Ramsey Sentences and the Meaning of Quantifiers

Abstract

In his famous paper “Theories” Frank Ramsey (1931, pp. 212–236; 1978, ch. 4) introduced a technique of examining a scientific theory by means of certain propositions, dubbed later “Ramsey Sentences.” They are the results of what is often called Ramsey elimination. This prima facie elimination is often presented as a method of dispensing with theoretical concepts in scientific theorizing. The idea is this: Assume that we are given a finitely axiomatized scientific theory $$T\left[ {{O_1},{O_2}...{H_1},{H_2}} \right]$$ (1) where O1, O2, .. are the primitive observation terms (individual constants, predicate constants, function constants, etc.) of (1) and H1, H2, ... its primitive theoretical terms. For simplicity, it will be assumed that (1) is a first-order theory. Since it is finitely axiomatizable, we may think of it as having the form of a single complex proposition, i.e. the conjunction of all the axioms. What can then be done is to generalize existentially with respect to (1). The result is a sentence of the form $$\left( {\exists {X_1}} \right)\left( {\exists {X_2}} \right)...T\left[ {{O_1},{O_2},...,{X_1},{X_2}} \right]$$ (2) In (2), the theoretical terms H1, H2, ... do not occur any longer. They have been replaced by the variables X1, X2, ... bound to initial existential quantifiers. In this sense at least, Ramsey sentences do effect on elimination. Unlike (1), (2) is not a first-order sentence but a second-order one.