Re: higher-order logic

From: Owen (oorionus_at_yahoo.com)
Date: 01/31/05 

Date: Mon, 31 Jan 2005 07:13:31 -0500

  "H. Enderton" <hbe@sonia.math.ucla.edu> wrote in message
news:ctjns5$hq6$1@daisy.noc.ucla.edu...
> alex goldman <hello@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
>
>

Relevant Pages

  • Re: higher-order logic
    ... > First order: quantify over individuals ... quantify over sets of individuals and relations on ... Second order logic quantifies over properties/relations of individuals, ... Only if sets correspond with properties, ...
    (sci.logic)
  • Re: Can we have Second Order Logic expressibility in FOL?
    ... > quantify over sets, and thus over relations and functions since they ... But isn't this the domain of second order logic - to be able ... In first order logic, we can quantify over anything as long as it's ... Similarly for first order definability of relations ...
    (sci.math)
  • Re: Can we have Second Order Logic expressibility in FOL?
    ... > quantify over sets, and thus over relations and functions since they ... But isn't this the domain of second order logic - to be able ... In first order logic, we can quantify over anything as long as it's ... Similarly for first order definability of relations ...
    (sci.logic)
  • Re: Small set Theory:final version.
    ... In a first order theory you can't, in the object language quantify over ... therefore I don't know what is the importance of these formalities. ...
    (sci.math)
  • Re: Small set Theory:final version.
    ... first order axiom schemata. ... In a first order theory you can't, in the object language quantify over ...
    (sci.math)