Jenny: habitante de paso de Carrera Séptima

As was already mentioned, in his (1934) Carnap outlines a program of what scientific philosophy as logic of science is. According to Carnap, this program is basically interested in a critical appraisal of calculi. Note that the calculi Carnap talks of are not to be confused with what we understand under calculi today, namely proof calculi (Beweiskalkül). The latter kind

of calculi only involve inferential rules. Carnap’s calculi, however, do also involve formation rules, that is rules which tell us how to built up complex expressions like sentences from simple expressions. These rules are hence what we would call “syntactical rules” in a proper sense today.

A system of a language contains the formation and inference rules of a language.

Speaking in the formal mode is only about the syntactical structure and logical form of expressions, but not about the meaning of sentences.

So everything which is internal to such a calculus is in the formal mode. Everything which is external to such a calculus is external to a framework.

As we saw, calculi are nothing over and above ordered quadruples like <A, F, T, Φ>, where A is the logical and non-logical vocabulary of L, F a set of syntactical rules (for terms and formulas) of L, Φ a set of formulas of L and T a set of inference or transformation rules of L. The syntactical rules govern how complex expressions (terms or formulas) from L are built up from the logical and non-logical expressions in A. Φ is the set of logical axioms and possibly also meaning postulates.30 The inference rules in T tell us something about the inferential relationships of the formulas that are built up according to F. The ordered pair <A, F> can be considered a language here.

Carnap gives the following example of such a framework.

“As an example of a system which is of a logical rather than a factual nature let us take the system of natural numbers. The framework for this system is constructed by introducing into the language new expressions with suitable rules: (1) numerals like “five” and sentence forms like “there are five books on the table”; (2) the general term “number” for the new entities, and sentence forms like “five is a number”; (3) expressions for properties of numbers (e.g., “odd”, “prime”), relations (e.g., “greater than”), and functions (e.g. “plus”), and sentence forms like “two plus three is five”; (4) numerical variables (“m), “n”, etc.) and quantifiers for universal sentences (“for every n, …”) and existential sentences (“there is an n such that …”) with the customary deductive rules” (1950: 208).

So frameworks are very near to calculi. But now we have only the special case, where frameworks are “logical” but not “factual.” That is, we only have frameworks which are analytic. Mind you that Carnap belongs to a tradition that classifies sentences according to their alethic status. The period in which this tradition had prevailed stretches from Kant’s critiques of the 1780’s to the twentieth-century analytic movement which ended broadly speaking in the 1950’s as a result of Quine’s work.31 Sentences are either analytically or synthetically true.

30 _{cf. Carnap (1952)}

31 _{Of course, it was only the classification which was adopted by the logical positivists. But this does not mean} that they agreed with everything that Kant did with it. In fact, the logical empiricists denied two of Kant’s influential thesis. First, they denied that the possibility of knowledge requires that there be synthetic a priori

The following diagram which Carnap regularly incorporates in the multitude of writings over his whole career to illustrate the distinction between the alethic status of sentences is certainly familiar to the reader.

analytic synthetic contradictory

P-valid P-contravalid

valid indeterminate contravalid

If a sentence S can be deduced from Φ with the rules from T alone, it is an L-consequence of

Φ. If S can be shown to be true on the basis of the rules in T alone, it is L-valid. If S can be shown to be false on the same basis it is L-contravalid or contradictory. If S is neither valid nor contravalid it is indeterminate. If S is neither analytic nor contradictory, that is if its truth cannot be determined by logic alone S, is called synthetic. So, for instance:

„‘There is an n, such that n is a number’. This statement follows from the analytic statement ‘five is a number’…” (Carnap 1950: 209).

But we also want frameworks which are not analytic, but synthetic like the framework of material things. How do we introduce such frameworks? Carnap writes the following:

“The thing language contains words only like “red”, “hard”, “stone”, “house”, etc., which are used for describing what things are like. Now we may introduce new variables, say “f”, “g”, etc., for which those words are substitutable and furthermore the general term “property”. New rules are laid down which admit sentences like “Red is a property”, “Red is a

color”, “These two pieces of paper have at least one color in common” (i.e., “There is an f such that f is a color and …”)” (1950: 211f.).

In his (1936) Carnap departs from the more radical principles of verifiability that he defended in his earlier writings. Roughly, the more radical principle holds that a statement must be (at least in principle) verifiable to be meaningful. Carnap backs out of this principle for the reasons that universal sentences cannot be meaningful and that science contains a lot of sentence which are not verifiable (Compare this with what Hahn says in the earlier quotes). However, he doesn’t entirely depart from the principle of verifiability of meaning. He only weakens it. According to Carnap (1936), an existential sentence must be testable to be meaningful. Carnap also alludes to this principle in his (1950) when he writes:

“To accept the thing world means nothing more than to accept a certain form of language, in other words, to accept rules for forming statements and for testing, accepting, or rejecting them” (1950: 208).

We can incorporate this demand by including a set C of confirmation rules into the ordered tuples, representing linguistic frameworks. Where C is a confirmation rule, “C(A,B)” says that evidence A confirms B. Note that the confirmation rules also hold for analytic sentences. But analytic sentences are trivially confirmed since they do not contain any factual content.

So we have the following suggestion:

(1) <A, F, T, Φ, C>

But this still won’t be sufficient. It won’t be sufficient since there are cases in which the languages are minimally different. Consider the following example which draws on a point made by Niebergall (2000: 29) in the context of reducibility of theories.

ZF is normally formulated in the first-order language L[ZF] with ‘∈’ as its sole non-logical predicate. Let the language L[ZF] be the ordered triple <A, F> with the conventions from above. Now the set A consists of only logical vocabulary - say a set of first-order variables x, y,... and ‘⊥’, ‘→’, ‘∀’ and ‘(‘ and ‘)’ - and the sole non-logical predicate “∈”. A ZF

framework, FR[ZF], is then an ordered pair <A, T, F, Φ, C> where Φ is a set of the axioms of ZF.32 Assume that L[ZFη] results from L[ZF] by interchanging the expression “∈” through “η”. Thus L[ZFη] differs from L[ZF] only in its non-logical vocabulary. Let A’ be the set of logical and non-logical vocabulary of L[ZF] and L[ZF] be the ordered 4-tuple <A’, F, T, C>. So we have <A’, T>≠<A, T>. Let FR[ZFη] be the framework resulting from this substitution. Then it trivially holds that FR[ZF] ≠ FR[ZFη]. But this is clearly counterintuitive. FR[ZF] and FR[ZFη] should be identical.

This intuitive difference is supported by Carnap’s defence of the method of Ramsification. Carnap (1966) once tried to give an explication of the analytic-synthetic distinction by means of Ramsey sentences.

But let me first sketch Ramsey’s account. In his (1929) Ramsey suggested the sentences that bear his name as a solution to the problem of how to treat theoretical terms. Theoretical terms pose a number of difficulties. For instance, it is unclear how they get their meaning, and how we can find out the empirical meaning of theoretical terms. His ingenious solution consisted in an elimination of the theoretical terms.

Let T be a theory and let the postulates of the theory be T[F1,…,Fn], where F1,…,Fn are the

theoretical terms of T. If the theoretical terms are replaced by variables X1,…,Xn we get a

formula which Lewis (1970b: 81) calls the realization formula of T. Every n-tuple of entities

that satisfies this formula is a realization of T. If T is realized, we get the Ramsey-sentence of T which is as follows: ∃X1,…,Xn[X1,…,Xn]. Ramsey finally suggested that we can substitute

this sentence for the original theory without any loss of content. Carnap exploited Ramsey’s method for an explication of the notion of analyticity in a theoretical language by merging some analytical sentence A which is supposed to be an analytical postulate for the theoretical terms of a theory with a sentence S which is supposed to contain the whole observational content of the theory. According to Carnap S is the Ramsey sentence of the theory. A and S together imply T. The easiest sentence which together with S implies T is the conditional sentence S→T. Or if we adopt the way of putting things from above, we have:

32 _{Don’t forget the inference rules. They are important, since we can derive different sentences from the axioms} if we use the inference rules from minimal or intuitionistic logic (cf. Aczel and Rathjen (2000/2001)). After all,

∃X1,…,Xn[X1,…,Xn] → T[F1,…,Fn]

This sentence which is called the Ramsey-Carnap sentence is then our sentence A. According

to Carnap this sentence does not contain any factual content. It just says that if there is a n- tuple of entities that satisfies the formula in the antecendens, then the theory is true. By means of this sentence, analytic and synthetic sentences in the theoretical language can be distinguished from each other pretty easily. Analytic sentences are simply the sentences which are logically implied by the Ramsey-Carnap sentence.

If we treat the elementhood relation as a theoretical term and formulate the Ramsey sentence of ZF, we get the sentence “∃X.ZF[X]”. If we formulate the Ramsey sentence of ZFη under the same assumption, we get the same sentence. The Ramsey sentence of ZF is the same as the Ramsey sentence of ZFη. If Φ in the ordered tuple

(2’) <A, T, F, Φ>

making up the framework FR[ZF] is the Ramsey sentence of the axioms of ZF, then FR[ZF] = FR[ZFη].

So (2’) is not yet sufficient. But it is not difficult to improve (2’). To avoid the problems we need to take the equivalence relation of the vocabulary [A].