For Carnap, a linguistic framework of science is comprised of two kinds of rules: By defining the rules of the framework, Carnap is able to give a syntactic formulation of logical consequence, where L-rules are logically determined and are independent of experience and defined as analytic a priori. Statements that are not logically determined are defined as synthetic statements.
In Logical Syntax, Carnap posits the idea of linguistic frameworks to allow for changes in the appraisal or value of a scientific theory. He employs this idea of frameworks to explain changes in mathematical and logic laws in the foundation of science, as this has occurred with the replacement of Newtonian to Einsteinian worldview.
Carnap argues for a Principle of Tolerance, which declares that there are no absolute rules of logic, meaning changes in L- rules will yield different frameworks with different appraisal methodologies and their choice is conventional.
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For Carnap, both L- and P- rules are subject to change within a framework to accommodate new scientific evidence. Scientific statements must comply with the Verification Principle, that is, they must be verifiable through experience.
Different segments in the Aufbau can be understood either as strict or principally verifiable. Challenges with strict verification caused Carnap to drop this view later, as uncorroborated theories were considered unscientific.
It creates a barrier that differentiates scientific physical statements from speculative metaphysics statements. Verification is regarded as meaning criteria, where verifiable scientific propositions are considered meaningful, while metaphysical propositions are meaningless since they are not funded in experience.
Throughout his career, Carnap has attempted in different formulations to employ the tools of probability and induction to the question of theory choice. Quine claims that as it follows from the first dogma against analyticity , even most simple sensory statements are interconnected, and cannot be claimed to be verifying or refuting in isolation 4 5. Popper stated that metaphysics should not be meaningless only by virtue of not being scientific. Here are the works of Carnap included in the bibliographic records of this encyclopedia:. Citation keys normally include author names followed by the publication year in brackets.
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Paul Patton 2. From Encyclopedia of Scientonomy. Jump to: navigation , search. To add a bibliographic record by this author, enter the citation key below:. Encyclopedia Britannica. In Zalta Ed. In Schilpp Ed. In Friedman and Creath Eds. Philosophy of Science 69 2 , In Fieser and Dowden Ed.
The Logical Syntax of Language : Rudolf Carnap :
Introduction 19 The vocabulary of languages for propositional logic contains i. It usually also includes a symbol for equality. Whereas a formal language L is a set of formulas, a formal system is a formal language equipped with a method of formal derivation. An L-structure or L-interpretation I is based on a non-empty set D — the domain, or universe of discourse — on which the non-logical symbols are interpreted by a function which associates: i.
The interpretation of the logical symbols is standard and common to all the L-structures. A theory T couched in language L is a set of sentences of L. Typically, one may want to formalize in L some version of arithmetic, of geometry, or of any mathematical or non-mathematical theory.
The Logical Syntax Of Language
Any remarks which are made on this subject fall into two categories. Some philosophers — most notably Quine — have used the conjunction of these two remarks to argue that second-order logic is no logic at all, but mathematics in the guise of logic. The informal comments by means of which Carnap makes his motivation explicit should not be confused with the formal application of the syntactical method. If he wants to formalize let us say arithmetic, he will probably equip language L with an individual constant to be interpreted by zero, a one-place function symbol to be interpreted by the successor function, a two-places function symbol for addition.
The point is that applying the syntactical method requires the interpretation to be disregarded, not the language to be a non-interpreted one. Even though Carnap considers the possibility of giving several interpretations to a calculus LSL, p. The informal proof that follows, conducted in the metalanguage, is no application of the syntactical method. Though the issue of the bounds of logic is still much disputed, standard textbooks make a sharp distinction between a logical framework a formal language L equipped with a method of formal derivation and the formalization of a mathematical theory in this framework by non-logical axioms making essential use of non-logical symbols of L.
This is in sharp contrast with Language I and Language II where the list of the logical symbols is devised in such a way that mathematical sentences can be formalized using only logical symbols.
These axioms — and hence the theorems derivable from them — are demonstrable in Language II. It gives evidence of the evolution of his thought in the philosophy of mathematics. Clearly, when Carnap wrote this, he neither considered zero and the successor function as primitive logical concepts nor any sentence of arithmetic as a logical axiom. In , he still followed Frege and Russell in taking the concept of logical symbol in a narrow sense and he took the trouble to explain his reasons for sticking to this philosophical position Carnap b.
In , when the principle of tolerance was applied, the decision to include mathematical symbols among the primitive symbols as Hilbert had done became a question of purely technical expedience. In so doing, he did not mean to abandon logicism but to reformulate it in a way that essentially eliminated the importance of any discussion of the distinction between logical symbols in the narrower sense and mathematical symbols.
Carnap , p. Here Carnap makes a most important distinction between derivation Ableitung and consequence Folge. So far, everything is clear. But commentators have usually tried to give a more literal interpretation of his surprising remark. And much of the attraction of the method of syntax [. Oberdan , p. Thomas Ricketts thus offers an alternative interpretation: Carnap rejects the notion of truth [in LSL] because, for good reasons, he believes the notion of truth to be both syntactically intractable and otiose in logic.
Ricketts , p. I was surprised when he said that he meant truth in the customary sense, including contingent factual truth. In his Foundations of Logic and Mathematics and in his Introduction to Semantics , Carnap does not replace the syntactical approach with a semantic one: he complements the syntactical calculi with semantic systems. The fact that the intuitionistic concept of the continuum cannot be formalized in Language I suggests that Brouwer himself would not have found Language I very appealing.
Language I is extensional. Primitive sentences are not given directly but through schemata, because no sentential variables are available in Language I. A sentence S1 is called analytic in I when it is the consequence of the null class of sentences [. Hence, it does not state anything about facts.
On the other hand, a contradictory sentence states too much to be capable of being true; for from a contradictory sentence each fact as well as its opposite can be deduced. A synthetic sentence is sometimes true — namely, when certain facts exist — and sometimes false; hence it says something as to what facts exists. The purpose of Language I, Language II, and the syntactical method is to provide tools for transforming such misleading and inexact formulations of the material mode of speech inhaltliche Redeweise into exact ones in the formal mode of speech.
The symbol of identity is used between Pr and Fu as well as between Z numerical expressions and S sentences. The result is a language with a much greater expressive power than Language I. In material interpretation, the rule means that if the property expressed by S1 is true of each natural number, then it is true of all. In Language II, the free variable in S1 is not necessarily a numerical variable; it may be, for instance, a predicate-variable. In this introduction, I can hardly do more than enumerate the main issues as stake in this extremely rich part of the book which has been much discussed in the literature.
In order to prove that the sentence which formulates the principle of induction in Language II is analytic in Language II, Carnap uses the principle of induction at the level of the syntax-language. Carnap then argues that the proof is neither circular because the syntax-language is not Language II nor absolute because it depends on the richness of the syntax-language. The same holds for the axiom of choice.
They suggest ways of treating classes, cardinal numbers, real numbers, and physical concepts in the framework of this language. This clearly aims at showing some of the advantages of using a language such as Language II as a framework for science. However, the purpose of LSL is not to decide such a general issue, so Carnap uses more cautious formulations and he does not go so far as to present Language II as his own proposal for the form of a language for science: It follows from all these suggestions that all the sentences of physics can be formulated in a language of the form of II.
In the case of general syntax, we do not know anything about the syntactical categories of the language under consideration, indeed general syntax must apply to any language whatsoever, including languages with no variables, no symbol for zero, no symbol for negation. We now realize that this is an enormous task on which logicians have worked hard to this day and which is still much discussed in the philosophy of logic.
In other words: the truth-value of each sentence constructed solely from them is determined by the rules of the language. It seems to me at present that these distinctions have to be made primarily in semantics, not in syntax. Both Language I and Language II are based only on primitive sentences and transformation rules which, in material interpretation, have a logico-mathematical meaning. Intuitively, the difference between P-valid and L-valid sentences is that an L-valid sentence remains valid under all possible substitutions of descriptive expressions. This is done in the framework of general syntax, i.
Or, as we would now say: the class of all the possible state-descriptions of the object-domain which make the sentence true.
E2 is the syntactical counterpart of the seemingly object-property E1. A sentence is said to be quasi-syntactical when it ascribes a quasi-syntactical property to an object LSL, p. Carnap argues that the translation of quasi-syntactical sentences into syntactical sentences is possible and he distinguishes several cases where such a translation has an important clarifying effect.
The thesis of extensionality becomes: every Introduction 37 intensional language may be translated into an extensional language LSL, p. Are these necessarily two separate languages? Pure syntax, the main subject matter of LSL, abstracts from considerations such as the shape of the symbols, their physical realization, or their occurrence in particular places.