Abstract
This paper provides an analysis of the notational difference between Beta Existential Graphs, the graphical notation for quantificational logic invented by Charles S. Peirce at the end of the 19th century, and the ordinary notation of first-order logic. Peirce thought his graphs to be “more diagrammatic” than equivalently expressive languages (including his own algebras) for quantificational logic. The reason of this, he claimed, is that less room is afforded in Existential Graphs than in equivalently expressive languages for different ways of representing the same fact. The reason of this, in turn, is that Existential Graphs are a non-linear, occurrence-referential notation. As a non-linear notation, each graph corresponds to a class of logically equivalent but syntactically distinct sentences of the ordinary notation of first-order logic that are obtained by permuting those elements (sentential variables, predicate expressions, and quantifiers) that in the graphs lie in the same area. As an occurrence-referential notation, each Beta graph corresponds to a class of logically equivalent but syntactically distinct sentences of the ordinary notation of first-order logic in which the identity of reference of two or more variables is asserted. In brief, Peirce’s graphs are more diagrammatic than the linear, type-referential notation of first-order logic because the function that translates the latter to the graphs does not define isomorphism between the two notations.
Highlights
According to Michael Dummett (1973), Frege’s discovery of quantifiers and variables to express generality was the most important discovery in logical theory since Aristotle
Why did not Peirce stick to his first successful formalism, like Frege did? Why did he move from the general algebra to EGs, if the former was already substantially capable of capturing what we reckon as first-order logic? In what sense are EGs “different” from the general algebra of logic and the ordinary notation of first-order logic? The primary purpose of the present paper is to provide a philosophical analysis of the difference between Beta graphs and the ordinary notation of first-order logic
Peirce considered all deductive reasoning, independently of the notation in which it is expressed, to be diagrammatic. He thought that “the more diagrammatically perfect a system of representation, the less room it affords for different ways of expressing the same fact” (R 482). In this paper it was explained why Existential Graphs, and especially the Beta department which corresponds to first-order quantificational logic, are more diagrammatically perfect than the ordinary notation of first-order logic, and in what sense less room is afforded in Beta graphs than in ordinary notation for different ways of representing the same fact
Summary
According to Michael Dummett (1973), Frege’s discovery of quantifiers and variables to express generality was the most important discovery in logical theory since Aristotle. Express one and the same fact (i.e., have the same truth-conditions).4 In many of his papers on the algebra of logic, Peirce was clear that a language with a greater amount of primitive operators is less “economic”, or as he said, less “analytical”, than a language with a lesser amount thereof, because in the richer (less economic, less analytical) language there will always be more room for different ways of expressing the same fact than there are in the poorer (more economic, more analytical) language. The notion of occurrence solves the problem exemplified in the preceding paragraph quite well: as a sentence type, (1) contains one type of the word “war” (and not two of them), and two occurrences of it (and not two tokens of it) This tripartite distinction between type expressions, token expressions, and occurrences of expressions is necessary to explain how EGs differ from the ordinary notation of first-order logic
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