The Criteria for the
Empirical Significance of Terms
—Summary—
Sebastian Lutz∗
2012–04–01
A criterion of empirical significance purportedly distinguishes between those
sentences or terms (i. e., non-logical symbols) that are connected to observations
and those that are not. The need for such a criterion has recently been shown by
discussions about the empirical significance of the theory of Intelligent Design
and string theory.
After proving, almost in passing, that the criterion for the empirical significance of sentences by Ayer (1946, 13) is trivial, and thereby initiating the long
descent of the concept of empirical significance into infamy, Church (1949, 53)
concludes that “any satisfactory solution of the difficulty will demand systematic use of the logistic method”. In this spirit, I will discuss the logical relations
between criteria for the empirical significance of terms as given within syntactic
approaches to the philosophy of science (e. g. by Carnap and Rozeboom) as well
as semantic approaches (e. g. by Luce and Wójcicki). It will turn out that the
criteria cannot serve in their intended role as a basis for criteria for the empirical
significance of sentences. Instead, the generalizations of two of the criteria to
criteria for sets of terms and a third, new criterion for sets allow for the distinction
between mathematical concepts, concepts that are mathematical but may become
surrepitiously non-mathematical, and completely useless concepts.
Carnap (1967, §38) suggests that a theoretical term Ti ∈ T of some theory
Σ in vocabulary V is empirically significant iff Σ entails an explicit definition of
Ti in observational terms O = V − T . Without much justification, Luce (1978,
§4) suggests to consider a relation symbol Ti in a first order theory Σ empirically
significant iff for each A  Σ, TiA is invariant under the automorphisms of A|O ,
where A|O is the reduct of A to O . It is a corollary of Svenonius’s theorem that
a relation is so symmetric iff Ti is piecewise definable, that is, Σ entails a finite
disjunction of explicit definitions of the term (Hodges 1993, Corollary 10.5.3). It
is known from the theory of definition that a sentence containing only explicitly
definable terms can be translated into an O -sentence, and I show that a sentence
∗
Theoretical Philosophy Unit, Utrecht University, The Netherlands. [email protected].
Sebastian Lutz
The Criteria for the Empirical Significance of Terms—Summary
containing only tems that are empirically significant according to Luce’s criterion
is empirically meaningful according to a criterion suggested by Suppes (1959)
in measure theory. Since some sentences that contain terms that are neither
definable nor empirically significant according to Luce are still translatable into
O -sentences, however, these criteria for terms can only provide sufficient criteria
for the empirical significance of sentences.
Based on arguments that can be extended to piecewise definability, Carnap
(1936) argues that explicit definability is too strict a criterion and suggests that Ti be
considered empirically significant iff it is reducible, that is, Σ entails ∀x̄[ω(x̄) →
Ti x̄] or ∀x̄[ω(x̄) → ¬Ti x̄] and not ¬∃x̄ω(x̄) for some O -formula ω and x̄ =
x1 , . . . , xki . For finite Σ = Σ(Ō, T̄ ) with the O -terms Ō = O1 , . . . , O m and the
T -terms T̄ = T1 , . . . , Ti , . . . , Tn , Martin (1966)
suggests to define Ti ’s RamseyΣ
constant as ∀x̄[RO ,i x̄ ↔ ∃X̄ Σ(Ō, X̄ )∧Xi x̄ ]. With this and the reverse Ramsey
constant ∀x̄[R Σ
x̄ ↔ ∃X̄ Σ(Ō, X̄ ) ∧ ¬Xi x̄ ], I show that for finite Σ and O O ,i
x̄ or 6 ∀x̄ RΣ
sentences of any order, Ti is reducible iff 6 ∀x̄R Σ
x̄. Wójcicki
O ,i
O ,i
(1966)Tdefines Ti to be empirically significant
T iff there is an O -structure AO such
that {TiB | B|O = AO and B  Σ} or {ûTiB | B|O = AO and B  Σ} are
not invariant under all permutations on |AO |. I discuss this criterion’s relation
to reducibility and note that it can be easily expressed with (reverse) Ramseyconstants. None of these criteria, however, can provide a plausible necessary
or sufficient criterion for the empirical significance of sentences. For it is easily
shown that some sentences that contain only reducible terms do not even fulfill
the weakest known criterion for the empirical significance of sentences, while
there are also sentences that contain only terms that are not reducible, but which
can nonetheless be translated into O -sentences.
Using the Ramsey-sentence RO (Σ)  ∃X̄ Σ(Ō, X̄ ), Rozeboom (1962, postulate 7) calls Ti empirically significant iff 6 RO (Σ) → RO ∪{Ti } (Σ). Wójcicki (1966)
calls Ti O -dependent iff there are models A and B of Σ with |A| = |B| such that
{TiC | C|O = A|O and C  Σ} =
6 {TiD | D|O = B|O and D  Σ}. I show that for
finite Σ, this criterion can be expressed as 6 RO (Σ) ∧ R{Ti } (Σ) → RO ∪{Ti } and is
thus more exclusive than Rozeboom’s. Carnap (1956) suggests a criterion that,
roughly, demands that Ti feature essentially in the derivation of an observation
sentence. It has largely escaped notice that for his proof that the criterion is
not too exclusive, Carnap assumes that Ti can be primitively interpreted. Using
O -structures instead of observation sentences, I show that Carnap’s criterion is
then equivalent to O -dependence. I further show that a sentence containing only
terms that are not empirically significant according to Rozeboom does not fulfill
the weakest known criterion for the empirical significance of terms.
This overview shows the very questionable status of the criteria of empirical
significance for terms, since none of them seems to provide a necessary and
sufficient criterion for the significance of sentences. What is more, all of the
criteria are defined relative to a theory, and thus, in effect, relative to a set of
2
Sebastian Lutz
The Criteria for the Empirical Significance of Terms—Summary
sentences accepted in the context of the application of the criterion. But the
descent of the criteria for sentences into infamy is intimately connected with
the search for a criterion for sentences that is not relative to a set of accepted
sentences. For Ayer’s criterion and all its subsequent amendments define not only
the empirical significance of a sentence relative to a set Π of sentences, but also
define which sets Π of sentences are acceptable. Thus the criteria for the empirical
significance of terms beg the question that the criteria for sentences are meant to
answer. What is more, Carnap’s criterion (Carnap 1956) even relies itself on a
criterion for the empirical significance of sentences, namely falsifiability relative
to a set of sentences, and thus cannot consistently provide a different one.
To generalize O -dependence, define
given
the set of O -possible I -structures
O -structure AO and Θ as I AO ,Θ = C|I | C|O = AO and C  Θ . Then define
the set I ⊆ T of terms to be O -dependent with respect to O and Θ if and only if
there are models A, B  Θ with |A| = |B| such that B|I 6∈ I A|O ,Θ (for finite sets
of sentences, this definition can be expressed neatly with the help of Ramsey sentences). The use of reducts of structures rather than restrictions of interpretations
is an avoidable convenience. I then use an argument by Demopoulos (2003, 387)
to show that when a set of terms is not O -dependent, the terms are not restriced
in their interpretation and thus are mathematical (incidentally, Demopoulos’s
argument is invalid in the context in which he applies it). I then show that there
is a set Θ of sentences and a set I of terms that is not O -dependent in Θ, but
O -dependent in Θ ∪ ∆, where ∆ is a set of V − I -sentences. This means that
O -independent sets of terms are mathematical but can become non-mathematical
without being explicitly restricted in their interpretation.
I show that a definition by Wójcicki (1966) is equivalent to the claim that
a set I of T -terms is O -isolated in Θ if and only if if and only if for any two
models A, B  Θ with |A| = |B| it holds that B|I ∈ I A|V −I ,Θ (for finite sets
of sentences, this definition can be expressed neatly with the help of Ramsey
sentences). I show that if I is O -isolated in Θ, then I is O -isolated in Θ ∪ ∆
for any set ∆ of V − I -sentences. That is, unlike O -independent sets of terms,
O -isolated sets of terms are mathematical and can only become non-mathematical
by being explicitly restricted in their interpretation.
Finally, one can define a set I of terms to occur O -vacuously in Θ if and only
if for each Ai ∈ I and A  Θ, {Ai } A|V −I ,Θ = {Ai } A|V −I ,> . O -vacuous terms,
however, are indeed useless and can be introduced into any theory by equivalent
reformulation.
In conclusion, the criteria for the empirical significance of terms are of no use
as basis for criteria for the empirical significance of sentences. However, generalizations of the weakest criteria for terms can be used to identify within theories sets
of mathematical terms, sets of mathematical terms that can become surreptitiously
non-mathematical, and sets of terms that are not even mathematical.
3
Sebastian Lutz
The Criteria for the Empirical Significance of Terms—Summary
References
Ayer, A. J. (1946). Language, Truth and Logic. Victor Gollanz, London, 2nd
edition.
Carnap, R. (1936). Testability and meaning. Philosophy of Science, 3(4):420–468.
Carnap, R. (1956). The methodological character of theoretical concepts. In
Feigl, H. and Scriven, M., editors, The Foundations of Science and the Concepts of
Psychology and Psychoanalysis, volume 1 of Minnesota Studies in the Philosophy of
Science. University of Minnesota Press, Minneapolis, MN.
Carnap, R. (1967). The Logical Structure of the World. Pseudoproblems of Philosophy.
University of California Press, Berkeley and Los Angeles. Translation by Rolf
A. George.
Church, A. (1949). Review of Ayer’s Language, Truth, and Logic, 2nd edition. The
Journal of Symbolic Logic, 14(1):52–53.
Demopoulos, W. (2003). On the rational reconstruction of our theoretical knowledge. The British Journal for the Philosophy of Science, 54:371–403.
Hodges, W. (1993). Model Theory, volume 42 of Encyclopedia of Mathematics and
its Applications. Cambridge University Press, Cambridge. Digitally printed in
2008.
Luce, R. D. (1978). Dimensionally invariant numerical laws correspond to meaningful qualitative relations. Philosophy of Science, 45(1):1–16.
Martin, R. M. (1966). On theoretical constructs and Ramsey constants. Philosophy
of Science, 33(1/2):1–13.
Rozeboom, W. W. (1962). The factual content of theoretical concepts. In Feigl,
H., Scriven, M., and Maxwell, G., editors, Scientific Explanation, Space, and
Time, volume 3 of Minnesota Studies in the Philosophy of Science, pages 273–357.
University of Minnesota Press, Minneapolis, MN.
Suppes, P. (1959). Measurement, empirical meaningfulness, and three-valued logic.
In Churchman, C. W. and Ratoosh, P., editors, Measurement: Definitions and
Theories, pages 129–143. Wiley, New York.
Wójcicki, R. (1966). Semantical criteria of empirical significance. Studia Logica,
19:75–102.
4