Intertheoretic Reduction, Confirmation, and Montague’s Syntax-Semantics Relation

Abstract

Intertheoretic relations are an important topic in the philosophy of science. However, since their classical discussion by Ernest Nagel, such relations have mostly been restricted to relations between pairs of theories in the natural sciences. This paper presents a case study of a new type of intertheoretic relation that is inspired by Montague’s analysis of the linguistic syntax-semantics relation. The paper develops a simple model of this relation. To motivate the adoption of our new model, we show that this model extends the scope of application of the Nagelian (or related) models and that it shares the epistemological advantages of the Nagelian model. The latter is achieved in a Bayesian framework.

Appendix: Proofs and Calculations for Sect. 4

We have calculated the pre-‘reductive’ probabilities of the conjunction of the positive instantiations of S and G in Sect. 4.1. The joint distribution, ${{\varvec{P}}}_{2}(S, G, E)$, of the (post-‘reductive’) graph in Fig. 8 is given by the expression

$$\begin{aligned} {{\varvec{P}}}_{2}(S)\, {{\varvec{P}}}_{2}(G)\, {{\varvec{P}}}_{2}(E \vert G). \end{aligned}$$

Using the methodology from Bovens and Hartmann (2003), the prior probability of the conjunction of ${\text{ S }}$ and ${\text{ G }}$ is obtained as follows:

$$\begin{aligned} {{\varvec{P}}}_{2}({\text{ S }}, {\text{ G }}) = \sum _{E} {{\varvec{P}}}_{2}({\text{ S }}, {\text{ G }}, E) = \pi \, \sigma + \bar{\pi }\, \sigma = \sigma \end{aligned}$$

(17)

We yield the posterior probability, ${{\varvec{P}}}_{2}^{\,*} := {{\varvec{P}}}_{2}({\text{ S }}, {\text{ G }}|{\text{ E }})$, of the conjunction of ${\text{ S }}$ and ${\text{ G }}$ thus:

$$\begin{aligned} {{\varvec{P}}}_{2}^{\,*} = \frac{{{\varvec{P}}}_{2}({\text{ S }}, {\text{ G }}, {\text{ E }})}{{{\varvec{P}}}_{2}({\text{ E }})} = \frac{\pi \, \sigma }{\pi \, \sigma + \rho \, \bar{\sigma }} \end{aligned}$$

(18)

To obtain the difference $\varDelta _{0}$, we calculate

$$\begin{aligned} {{\varvec{P}}}_{2}({\text{ S }},{\text{ G }}) - {{\varvec{P}}}_{1}({\text{ S }},{\text{ G }}) = \sigma - \sigma ^{2} = \sigma \, \bar{\sigma }\,. \end{aligned}$$

This proves the following proposition:

Proposition 3

$\varDelta _{0} = 0$ iff $\sigma = 0$ or 1; $\varDelta _{0} > 0$ iff $\sigma \in (0, 1)$.

The difference, $\varDelta _{1}$, between the conjunction’s pre- and post-‘reductive’ posterior probabilities is obtained as follows:

$$\begin{aligned} \varDelta _{1} := {{\varvec{P}}}_{2}^{\,*} - {{\varvec{P}}}_{1}^{\,*} = \frac{\pi \, \sigma - \pi \, \sigma ^{2}}{\pi \,\sigma + \rho \,\bar{\sigma }} = \frac{\pi \, \sigma \, \bar{\sigma }}{\pi \,\sigma + \rho \, \bar{\sigma }} \end{aligned}$$

(19)

From the difference measure

$$\begin{aligned} d_{2} := {{\varvec{P}}}_{2}({\text{ S }}, {\text{ G }}|{\text{ E }}) - {{\varvec{P}}}_{2}({\text{ S }}, {\text{ G }}) = \frac{\sigma \, \bar{\sigma }\, (\pi - \rho )}{\pi \,\sigma + \rho \,\bar{\sigma }}\,, \end{aligned}$$

(20)

we calculate the difference, $\varDelta _{2}$, between the conjunction’s degree of confirmation before and after the establishment of Montague’s syntax-semantics relation as follows:

$$\begin{aligned} \varDelta _{2} := d_{2} - d_{1} = \frac{\sigma \,\bar{\sigma }\, (\pi - \rho ) - \sigma ^{2}\, \bar{\sigma }\, (\pi - \rho )}{\pi \,\sigma + \rho \,\bar{\sigma }} = \frac{\sigma \, \bar{\sigma }^{2}\, (\pi - \rho )}{\pi \,\sigma + \rho \,\bar{\sigma }} \end{aligned}$$

(21)

This completes our proofs and calculations for Sect. 4.