“By the logical syntax of a language, we mean the formal theory of the linguistic forms of that language -- the systematic statement of the formal rules which govern it together with the development of the consequences which follow from these rules. A theory, a rule, a definition, or the like is to be called formal when no reference is made in it either to the meaning of the symbols (for examples, the words) or to the sense of the expressions (e. g. the sentences), but simply and solely to the kinds and order of the symbols from which the expressions are constructed.”

Source: Logical Syntax of Language, 1934/1937, p. 1

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Rudolf Carnap21

German philosopher 1891–1970

Rudolf Carnap (1891–1970) German philosopher

Source: Logical Syntax of Language, 1934/1937, p. 8

“It remains to mention some of the ways in which people have spoken misleadingly of logical form. One of the commonest of these is to talk of ' the logical form' of a statement; as if a statement could never have more than one kind of formal power; as if statements could, in respect of their formal powers, be grouped in mutually exclusive classes, like animals at a zoo in respect of their species. But to say that a statement is of some one logical form is simply to point to a certain general class of, e. g., valid inferences, in which the statement can play a certain role. It is not to exclude the possibility of there being other general classes of valid inferences in which the statement can play a certain role.”

P. F. Strawson (1919–2006) British philosopher

Source: Introduction to Logical Theory (1952), p. 53 as cited in: Ian Hacking (1975) Why Does Language Matter to Philosophy?, p. 83.

“Not only the tools of manual labour, but also the tools of human thought — words — are subject to the laws of historical development. The history of the meanings of words is outside the area of interest of formal logic, and could not be fruitfully studied by the methods of that discipline.
The history of language in what is its most essential content is the history of language as a social instrument of thought; it is historical epistemology which cannot be studied within the scope of any other discipline.
The linguist is of necessity only marginally interested in all conventional terminology, whereas certain votaries of formal logic are inclined to investigate domains which are alien to linguistics and even to some extent in contradiction to its basic assumptions!”

Witold Doroszewski (1899–1976) Lexicographer and linguist

As cited in Schaff (1962;7).
"Comments on Semantics", 1952

“Modern positivism…expresses criticism against the naïve use of certain terms… by the general postulate that the question whether a given sentence has any meaning… should always be thoroughly and critically examined. This… is derived from mathematical logic. The procedure of natural science is pictured as an attachment of symbols to the phenomena. The symbols can, as in mathematics, be combined according to certain rules… However, a combination of symbols that does not comply with the rules is not wrong but conveys no meaning.
The obvious difficulty in this argument is the lack of any general criterion as to when a sentence should be considered meaningless. A definite decision is possible only when the sentence belongs to a closed system of concepts and axioms, which in the development of natural science will be rather the exception than the rule. In some case the conjecture that a certain sentence is meaningless has historically led to important progress… new connections which would have been impossible if the sentence had a meaning. An example… sentence: "In which orbit does the electron move around the nucleus?"”

Werner Heisenberg (1901–1976) German theoretical physicist

But generally the positivistic scheme taken from mathematical logic is too narrow in a description of nature which necessarily uses words and concepts that are only vaguely defined.
Physics and Philosophy (1958)

“The subject of a science is never well circumscribed and there is little use sharpening its definition. However, nobody will deny that physics deals with nature and sociology with human society in some of their aspects. With logic, it is another matter. Logic is usually understood nowadays as a study of certain formal systems, though in former times there were philosophers who held that the subject matter of logic was the formal rules of human thought. In the latter sense it would be an empirical rather than a formal science, though its empirical subject matter would still be fundamentally different from that of psychology of thinking. One interpretation of logic does not exclude the other. Formal approaches are often easier than empirical ones, and for this reason one can understand why logic as a study of formal systems has till now made more progress than logic as a study of the formal rules of thought, even if restricted to scientific thought.”

Hans Freudenthal (1905–1990) Dutch mathematician

Source: Realistic models in probability (1968), p. 1

“A treatise called Astadhyayi (or Astaka) is Panini's major work. It consists of eight chapters, each subdivided into quarter chapters. In this work Panini distinguishes between the language of sacred texts and the usual language of communication. Panini gives formal production rules and definitions to describe Sanskrit grammar. Starting with about 1700 basic elements like nouns, verbs, vowels, consonants he put them into classes. The construction of sentences, compound nouns etc., is explained as ordered rules operating on underlying structures in a manner similar to modern theory. In many ways Panini's constructions are similar to the way that a mathematical function is defined today.”

Pāṇini ancient Sanskrit grammarian

An Analytical Study of 'Sanskrit' and 'Panini' as Foundation of Speech Communication in India and the World

“Conceptual graphs are system of logic based on the existential graphs of Charles Sanders Peirce and the semantic networks of artificial intelligence. The purpose of the system is to express meaning in a form that is logically precise, humanly readable, and computationally tractable. With their direct mapping to language, conceptual graphs can serve as an intermediate language for translating computer-oriented formalisms to and from natural languages. With their graphic representation, they can serve as a readable, but design and specification language.”

John F. Sowa (1940) artificial intelligence researcher

Source: Conceptual graphs for knowledge representation, 1993, p. 3-51. cited in: Bernhard Ganter, Gerd Stumme, Rudolf Wille (2005) Formal Concept Analysis: Foundations and Applications. p. 87

“…Once you entomb mathematics in an artificial language à la Hilbert, once you set up a completely formal axiomatic system, then you can forget that it has any meaning and just look at it as a game that you play with marks on paper that enable you to deduce theorems from axioms. You can forget about the meaning of the game, the game of mathematical reasoning, it's just combinatorial play with symbols! There are certain rules, and you can study these rules and forget that they have any meaning!”

Gregory Chaitin (1947) Argentinian mathematician and computer scientist

1999 Lecture—"A Century of Controversy over the Foundations of Mathematics" at U. Massachusetts at Lowell, quoted in [2012, Conversations with a Mathematician: Math, Art, Science and the Limits of Reason, Springer, https://books.google.com/books?id=DczTBwAAQBAJ&pg=PA15] p. 15

“By virtue of the universal concept, thought attains mastery over the particular cases. However, the most formalized universe of logic still refers to the most general structure of the given, experienced world; the pure form is still that of the content which it formalizes. The idea of formal logic itself is a historical event in the development of the mental and physical instruments for universal control and calculability. In this undertaking man had to create theoretical harmony out of actual discord, to purge thought from contradictions, to hypostatize identifiable and fungible units in the complex process of society and nature. Under the rule of formal logic, the notion of the conflict between essence and appearance is expendable if not meaningless; the material content is neutralized; the principle of identity is separated from the principle of contradiction (contradictions are the fault of incorrect thinking); final causes are removed from the logical order. Well defined in their scope and function, concepts become instruments of prediction and control. Formal logic is thus the first step on the long road to scientific thought”

Herbert Marcuse book One-Dimensional Man

Source: One-Dimensional Man (1964), p. 137

“For remember that in general we don't use language according to strict rules — it hasn't been taught us by means of strict rules, either.”

Ludwig Wittgenstein (1889–1951) Austrian-British philosopher

Source: 1930s-1951, The Blue Book (c. 1931–1935; published 1965), p. 25

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