_2|E^{w^\prime}_m)-p(T_1|E^{w^\prime}_m)<\epsilon\) for all \(m\geq n_\epsilon\) as assumed above.

2. 2.

   \([p(T_2)-p(T_1)]-[p(T_2 |E^{w^\prime}_m)-p(T_1|E^{w^\prime}_m)]>[p(T_2)-p(T_1)]-\epsilon\)

3. 3.

   \([p(T_2)-p(T_1)]-[p(T_2 |E^{w^\prime}_m)-p(T_1|E^{w^\prime}_m)]>\epsilon\)

4. 4.

   \([p(T_2)-p(T_1)]-[p(T_2 |E^{w^\prime}_m)-p(T_1|E^{w^\prime}_m)]\times p(E^{w^\prime}_m)>\epsilon\) (from 3.)

5. 5.

   \([p(T_2)-p(T_1)]-[p(T_2 \wedge E^{w^\prime}_m)-p(T_1\wedge E^{w^\prime}_m)]> \epsilon\)

6. 6.

   \([p(T_2)-p(T_2 \wedge E^{w^\prime}_m)]-[ p(T_1)-p(T_1\wedge E^{w^\prime}_m)]>\epsilon\)

7. 7.

   \([p(T_2 )+p(E^{w^\prime}_m)-p(T_2 \wedge E^{w^\prime}_m)]-[p(T_1)+p(E^{w^\prime}_m)-p(T_1 \wedge E^{w^\prime}_m)]>\epsilon\)

8. 8.

   \([p(T_2\vee E^{w^\prime}_m)]-[p(T_1\vee E^{w^\prime}_m)]>\epsilon\)

9. 9.

   \([1-p(T_1\vee E^{w^\prime}_m)]-[1-p(T_2 \vee E^{w^\prime}_m)]>\epsilon\)

10. 10.

    \(p(\neg T_1\wedge \neg E^{w^\prime}_m)-p(\neg T_2 \wedge \neg E^{w^\prime}_m)> \epsilon\)

11. 11.

    \(p(\neg T_1\wedge \neg E^{w^\prime}_m)-p(\neg T_2 \wedge \neg E^{w^\prime}_m)>\epsilon\)

12. 12.

    \(p(\neg T_1|\neg E^{w^\prime}_m)-p(\neg T_2 |\neg E^{w^\prime}_m)>\frac{\epsilon}{p(\neg E^{w^\prime}_m)}>\epsilon\)

From which we can infer that:

By Theorem 3.3 we can infer that there is a \(\delta_{\epsilon}\) such that:

Again by the Gaifman–Snir Theorem we get that there is a \(n_{\delta_{\epsilon}}\) such that for all \(m\geq n_{\delta_{\epsilon}}\)

or equivalently: (II) \(p(T_1| E^{w^\prime}_m)> p(T_2 |E^{w^\prime}_m)-\delta_{n_1}\)

Again by Theorem 3.3 and (I) and (II) we can conclude: For all m ≥ n*, where \(n^*= \hbox{max}.\{n_{\delta_{\epsilon}}, n_{\epsilon} \}\) it holds that:

Which implies that there a n such that for all For all m ≥ n:

Proof of Theorem 4.1 for Shogenji’s Coherence Measure

Let \(e_1, \ldots, e_n,\ldots\) be a sequence of sentences of \(\mathcal{L}\) which separates \(Mod_\mathcal{L}\), and let e ^w _i  = e _i , if \(w\,\vDash\, e_i\) and \(\neg e_i\) otherwise. Let p be a strict (or regular) probability function on \(\mathcal{L}\). Let p* be the unique determined probability function on the smallest σ-field \(\mathcal{A}\) containing the field \(\{Mod(A):A\in \mathcal{L}\}\) satisfying p*(Mod(A)) = p(A) for all \(A\in \mathcal{L}, \) where \(Mod(A)=\{w\in Mod_\mathcal{L}: w\,\vDash\, A\}\) and \(Mod_\mathcal{L}\) is the set of all maximal-consistent sets of sentences of \(\mathcal{L}\) including instances.

Then according to the Gaifman–Snir Theorm (Gaifman and Snir 1982) there is a \(X\subseteq Mod_\mathcal{L}\) with p*(X) = 1, such that the following holds for every \(w\in X\) and all theories T ₁ and T ₂ of \(\mathcal{L}:\)

where \(\mathcal{I}(T_1,w)=1\), if \(w\,\vDash\, T_1\) and 0 otherwise.

1. 1.

   Suppose additionally that \(w^\prime \in X,\, w^\prime \,\vDash\, T_1\) and \(w^\prime \,\vDash\, \neg T_2\).

   We know that \(lim_{n\rightarrow \infty}[p(T_1|E^{w^\prime}_n)]=1\) and \(lim_{n\rightarrow \infty}[p(T_2 |E^{w^\prime}_n)]=0\) by the Gaifman–Snir Theorem.

   So we can infer that:

   \(lim_{n\rightarrow \infty}[p(T_1|E^{w^\prime}_n)\times \frac{1}{p(T_1)}]= \frac{1}{p(T_1)}>1>lim_{n\rightarrow \infty}[p(T_2 |E^{w^\prime}_n)\times \frac{1}{p(T_2)}]= 0\)

   Let \(\epsilon=\frac{\frac{1}{p(T_1)}}{2}. \) Then it holds that: \(\exists n\forall m\geq n : |\frac{1}{p(T_1)}-\mathcal{C}_{S,p}(T_1,E^{w^\prime}_m)|<\epsilon\) and \(\mathcal{C}_{S,p}(T_1,E^{w^\prime}_m)>1 \) and \(\exists n^\prime \forall m\geq n^\prime: |0-\mathcal{C}_{S,p}(T_2 ,E^{w^\prime}_m)|<\epsilon\) and \(\mathcal{C}_{S,p}(T_2,E^{w^\prime}_m)<1\).

   Now let n ₁ = max.\(\{n, n^\prime\}. \) Then it holds for all m ≥ n ₁:

   $$ {\mathcal{C}}_{S,p}\left(T_1,E^{w^\prime}_m\right)>1> {\mathcal{C}}_{S,p}\left(T_2 ,E^{w^\prime}_m\right) $$
2. 2.

   Additionally assume that \(w^\prime \in X\) and \(w^\prime \,\vDash\, T_1\wedge T_2\) and \(T_1\,\vDash\, T_2\) but \(T_2\nvdash T_1. \) Since p is by assumption a strict probability function is must hold that cont _p (T ₁) > cont _p (T ₂) and hence that p(T ₂) > p(T ₁).

   Thereby we know that: \(lim_{n\rightarrow \infty}[p(T_1|E^{w^\prime}_n)]=1\) and \(lim_{n\rightarrow \infty}[p(T_2 |E^{w^\prime}_n)]=1\) by the Gaifman–Snir Theorem and the assumption that \(w^\prime \,\vDash\, T_1 \wedge T_2\).

   This implies that \(lim_{n\rightarrow \infty}[p(T_1|E^{w^\prime}_n)\times \frac{1}{p(T_1)}]= \frac{1}{p(T_1)}>lim_{n\rightarrow \infty}[p(T_2 |E^w_n)\times \frac{1}{p(T_2)}]= \frac{1}{p(T_2)}\)

   Since cont _p (T ₁) > cont _p (T ₂) we can infer that p(T ₁) < p(T ₂) and \(\frac{1}{p(T_1)}> \frac{1}{p(T_2)}\)

   Now let \(\epsilon=\frac{\frac{1}{p(T_1)}-\frac{1}{p(T_2)}}{2}\). Then it holds that: \(\exists n\forall m\geq n : |\frac{1}{p(T_1)}-\mathcal{C}_{S,p}(T_1,E^w_m)|<\epsilon\) and \(\exists n^\prime \forall m\geq n^\prime |\frac{1}{p(T_2)}-\mathcal{C}_{S,p}(T_2 ,E^w_m)|<\epsilon\).

   Now let n ₁ = max. \(\{n, n^\prime\}. \) Then it holds for all m ≥ n ₁:

   $$ {\mathcal{C}}_{S,p}\left(T_1,E^w_m\right)> {\mathcal{C}}_{S,p}\left(T_2 ,E^w_m\right) $$

Proof of Theorem 4.1 for Fitelson’s Coherence Measure

Let \(e_1,\ldots , e_n,\ldots\) be a sequence of sentences of \(\mathcal{L}\) which separates \(Mod_\mathcal{L}\), and let e ^w _i  = e _i , if \(w\,\vDash\, e_i\) and \(\neg e_i\) otherwise. Let p be a strict (or regular) probability function on \(\mathcal{L}\). Let p* be the unique determined probability function on the smallest σ-field \(\mathcal{A}\) containing the field \(\{Mod(A):A\in \mathcal{L}\}\) satisfying p*(Mod(A)) = p(A) for all \(A\in \mathcal{L}, \) where \(Mod(A)=\{w\in Mod_\mathcal{L}: w\,\vDash\, A\}\) and \(Mod_\mathcal{L}\) is the set of all maximal-consistent sets of sentences of \(\mathcal{L}\) including instances. Then according to the Gaifman–Snir Theorm (Gaifman and Snir 1982) there is a \(X\subseteq Mod_\mathcal{L}\) with p*(X) = 1, such that the following holds for every \(w\in X\) and all theories T ₁ and T ₂ of \(\mathcal{L}\).

where \(\mathcal{I}(T_1,w)=1\), if \(w\,\vDash\, T_1\) and 0 otherwise.

1. 1.

   Assume additionally that \(w^\prime \in X\) and and \(w^\prime \,\vDash\, T_1\) and \(w^\prime \,\vDash\, \neg T_2. \)

   First note that:

   $$ \begin{aligned}&{\mathop{lim}\limits_{n\rightarrow\infty}}\left[{\mathcal{C}}_{F,p}(T_i,E_n^{w^\prime})\right]\\ &=\frac{1}{2}\left[{\mathop{lim}\limits_{n\rightarrow\infty}}\left[\frac{p(T_i|E^{w^\prime}_n)-p(T_i|\neg E^{w^\prime}_n)}{p(T_i|E^{w^\prime}_n)+p(T_i|\neg E^{w^\prime}_n)}\times\frac{\frac{1}{p(T_i)}}{\frac{1}{p(T_i)}} \right]\right.\\ &\quad+\left. {\mathop{lim}\limits_{n\rightarrow\infty}}\left[\frac{p(E^{w^\prime}_n|T_i)-p(E^{w^\prime}_n|\neg T_i)}{p(E^{w^\prime}_n|T_i)+p(E^{w^\prime}_n|\neg T_i)}\times\frac{\frac{1}{p(E^{w^\prime}_n)}}{\frac{1}{p(E^{w^\prime}_n)}}\right]\right]\\ &=\frac{1}{2}\left[{\mathop{lim}\limits_{n\rightarrow\infty}}\left[\frac{{\mathcal{C}}_{S,p}(T_i,E^{w^\prime}_n)-{\mathcal{C}}_{S,p}(T_i,\neg E^{w^\prime}_n)}{{\mathcal{C}}_{S,p}(T_i,E^{w^\prime}_n)+{\mathcal{C}}_{S,p}(T_i,\neg E^{w^\prime}_n)} \right]\right.\\&\quad+\left.{\mathop{lim}\limits_{n\rightarrow\infty}}\left[\frac{{\mathcal{C}}_{S,p}(T_i,E^{w^\prime}_n)-{\mathcal{C}}_{S,p}(\neg T_i,E^{w^\prime}_n)}{{\mathcal{C}}_{S,p}(T_i,E^{w^\prime}_n)+{\mathcal{C}}_{S,p}(\neg T_i,E^{w^\prime}_n)}\right] \right] \end{aligned} $$

   and since we know that under the above assumptions by Theorem 4.1 the following holds:

   1. 1.

      \(lim_{n\rightarrow \infty}[\mathcal{C}_{S,p}(T_1,E^{w^\prime}_n)]=\frac{1}{p(T_1)}\)

   2. 2.

      \(lim_{n\rightarrow \infty}[\mathcal{C}_{S,p}(T_1,\neg E^{w^\prime}_n)]\leq \frac{1}{p(T_1)}\)

   3. 3.

      \(lim_{n\rightarrow \infty}[\mathcal{C}_{S,p}(\neg T_1,E^{w^\prime}_n)]=0\)

   4. 4.

      \(lim_{n\rightarrow \infty}[\mathcal{C}_{S,p}(T_2,E^{w^\prime}_n)]=0\)

   5. 5.

      \(lim_{n\rightarrow \infty}[\mathcal{C}_{S,p}(T_2,\neg E^{w^\prime}_n)]\geq 0\)

   6. 6.

      \(lim_{n\rightarrow \infty}[\mathcal{C}_{S,p}(\neg T_2,E^{w^\prime}_n)]=\frac{1}{p(\neg T_2)}\)

   we can conclude that:

   $$ \mathop {lim}\limits_{{n \to \infty }}[{\mathcal{C}}_{F,p}(T_1,E_n^{w^\prime})]>0>\mathop {lim}\limits_{{n \to \infty }}[{\mathcal{C}}_{F,p}(T_2,E_n^{w^\prime}) $$

   Which implies that:

   $$ \exists n \forall m\geq n: {\mathcal{C}}_{F,p}\left(T_1,E_m^{w^\prime}\right)>0>{\mathcal{C}}_{F,p}\left(T_2,E_m^{w^\prime}\right) $$
2. 2.

   Additionally assume that \(w^\prime \in X\) and \(w^\prime \,\vDash\, T_1\wedge T_2\) and \(T_1\,\vDash\, T_2\) but \(T_2\nvdash T_1. \) Since p is by assumption a strict probability function is must hold that cont _p (T ₁) > cont _p (T ₂) and hence that p(T ₂) > p(T ₁).

   We already know that:

   $$ \begin{aligned} &{\mathop{lim}\limits_{n\rightarrow\infty}}\left[{\mathcal{C}}_{F,p}(T_i,E_n^{w^\prime})\right]\\&=\frac{1}{2}\left[{\mathop{lim}\limits_{n\rightarrow\infty}}\left[\frac{{\mathcal{C}}_{S,p}(T_i,E^{w^\prime}_n)-{\mathcal{C}}_{S,p}(T_i,\neg E^{w^\prime}_n)}{{\mathcal{C}}_{S,p}(T_i,E^{w^\prime}_n)+{\mathcal{C}}_{S,p}(T_i,\neg E^{w^\prime}_n)}\right]\right.\\ &\quad+\left. {\mathop{lim}\limits_{n\rightarrow\infty}}\left[\frac{{\mathcal{C}}_{S,p}(T_i,E^{w^\prime}_n)-{\mathcal{C}}_{S,p}(\neg T_i,E^{w^\prime}_n)}{{\mathcal{C}}_{S,p}(T_i,E^{w^\prime}_n)+{\mathcal{C}}_{S,p}(\neg T_i,E^{w^\prime}_n)} \right]\right] \end{aligned} $$

   and since we know that under the above assumptions by Theorem 4.1 the following holds:

   1. 1.

      \(lim_{n\rightarrow \infty}[\mathcal{C}_{S,p}(T_1,E^{w^\prime}_n)]=\frac{1}{p(T_1)}\)

   2. 2.

      \(lim_{n\rightarrow \infty}[\mathcal{C}_{S,p}(\neg T_1,E^{w^\prime}_n)]=0\)

   3. 3.

      \(lim_{n\rightarrow \infty}[\mathcal{C}_{S,p}(T_2,E^{w^\prime}_n)]=\frac{1}{p(T_2)}\)

   4. 4.

      \(lim_{n\rightarrow \infty}[\mathcal{C}_{S,p}(\neg T_2,E^{w^\prime}_n)]=0\)

   Accordingly, in order to prove Theorem 4.3 it is sufficient to prove that:

   $$ \mathop {lim}\limits_{{n \to \infty }}\left[{\mathcal{C}}_{S,p}\left(T_2,\neg E^{w^\prime}_n\right)\right]\geq \mathop {lim}\limits_{{n \to \infty }}\left[{\mathcal{C}}_{S,p}\left(T_1,\neg E^{w^\prime}_n\right)\right] $$

   We already know that:

   $$ \mathop {lim}\limits_{{n \to \infty }}\left[{\mathcal{C}}_{S,p}\left(T_1,E^{w^\prime}_n\right)\right]=\frac{1}{p(T_1)}>\mathop {lim}\limits_{{n \to \infty }}\left[{\mathcal{C}}_{S,p}\left(T_2,E^{w^\prime}_n\right)\right]=\frac{1}{p(T_2)} (\hbox{since}\; p(T_1)<p(T_2)) $$

   This implies

   $$ \begin{aligned} & {\mathop{lim}\limits_{n\rightarrow\infty}}\left[p\left(E^{w^\prime}_n|T_1\right)\right]>{\mathop{lim}\limits_{n\rightarrow\infty}}\left[p\left(E^{w^\prime}_n|T_2\right)\right]\\ & \quad \Leftrightarrow{\mathop{lim}\limits_{n\rightarrow\infty}}\left[1-p\left(E^{w^\prime}_n|T_2\right)\right]>{\mathop{lim}\limits_{n\rightarrow\infty}}\left[1-p\left(E^{w^\prime}_n|T_1\right)\right]\\ & \quad \Leftrightarrow \mathop {lim}\limits_{{n \to \infty }}\left[p\left(\neg E^{w^\prime}_n|T_2\right)\right]>{\mathop{lim}\limits_{n\rightarrow\infty}}\left[p\left(\neg E^{w^\prime}_n|T_1\right)\right]\\ & \quad \Leftrightarrow \mathop {lim}\limits_{{n \to \infty }}\left[p\left(\neg E^{w^\prime}_n|T_2\right)\times \frac{1}{p\left(\neg E^{w^\prime}_n\right)}\right]>{\mathop{lim}\limits_{n\rightarrow\infty}}\left[p\left(\neg E^{w^\prime}_n|T_1\right)\times\frac{1}{p\left(\neg E^{w^\prime}_n\right)}\right]\Leftrightarrow\\& \mathop {lim}\limits_{{n \to \infty }}\left[{\mathcal{C}}_{S,p}\left(T_2,\neg E^{w^\prime}_n\right)\right]\geq {\mathop{lim}\limits_{n\rightarrow\infty}}\left[{\mathcal{C}}_{S,p}\left(T_1,\neg E^{w^\prime}_n\right)\right] \end{aligned} $$

   We can conclude

   $$ \mathop {lim}\limits_{{n \to \infty }}\left[\frac{{\mathcal{C}}_{S,p}\left(T_1,E^{w^\prime}_n\right)-{\mathcal{C}}_{S,p}\left( T_1,\neg E^{w^\prime}_n\right)}{{\mathcal{C}}_{S,p}\left(T_1,E^{w^\prime}_n\right)+{\mathcal{C}}_{S,p}\left( T_1,\neg E^{w^\prime}_n\right)}\right]>\mathop {lim}\limits_{{n \to \infty }}\left[\frac{{\mathcal{C}}_{S,p}\left(T_2,E^{w^\prime}_n\right)-{\mathcal{C}}_{S,p}\left( T_2,\neg E^{w^\prime}_n\right)}{{\mathcal{C}}_{S,p}\left(T_2,E^{w^\prime}_n\right)+{\mathcal{C}}_{S,p}\left( T_2,\neg E^{w^\prime}_n\right)}\right] $$

   and

   $$ \mathop {lim}\limits_{{n \to \infty }}\left[\frac{{\mathcal{C}}_{S,p}\left(T_2,E^{w^\prime}_n\right)-{\mathcal{C}}_{S,p}\left(\neg T_2,E^{w^\prime}_n\right)}{{\mathcal{C}}_{S,p}\left(T_2,E^{w^\prime}_n\right)+{\mathcal{C}}_{S,p}\left(\neg T_2,E^{w^\prime}_n\right)}\right]=\mathop {lim}\limits_{{n \to \infty }}\left[\frac{{\mathcal{C}}_{S,p}\left(T_2,E^{w^\prime}_n\right)-{\mathcal{C}}_{S,p}\left(\neg T_2,E^{w^\prime}_n\right)}{{\mathcal{C}}_{S,p}\left(T_2,E^{w^\prime}_n\right)+{\mathcal{C}}_{S,p}\left(\neg T_2,E^{w^\prime}_n\right)}\right] $$

   Which proves that:

   $$ \exists n \forall m\geq n: {\mathcal{C}}_{F,p}\left(T_1,E_m^{w^\prime}\right)>{\mathcal{C}}_{F,p}\left(T_2,E_m^{w^\prime}\right) $$

Proof of Theorem 4.2

Let \(e_1, \ldots, e_n, \ldots\) be a sequence of statements of \(\mathcal{L}\) which separates \(Mod_\mathcal{L}\), and let e ^w _i  = e _i if \(w\,\vDash\, e_i\) and \(\neg e_i\) otherwise. Let p be a strict (or regular) probability function on \(\mathcal{L}\). Let p* be the unique probability function on the smallest σ-field \(\mathcal{A}\) containing the field \(\{Mod(A):A\in \mathcal{L}\}\) satisfying p*(Mod(A)) = p(A) for all \(A\in \mathcal{L}, \) where \(Mod(A)=\{w\in Mod_\mathcal{L}: w\,\vDash\, A\}\) and \(Mod_\mathcal{L}\) is the set of all maximal-consistent sets of statements of \(\mathcal{L}\) including instances.

Then there is an \(X\subseteq Mod_\mathcal{L}\) with p*(X) = 1 such that the following holds for every \(w\in X\) and all theories T ₁ and T ₂ of \(\mathcal{L}\).

where \(\mathcal{I}(T_1,w)=1\), if \(w\,\vDash\, T_1\) and 0 otherwise.

Hence, if \(w\,\vDash\, \neg T_i\), then:

This implies that \(lim_{n\rightarrow \infty}[\mathcal{C}_{O,p}(T_i,E^w_m)]=0\) since \(0\leq\mathcal{C}_{O,p}(T_i,E^w_n)\leq p(T_i|E^w_n)\), for all n.

Furthermore, \(lim_{n\rightarrow \infty}[\mathcal{C}_{S,p}(T_i,E^w_m)]=0\), since \(\mathcal{C}_{S,p}(T_i,E^w_n)={{p(T_i|E^w_n)}\over {p(T_1)}}\), for all n.

In addition, we know from proof of theorem 4.1 for Fitelson’s measure \(\mathcal{C}_{F,p}\) that

We already said that \(lim_{n\rightarrow \infty}[\mathcal{C}_{S,p}(T_i,E^w_m)]=0\) and \(lim_{n\rightarrow \infty}[\mathcal{C}_{S,p}(\neg T_i,E^w_m)]=1\).

Furthermore, \(lim_{n\rightarrow \infty}[\mathcal{C}_{S,p}(T_i, E^w_m)]=0\) iff \(lim_{n\rightarrow \infty}[p(T_i\wedge E^w_m)]=0\) and hence, \(lim_{n\rightarrow \infty}[p(T_i\wedge \neg E^w_m)]=p(T_i)\). Thus \(lim_{n\rightarrow \infty}[\mathcal{C}_{S,p}(T_i,\neg E^w_m)]={{1}\over {lim_{n\rightarrow \infty}[p(\neg E ^w_m)]}}\), where \(lim_{n\rightarrow \infty}[p(\neg E ^w_m)]>0. \) Hence,

Reprints and permissions