" Using the traditional syntactic approach to scientific theories, we define a theory to be a deductively closed set of sentences of a formal language. Given a theory T, a theory formulation of T is a (usually finite) subset S of T, often interpreted as the set of basic axioms, such that the deductive closure of S is the whole of T. How are theory and observations related? Typically, observation sentences of the form "The liquid in this vessel is blue!" are not directly entailed by a theory (i.e. contained in the set T), because, first, they lack the generality characteristic of a theory and, second, their truth-value -- unlike the truth-value of a typical theoretical sentence -- is dependent upon the occasion of utterance. Quine portrays the link between a theory and such sentences as a two-step relation. As a first step, observation sentences are pegged to specific spatio-temporal co-ordinates so as to make their truth-value independent of the occasion of utterance. But as theories typically make little reference to particulars, pegged observation sentences are still insufficiently general to be directly entailed by a theory. Rather, theories imply particulars via other particulars, that is to say, via boundary conditions. But, instead of saying that the conjunction of a theory and a set of boundary conditions implies certain pegged observation sentences, we may simply say -- and this is the second step of the two-step relation -- that the theory implies appropriate observation conditionals, where an observation conditional is a conditional sentence whose antecedent is a conjunction of pegged observation sentences and whose consequent is a pegged observation sentence (Quine, 1975) .
In this terminology, two theories are defined to be empirically equivalent if they entail the same body of observation conditionals. Now the empirical underdetermination thesis requires that there exist rival theories which are empirically equivalent, but logically incompatible. However, if the empirical underdetermination thesis is to be an interesting and nontrivial claim, it must actually require more than that. It must, firstly, rule out the possibility that two purportedly irreconcilable rival theories simply turn out to be divergent extensions of a single theory, which itself is not subject to empirical underdetermination. And it must, secondly, rule out the possibility that two such rival theories turn out to be notational variants of each other, where one is, for example, the result of interchanging the terms 'electron' and 'neutron' throughout the other. To accommodate the first point, the underdetermination thesis must require not just that there are some pairs of suitable rival theories, but that, for any theory, a suitable rival exists. To accommodate the second point, we introduce a general method of constructing new notational variants of a theory by defining a reconstrual of predicates to be a function whose domain is the set of predicates of the relevant language and whose converse domain is the set of open sentences of the language such that each n-place predicate is mapped to a sentence with n free variables. Now the empirical underdetermination thesis can be stated thus: given any scientific theory, there exists a rival theory which is empirically equivalent to the given theory and which cannot be rendered logically compatible with it by means of a reconstrual of predicates(1) (Quine, 1975).
At first sight, we may find this claim puzzling. Surely, we may ask, if our 'irreconcilable' rival theories are empirically equivalent, they must have a considerable number of sentences in common, in particular all the observation conditionals which each of the theories implies. So, given a set of rival theories, can we not simply take their intersection as a new theory(2) which is both empirically equivalent to each of the given theories and immune to empirical underdetermination? Indeed, if the set of desired observation conditionals to be entailed by a theory is finite, we can easily define the theory to be the deductive closure of the conjunction of these observation conditionals; such a theory is obviously unique and unaffected by empirical underdetermination. Similarly, if the set of observation conditionals is infinite, but exhibits so much structure that its deductive closure can be expressed as the deductive closure of a finite theory formulation, it is also possible to construct a tightly fitting theory. In its most general form, the empirical underdetermination thesis is therefore wrong.
However, given the complexity of the world, we may expect that, in many cases, (the deductive closure of) the set of all relevant observation conditionals is not axiomatizable in terms of a conceptually neat, let alone finite, theory formulation. Rather, any conceptually manageable (in particular, finite) theory formulation may well entail (the deductive closure of) the desired observation conditionals as well as some other (non-observational) sentences. And this is precisely why we can envisage that the problem of empirical underdetermination may crop up.
(1) To be precise, we shall say that two theories can be rendered logically compatible by means of a reconstrual of predicates if there exists a reconstrual of predicates under which one of the two theories is mapped onto a (not necessarily proper) subset of the other.
(2) Since each of the given theories is deductively closed, so is their intersection. "