Projective unitary group

In philosophy, Ramsey sentences refer to an attempt by logical positivist philosopher Rudolf Carnap to reconstruct theoretical propositions such that they gained empirical content.

For Carnap, questions such as: “Are electrons real?” and: “Can you prove electrons are real?” were not legitimate questions implying great philosophical/metaphysical import. They were meaningless "pseudo-questions without cognitive content,” asked from outside a language framework of science. Inside this framework, entities such as electrons or sound waves, and relations such as mass and force not only exist and have meaning, but are "useful" to the scientists who work with them. To accommodate such internal questions in a way that would justify their theoretical content empirically – and to do so while maintaining a distinction between analytic and synthetic propositions – Carnap set out to develop a systematized way to consolidate theory and empirical observation in a meaningful language formula.

Carnap began by differentiating observable things from non-observable things. Immediately, a problem arises: neither the German nor the English language naturally distinguish predicate terms on the basis of an observational categorization. As Carnap admitted, "The line separating observable from non-observable is highly arbitrary." For example, the predicate "hot" can be perceived by touching a hand to a lighted coal. But "hot" might take place at such a microlevel (e.g., the theoretical "heat" generated by the production of proteins in a eukaryotic cell) that it is virtually non-observable (at present). Physicist-philosopher Moritz Schlick characterized the difference linguistically, as the difference between the German verbs "kennen" (knowing as being acquainted with a thing – perception) and "erkennen" (knowing as understanding a thing – even if non-observable). This linguistic distinction may explain Carnap’s decision to divide the vocabulary into two artificial categories: a vocabulary of non-observable ("theoretical") terms (hereafter "V_T"): i.e., terms we know of but are not acquainted with (erkennen), and a vocabulary of observable terms ("V_O"), those terms we are acquainted with (kennen) and will accept arbitrarily. Accordingly, the terms thus distinguished were incorporated into comparable sentence structures: T-terms into Theoretical sentences (T-sentences); O-terms into Observational sentences (O-sentences).

The next step for Carnap was to connect these separate concepts by what he calls "Correspondence Rules" (C-rules), which are "mixed" sentences containing both T- and O-terms. Such a theory can be formulated as: T + C = df: the conjunction of T-postulates + the conjunction of C-rules – i.e., $[(T_{1}\land T_{2}\land \dots \land T_{n})+(C_{1}\land C_{2}\land \dots \land C_{m})]$. This can be further expanded to include class terms such as for the class of all molecules, relations such as "betweenness," and predicates: e.g., TC ( t₁, t₂, . . ., t_n, o₁, o₂, . . ., o_m). Though this enabled Carnap to establish what it means for a theory to be "empirical," this sentence neither defines the T-terms explicitly nor draws any distinction between its analytic and its synthetic content, therefore it was not yet sufficient for Carnap’s purposes.

In the theories of Frank P. Ramsey, Carnap found the method he needed to take the next step, which was to substitute variables for each T-term, then to quantify existentially all T-terms in both T-sentences and C-rules. The resulting "Ramsey sentence" effectively eliminated the T-terms as such, while still providing an account of the theory’s empirical content. The evolution of the formula proceeds thus:

Step 1 (empirical theory, assumed true): TC ( t₁ . . . t_n, o₁ . . . o_m)

Step 2 (substitution of variables for T-terms): TC (x₁ . . . x_n, o₁ . . . o_m)

Step 3 ($\exists$-quantification of the variables): $\exists x_{1}...\exists x_{n}TC(x_{1}...x_{n},o_{1}...o_{m})$.

Step 3 is the complete Ramsey sentence, expressed "^RTC," and to be read: "There are some (unspecified) relations such that TC (x₁ . . . x_n, o₁ . . . o_m) is satisfied when the variables are assigned these relations. (This is equivalent to an interpretation as an appropriate model: there are relations r₁ . . . r_n such that TC (x₁ . . . x_n, o₁ . . . o_m) is satisfied when x_i is assigned the value r_i, and $1\leq i\leq m$.)

In this form, the Ramsey sentence captures the factual content of the theory. Though Ramsey believed this formulation was adequate to the needs of science, Carnap disagreed, with regard to a comprehensive reconstruction. In order to delineate a distinction between analytic and synthetic content, Carnap thought the reconstructed sentence would have to satisfy three desired requirements:

1. The factual (F_T) component must be observationally equivalent to the original theory (TC).
2. The analytic (A_T) component must be observationally uninformative.
3. The combination of F_T and A_T must be logically equivalent to the original theory – that is, $F_{T}+A_{T}\Leftrightarrow TC$.

Requirement 1 is satisfied by ^RTC in that the existential quantification of the T-terms does not change the logical truth (L-truth) of either statement, and the reconstruction FT has the same O-sentences as the theory itself, hence ^RTC is observationally equivalent to TC : (i.e., for every O-sentence: O, $[TC\models O\Leftrightarrow ^{R}TC\models O]$ ). As stated, however, requirements 2 and 3 remain unsatisfied. That is, taken individually, A_T does contain observational information (such-and-such a theoretical term is observed to do such-and-such, or hold such-and-such a relation); and A_T does not necessarily follow from F_T. Carnap’s solution is to make the two statements conditional. If there are some relations such that [TC (x1 . . . xn, o1 . . . om)] is satisfied when the variables are assigned some relations, then the relations assigned to those variables by the original theory will satisfy [TC (t1 . . . tn, o1 . . . om)] – or: ^RTC → TC. This important move satisfies both remaining requirements and effectively creates a distinction between the total formula’s analytic and synthetic components. Specifically, for requirement 2: The conditional sentence does not make any information claim about the O-sentences in TC, it states only that "if" the variables in are satisfied by the relations, "then" the O-sentences will be true. This means that every O-sentence in TC that is logically implied by the sentence ^RTC → TC is L-true (i.e., every O-sentence in AT is true or not-true: the metal expands or it does not; the chemical turns blue or it does not, etc.). Thus TC can be taken as the non-informative (i.e., non-factual) component of the statement, or A_T. Requirement 3 is satisfied by inference: given A_T, infer F_T → A_T. This makes A_T + F_T nothing more than a reformulation of the original theory, hence A_T Ù F_T ó TC.

Finally, the all-important requirement for an analytic–synthetic distinction is clearly met by using two distinct processes in the formulation: drawing an empirical connection between the statement’s factual content and the original theory (observational equivalence), and by requiring the analytic content to be observationally non-informative. Of course, Carnap’s reconstruction as it is given here is not intended to be a literal method for formulating scientific propositions. To capture what Pierre Duhem would call the entire "holistic" universe relating to any specified theory would require long and complicated renderings of ^RTC → TC. Instead, it is to be taken as demonstrating logically that there is a way that science could formulate empirical, observational explications of theoretical concepts – and in that context the Ramsey and Carnap construct can be said to provide a formal justificatory distinction between scientific observation and metaphysical inquiry. The Ramsey and Carnap formulation is, of course, not inviolate. Among its critics are John Winnie, who extended the requirements to include an "observationally non-creative" restriction on Carnap’s A_T – and both Quine and Hempel attacked Carnap’s initial assumptions by emphasizing the ambiguity that persists between observable and non-observable terms. Nonetheless, the Ramsey and Carnap construct was an interesting attempt to draw a substantive line between science and metaphysics.

Works cited

• Carnap, R. "Theoretical Concepts in Science," with introduction by Psillos, S. Studies in History and Philosophy of Science Part A
• Carnap, R. (1966) An Introduction to the Philosophy of Science (esp. Parts III, and V), ed. Martin Gardner. Dover Publications, New York. 1995.
• Carnap, R. (1950) "Empiricism, Semantics, and Ontology," in Moser & Nat, Human Knowledge Oxford Univ. Press. (2003).
• Demopoulos, W. "Carnap on the Reconstruction of Scientific Theories," The Cambridge Companion to Carnap, eds. R. Creath and M. Friedman.
• Moser, P.K. and vander Nat, A. (2003) Human Knowledge Oxford Univ. Press.
• Schlick, Moritz (1918) General Theory of Knowledge (Allegemeine Erkenntnislehre). Trans. Albert Blumberg. Open Court Publishing, Chicago/La Salle, IL. (2002).
• Hallvard Lillehammer, D. H. Mellor (2005), Ramsey's legacy, Oxford University Press, pg 109.
• Stathis Psillos, "CARNAP, THE RAMSEY-SENTENCE AND REALISTIC EMPIRICISM", 2000

External links

• Epistemic Structural Realism and Ramsey Sentences