higher-order logic
in [Theory]
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From: alex goldman on 27 Jan 2005 08:16 I'm interested in learning more about higher-order logic: its computational aspects, inference and learning algorithms, and perhaps uncertainty representation. The books on higher-order logic that I found appear to actually be tutorials on using specific proof assistance software products (HOL or Isabelle), which is not what I want. Does anyone have any suggestions or links to resources? (No research paper plugs please, however)
From: Jose Juan Mendoza Rodriguez on 28 Jan 2005 06:10 alex goldman wrote: > I'm interested in learning more about higher-order logic > (...) > Does anyone have any suggestions or links to resources? This is a very readable introduction to higher-order intuionistic logic: Simon Thompson, 'Type Theory and Functional Programming' http://www.cs.kent.ac.uk/people/staff/sjt/TTFP/ -- Jose Juan Mendoza Rodriguez let me=josejuanmr in let privacy=iespana in let net=es in me(a)privacy.net
From: alex goldman on 30 Jan 2005 13:22 Jose Juan Mendoza Rodriguez wrote: > > alex goldman wrote: > >> I'm interested in learning more about higher-order logic >> (...) >> Does anyone have any suggestions or links to resources? > > This is a very readable introduction to higher-order > intuionistic logic: > > Simon Thompson, 'Type Theory and Functional Programming' > http://www.cs.kent.ac.uk/people/staff/sjt/TTFP/ > Thanks. I've read some of it, however, while I asked for pointers to literature on higher-order logic, this paper discusses first-order logic. And, while it's not a research paper plug, it does seem to be Constructive Mathematics advocacy and mainly concerns itself with programming aides, does it not? I was hoping to find a non-advocacy review of main results in higher-order logic (as opposed to first-order logic). By the way, is the term "higher-order logic" ever used in a sense different from "second-order logic"? Since second-order logic allows quantification over relations, would third-, etc. order logics allow quantification over quantifiers or something similarly outlandish?
From: H. Enderton on 30 Jan 2005 17:42 alex goldman <hello(a)spamm.er> wrote: >By the way, is the term "higher-order logic" ever used in a sense different >from "second-order logic"? Since second-order logic allows quantification >over relations, would third-, etc. order logics allow quantification over >quantifiers or something similarly outlandish? It's simpler than that. First order: quantify over individuals Second order: quantify over sets of individuals and relations on individuals. Third order: quantify over sets of sets of individuals (and relations on relations on individuals). Quibble: Church actually used a slightly more refined way of counting third, fourth, ... order. > I'm interested in learning more about higher-order logic > (...) >I was hoping to find a non-advocacy review of main results in higher-order >logic (as opposed to first-order logic). You said (in the original post) no plugs, so I won't say anything about Chapter 4 of my logic book. --Herb Enderton
From: Owen on 31 Jan 2005 07:13
"H. Enderton" <hbe(a)sonia.math.ucla.edu> wrote in message news:ctjns5$hq6$1(a)daisy.noc.ucla.edu... > alex goldman <hello(a)spamm.er> wrote: >>By the way, is the term "higher-order logic" ever used in a sense different >>from "second-order logic"? Since second-order logic allows quantification >>over relations, would third-, etc. order logics allow quantification over >>quantifiers or something similarly outlandish? > > It's simpler than that. > First order: quantify over individuals > Second order: quantify over sets of individuals and relations on individuals. > Third order: quantify over sets of sets of individuals (and relations on > relations on individuals). I don't agree. Second order logic quantifies over properties/relations of individuals, and so on. There is no need to specify sets at all. Only if sets correspond with properties, are you correct. (there are exceptions: the Russell set does not correspond with any property) {x:~(x e x} does not exist! Third order predications quantify over properties of properties, etc. > > Quibble: Church actually used a slightly more refined way of counting > third, fourth, ... order. > >> I'm interested in learning more about higher-order logic >> (...) >>I was hoping to find a non-advocacy review of main results in higher-order >>logic (as opposed to first-order logic). > > You said (in the original post) no plugs, so I won't say anything > about Chapter 4 of my logic book. > > --Herb Enderton > > |