[FOM] Predicative second-order logic

John Bell jbell at uwo.ca
Mon Nov 11 13:35:36 EST 2019

I introduced what Riki has termed" predicative second order logic" under the
name "definable subset logic" in Chapter III of my  1969 Oxford D. Phil
thesis. There I formulate a  (necessarily) infinitary axiomatization of the
system and establish completeness using Boolean algebraic methods. The
dissertation may be found on my website below.

--- John Bell

Professor John L. Bell FRSC
Dept. of Philosophy
Western University

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Today's Topics:

   1. Re: first/second order logic (Richard Heck)


Message: 1
Date: Sat, 9 Nov 2019 04:12:20 -0500
From: Richard Heck <richard_heck at brown.edu>
To: Foundations of Mathematics <fom at cs.nyu.edu>, sambin at math.unipd.it
Subject: Re: [FOM] first/second order logic
Message-ID: <1808b944-0d69-29ee-08c8-82558f531613 at brown.edu>
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On 11/8/19 8:39 AM, sambin at math.unipd.it wrote:
> Quoting Richard Heck <richard_heck at brown.edu
> <mailto:richard_heck at brown.edu>>:
>> On 11/6/19 10:27 AM, Harvey Friedman wrote:
>>> QUESTION. Is there an interesting completeness theorem for 
>>> nontrivial fragments of second order logic? Obviously, first order 
>>> logic is a nontrivial fragment that does have a completeness 
>>> theorem. But what if we look at SIMPLE fragments of second order 
>>> logic. Maybe there are really interesting such with a completeness 
>>> theorem. Or if there has been a good start on this, then how far can it
be pushed?
>> Well, predicative second-order logic is natural and is complete, 
>> isn't it, with respect to some reasonably natural notion of what a 
>> model is??
> Dear Riki,
> what do you have in mind for "predicative second-order logic"?
There is no doubt a good deal of precision here I've not considered. But
what I mean here is what Albert Visser has in mind when he talks about the
predicative extension of a given theory, and proves various results about
it. Roughly, the predicative second-order extension of some first-order
theory T is T plus all predicative comprehension axioms for T (where the
language of T has been extended in the obvious way).

> And hence also the question: what is the natural notion of model for it?
Let M be a model for a first-order theory T. Let PV(T) be the predicative
extension of T (a la Visser). Let M be a model of T. Then let PV(M) be the
extension of M to a second-order model of PV(T) where the domain of the
second-order variables consists of the sets S definable over M, i.e., for
which there is some formula A(x) of the language of T such that S is the
extension of A(x) in M (possibly with parameters, if we wish to allow for
that, though that is inessential).
What I just said describes a natural notion of 'predicative second-order
extension of a model of T', and it's clear (yes? not thought through this
part in detail...) that predicative second-order logic is complete with
respect to that class of models.

I.e. and roughly: If T is a first-order theory, then let PV(T) be T plus
predicative second-order logic. Then if M is any model of T, then there is
an extension of M---where the second-order domain contains exactly the
(first-order) definable sets over M---that verifies PV(T). That's a kind of
completeness theorem, at least.



Richard Kimberly (Riki) Heck
Professor of Philosophy
Brown University

Pronouns: they/them/their

Email:           rikiheck at brown.edu
Website:         http://rkheck.frege.org/
Blog:            http://rikiheck.blogspot.com/
Amazon:          http://amazon.com/author/richardgheckjr
Google Scholar:  https://scholar.google.com/citations?user=QUKBG6EAAAAJ
ORCID:           http://orcid.org/0000-0002-2961-2663
Research Gate:   https://www.researchgate.net/profile/Richard_Heck

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