Some Thoughts on Carnap
By tdbergThe language of Physics, a language that both reports and predicts, is an inherently empirical language that must also make an appeal to the unobservable . As such, within the realm of Physics, distinctions need be clarified between two types of terms; those that are observable, and those that are theoretical, and, consequently, between two types of statements; those that are analytic and those that are synthetic. Observable terms and analytical statements are those that are confirmed empirically; they are a posteriori. Theoretical terms and synthetic statements, on the other hand, are those that are nonobservable, and are therefore known a priori. In light of this, Rudolf Carnap sought to define these distinctions through the development of a rational reconstruction. His framework would systemize a body of truths, by bridging the gap between the theoretical and the observable, in a nonintuitive manner.
Since theoretical laws are laws that deal exclusively with unobservable entities such as electrons, molecules, spacetime coordinates, and atoms, they are more general than empirical ones, and therefore deal with concepts that are more abstract. As such, the method of justifying a theoretical law must be indirect. A physicist does not test the theoretical law itself, but rather the empirical laws that are amongst its consequences. In other words, “theoretical law helps to explain empirical laws already formulated, and to permit the derivation of new empirical laws.” (Carnap, Philosophical Foundations of Physics) An example of this is Einstein’s Theory of Relativity. Einstein’s theory was not only concise and elegant; it additionally led to the discovery of new empirical laws such as the “bending of light rays in the neighborhood of the sun.” (Carnap, Philosophical Foundations of Physics) It was a theory of enormous analytical power that, according to Carnap, significantly influenced the role that mathematics might play in the logic of physical theory. In other words, through the separation of “the logicalformal from its objective or intuitive content” (Einstein, Geometry and Experience), a certain amount of logical security could be attained in the field of physics.
Although the necessity of this connection, between the empirical and the theoretical, can be readily discerned, difficulty can arise when one tries to formulate it. If the language of theoretical law deals exclusively with the unobservable, how can one deduce from such laws, laws of empiricism? Or, more importantly, if our theoretical sentences contain only theoretical terms, how do we derive empirical statements from them that contain only observable terms? Carnap’s answer was marked by a set of Correspondence Rules that linked the theoretical terms to the observable terms, providing the means for an indirect proof.
In order to explicate these Correspondence Rules further, one must first take into account how Carnap defines what is observable, and what is theoretical, and how those definitions relate to empirical laws. Carnap explains the difference between analytic and synthetic statements in the following way: If s is true in virtue of meaning, it is analytic; if s is not analytic, it is synthetic. For example, consider the WFF —∃
x(Yx ^ Bx), where Y represents yellow all over, and B represents black all over. Clearly, this WFF is not provable logically through a formal method, since one must make an appeal to meaning in order to decide. On the contrary, consider the WFF —∃
x(Yx ^ Yx), where Y represents yellow all over. Clearly, no appeal to meaning is necessary here, since nothing can be both Y and not Y. This differentiation is important since Carnap strongly believed that any given reconstruction would have to separate what is factual or observational, from what is conventional or theoretical. As such, the first phase of Carnap’s reconstruction was based on this idea of separation, which he achieved by identifying the vocabulary of the theoretical as T terms (T1…Tn), and the vocabulary of the observational as O terms (O1….On). Carnap further considered his process to be axiomization plus; the axioms would initiate a comparison to other truths, thereby revealing their epistemological basis. This ideology can be convincingly compared to Henri Poincare’s division of the physical from the geometrical, since for Poincare, the physical is factual, and the geometrical is conventional.
The next phase of Carnap’s reconstruction allowed for sentences to be constructed from these now distinguished terms, by isolating the things that had synthetic (O terms) properties, from the things that had analytic (T terms) properties. The conversion of the semantical into the syntactical, created a delineation that was less than arbitrary, allowing the T terms to be transformed into O terms. This transformation was facilitated through the formulation of a Ramsey Sentence, whose creation was conducted in the following way: if T represents the conjuncture of all theoretical terms, and C the conjuncture of all the correspondence rules, then TC represents the theory of both: TC (O1….On, T1…Tn). As such, if we replace every theoretical term in TC with a predicate variable, and add the corresponding amount of existential quantifiers, we get:
∃
x1…∃
xn TC((O1…On, X1…Xn)
The resulting substance of the Ramsey sentence captures the factual content of a theory, which thereby satisfies the original theory. Or, in other words, (TC) and R(TC) are O equivalent. As such, TC logically implies O iff R(TC) logically implies O. This further allows for the formation of the Carnap Sentence:
if R(TC) then TC, or,∃
x1…∃
xn TC((O1…On, X1…Xn) > TC (O1….On, T1…Tn)).
The Carnap Sentence implies only tautologies, and its factual component is the observational equivalent to the original theory. This is can be proven via modus pones:
C(TC) > (TC)

C(TC)
(TC)
This final phase of Carnap’s reconstruction dictated that there are no terms that are both observational and theoretical, and that, if a statement is derivable from the Carnap Sentence, it is analytic. Accordingly, the Carnap sentence was also considered to be O uninformative. The property of being O uninformative insured that C(TC) derived only logical truths, and that the analytic sentences of (TC) coincided with the consequence class of C(TC), or, in other words, X is analytic in (TC) if X is a logical consequence of C(TC). Moreover, this consequence class of C(TC) became defined as the largest subclass of O uninformative sentences, which were also, according to John Winnie, O noncreative. An arbitrary sentence X is O noncreative in (TC) iff :
(TC) logically implies X
For any set Y, (TC) logically implies Y
Any O consequence of X^Y is an O consequence of Y
This fact, of being O uninformative and O noncreative, made Carnap Sentences the best model for separating the analytic from the synthetic.
We can see this method applied to real world physics by looking at the example of solubility. Since Solubility is a dispositional property, it must be described via its manifest properties. For example, if x is placed in water, and x dissolves, then x is soluble: ∃
x (Ax > (Bx > Tx)). In this example, A and B are manifest properties, and T is a dispositional (or theoretical) property. As such, we must separate the sentences that have the presence of T, herein represented as φ
x, from the sentences that have the absence of T, herein represented by Σ
x. Accordingly, we can derive the following:
“∃
x(φ
x > Tx ) ^ (Σ
x > Tx).
Through further logical derivation we can formulate the observational content to be:
∀
x(φ
x > –Σ
x)
Next, we replace the T terms with a predicate variable of the corresponding arity, which renders the following:
∀
(φ
x > Yx ) ^ (Σ
x > Yx)
The addition of an existential quantifier yields the following Ramsey Sentence:
∃
Yx”x(φ
x > Yx ) ^ (Σ
x > Yx)
This is logically equivalent to the observational content:
∀
x(φ
x > –Σ
x)
Carnap strategically extended Hilbert’s concept, by paying homage to the idea that a deduction does not depend on what things mean, but rather on what their logical form is. Although Carnap found Hilbert’s axiom system, with its uninterpreted terms, to be useful, he also realized that when “geometry is applied to physics, these terms must be connected to something in the physical world.”(Carnap, Philosophical Foundations of Physics) However, in order to attach these uninterpreted terms to things in the observable world, one must first establish rules for making the connection. Carnap did this through the creation of his Correspondence Rules. These rules logically bridged the gap between the analytic and synthetic, and eliminated Hilbert’s previous appeal to intuition. Subsequently, a physicist could now say that geometrical lines are defined by rays of light, or that “the temperature of a gas is proportional to the mean kinetic energy of it’s molecules.” (Carnap, Philosophical Foundations of Physics) As long as a physicist didn’t develop Correspondence Rules that were incompatible with each other, or with their theoretical laws, the process could be never ending. This allowed for the continued increase in the amount of interpretation specified by the theoretical terms, as scientists developed new procedures. Through the presence of these Correspondence Rules, and their subsequent connection to the observational, a theory could eventually be defined as an empirical law.
December 1st, 2015 at 11:24 am
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