Information = Comprehension × Extension • Selection 4
Selection 3 showed how it was possible to combine symbols in such a way as to end up with species of representation outside the class of genuine symbols and introduced the concepts of conjunctive terms and disjunctive terms to describe two ways of doing this. The essence of wit being quickly grasping the middle term, Peirce’s wit fastens on those terms to highlight the links between manners of representation and modes of inference.
Selection 4 finds Peirce in the middle of articulating the connection between indexical reference and inductive inference, using examples of disjunctive terms as pivotal cases.
Accordingly, if we are engaged in symbolizing and we come to such a proposition as “Neat, swine, sheep, and deer are herbivorous”, we know firstly that the disjunctive term may be replaced by a true symbol. But suppose we know of no symbol for neat, swine, sheep, and deer except cloven-hoofed animals. There is but one objection to substituting this for the disjunctive term; it is that we should, then, say more than we have observed. In short, it has a superfluous information. But we have already seen that this is an objection which must always stand in the way of taking symbols. If therefore we are to use symbols at all we must use them notwithstanding that. Now all thinking is a process of symbolization, for the conceptions of the understanding are symbols in the strict sense. Unless, therefore, we are to give up thinking altogether we must admit the validity of induction. But even to doubt is to think. So we cannot give up thinking and the validity of induction must be admitted.
(Peirce 1866, p. 469)
Reference
- Peirce, C.S. (1866), “The Logic of Science, or, Induction and Hypothesis”, Lowell Lectures of 1866, pp. 357–504 in Writings of Charles S. Peirce : A Chronological Edition, Volume 1, 1857–1866, Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982.
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- This Blog • Survey of Pragmatic Semiotic Information
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- C.S. Peirce • Upon Logical Comprehension and Extension
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Information = Comprehension × Extension • Selection 3
Selection 3 opens with Peirce remarking a critical property of genuine symbols — the class of symbols is not closed under combinations. In particular, there are logical conjunctions of symbols and logical disjunctions of symbols which are not themselves genuine symbols.Applying this paradigm to terms, Peirce introduces two sets of examples under the headings of conjunctive terms and disjunctive terms designed to illustrate the correspondence between manners of representation and modes of inference.
Yet there are combinations of words and combinations of conceptions which are not strictly speaking symbols. These are of two kinds of which I will give you instances. We have first cases like:
man and horse and kangaroo and whale,
and secondly, cases like:
spherical bright fragrant juicy tropical fruit.
The first of these terms has no comprehension which is adequate to the limitation of the extension. In fact, men, horses, kangaroos, and whales have no attributes in common which are not possessed by the entire class of mammals. For this reason, this disjunctive term, man and horse and kangaroo and whale, is of no use whatever. For suppose it is the subject of a sentence; suppose we know that men and horses and kangaroos and whales have some common character. Since they have no common character which does not belong to the whole class of mammals, it is plain that mammals may be substituted for this term. Suppose it is the predicate of a sentence, and that we know that something is either a man or a horse or a kangaroo or a whale; then, the person who has found out this, knows more about this thing than that it is a mammal; he therefore knows which of these four it is for these four have nothing in common except what belongs to all other mammals. Hence in this case the particular one may be substituted for the disjunctive term. A disjunctive term, then, — one which aggregates the extension of several symbols, — may always be replaced by a simple term.
Hence if we find out that neat are herbivorous, swine are herbivorous, sheep are herbivorous, and deer are herbivorous; we may be sure that there is some class of animals which covers all these, all the members of which are herbivorous. Now a disjunctive term — such as neat swine sheep and deer, or man, horse, kangaroo, and whale — is not a true symbol. It does not denote what it does in consequence of its connotation, as a symbol does; on the contrary, no part of its connotation goes at all to determine what it denotes — it is in that respect a mere accident if it denote anything. Its sphere is determined by the concurrence of the four members, man, horse, kangaroo, and whale, or neat swine sheep and deer as the case may be.
Now those who are not accustomed to the homologies of the conceptions of men and words, will think it very fanciful if I say that this concurrence of four terms to determine the sphere of a disjunctive term resembles the arbitrary convention by which men agree that a certain sign shall stand for a certain thing. And yet how is such a convention made? The men all look upon or think of the thing and each gets a certain conception and then they agree that whatever calls up or becomes an object of that conception in either of them shall be denoted by the sign. In the one case, then, we have several different words and the disjunctive term denotes whatever is the object of either of them. In the other case, we have several different conceptions — the conceptions of different men — and the conventional sign stands for whatever is an object of either of them. It is plain the two cases are essentially the same, and that a disjunctive term is to be regarded as a conventional sign or index. And we find both agree in having a determinate extension but an inadequate comprehension.
(Peirce 1866, pp. 468–469)
Reference
- Peirce, C.S. (1866), “The Logic of Science, or, Induction and Hypothesis”, Lowell Lectures of 1866, pp. 357–504 in Writings of Charles S. Peirce : A Chronological Edition, Volume 1, 1857–1866, Peirce Edition Project, Indiana University Press, Bloomington, IN, 1982.
Resources
- This Blog • Survey of Pragmatic Semiotic Information
- OEIS Wiki • Information = Comprehension × Extension
- C.S. Peirce • Upon Logical Comprehension and Extension
cc: Conceptual Graphs • Cybernetics • Structural Modeling • Systems Science
cc: FB | Inquiry Into Inquiry • Laws of Form • Mathstodon • Academia.edu
cc: Research Gate#Abduction #CSPeirce #Comprehension #Deduction #Extension #Hypothesis #IconIndexSymbol #Induction #Inference #InformationComprehensionExtension #Inquiry #Intension #Logic #PeirceSCategories #PragmaticSemioticInformation #Pragmatism #ScientificMethod #Semiotics #SignRelations
Jon Awbrey (@Inquiry@mathstodon.xyz)
Information = Comprehension × Extension • Preamble • https://inquiryintoinquiry.com/2024/10/04/information-comprehension-x-extension-preamble/ Eight summers ago I hit on what struck me as a new insight into one of the most recalcitrant problems in P…Mathstodon