Information = Comprehension × Extension • Selection 3 • https://inquiryintoinquiry.com/2024/10/07/information-comprehension-x-exten… All, 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. Regards, Jon cc: https://www.academia.edu/community/Lx81Ga cc: https://mathstodon.xyz/@Inquiry/113249701127551380

Information = Comprehension × Extension • Selection 5 • https://inquiryintoinquiry.com/2024/10/09/information-comprehension-x-exten… All, Peirce now turns to his example of a conjunctive term, which he uses to show the connection between iconic reference and abductive inference. ❝A similar line of thought may be gone through in reference to hypothesis. In this case we must start with the consideration of the term: ❝spherical, bright, fragrant, juicy, tropical fruit. ❝Such a term, formed by the sum of the comprehensions of several terms, is called a conjunctive term. A conjunctive term has no extension adequate to its comprehension. Thus the only spherical bright fragrant juicy tropical fruit we know is the orange and that has many other characters besides these. Hence, such a term is of no use whatever. ❝If it occurs in the predicate and something is said to be a spherical bright fragrant juicy tropical fruit, since there is nothing which is all this which is not an orange, we may say that this is an orange at once. ❝On the other hand, if the conjunctive term is subject and we know that every spherical bright fragrant juicy tropical fruit necessarily has certain properties, it must be that we know more than that and can simplify the subject. Thus a conjunctive term may always be replaced by a simple one. ❝So if we find that light is capable of producing certain phenomena which could only be enumerated by a long conjunction of terms, we may be sure that this compound predicate may be replaced by a simple one. And if only one simple one is known in which the conjunctive term is contained, this must be provisionally adopted.❞ (Peirce 1866, p. 470) 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. Regards, Jon cc:https://www.academia.edu/community/5kdJBG cc: https://mathstodon.xyz/@Inquiry/113249701127551380