Hi John, Thank you for sending the links to the excerpts from Church's work in logic. His explanation of the assumptions behind extensional approaches in formal logic and the philosophical theory of logic are remarkably clear. If you have additional thoughts to add that help to explain why it is that nominalists such as J.S. Mill and Nelson Goodman strongly prefer extensional systems--and have significant reservations about using intensional systems in philosophy--I'd be interested to hear what you think. In particular, I'd like to hear more about the connections that you see between (1) the motives for developing intensional systems, (2) Church's remarks about the treatment of things such as functions and relations as objects in these systems (e.g., in the lambda operator), and (3) the treatment of the infinite character of some collections and the continuity of some operations at both the object level and the meta-level within intensional systems.
--Jeff Jeffrey Downard Associate Professor Department of Philosophy Northern Arizona University (o) 928 523-8354 ________________________________________ From: John F Sowa <[email protected]> Sent: Thursday, April 20, 2017 6:14 AM To: [email protected]; Jon Awbrey Subject: Re: [PEIRCE-L] Re: Laws of Nature as Signs Jon, That is an extensional definition of a relation: > Following the pattern of the functional case, let the notation > “L ⊆ X × Y” bring to mind a mathematical object specified by > three pieces of data, the set X, the set Y, and a particular > subset of their cartesian product X × Y}. As before we have > two choices, either let L = (X, Y, graph(L)) or let “L” denote > graph(L) and choose another name for the triple. Nominalists prefer extensional definitions. But Peirce would usually state intensional definitions (rules) for the functions or relations he was considering. Alonzo Church (1941) stated the intensional definition: > A function is a rule of correspondence by which when anything is > given (as argument) another thing (the value of the function for > that argument) may be obtained. That is, a function is an operation > which may be applied on one thing (the argument) to yield another > thing (the value of the function). For further discussion of the distinction between intensions extensions, see pp. 1 to 3 of Church's book: http://www.jfsowa.com/logic/alonzo.htm By the way, Church was not a nominalist. See the transcript of his talk "On the ontological status of women and abstract entities": http://www.jfsowa.com/ontology/church.htm John
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