[FOM] First Order Logic

Charlie silver_1 at mindspring.com
Sun Sep 8 11:48:13 EDT 2013

	  Or, virtually the same, Omega Logic (also in Shapiro's book), though admittedly, 
it is *not* based on a primitive, intuitively acceptable relation like the "ancestral".

On Sep 7, 2013, at 3:46 AM, aa at tau.ac.il (Arnon Avron) wrote:

> On Sun, Aug 25, 2013 at 03:59:02PM -0400, Harvey Friedman wrote:
>> Furthermore, first order logic is apparently the unique vehicle for such
>> foundational purposes. (I'm not talking about arbitrary interesting
>> foundational purposes). However, we still do not know quite how to
>> formulate this properly in order to establish that first order logic is in
>> fact the unique vehicle for such foundational purposes.
> Very similar declarations have been made by Harvey about 7 years ago.
> Thus in his posting on Tue Oct 24 22:00:59 EDT 2006 
> (http://www.cs.nyu.edu/pipermail/fom/2006-October/011020.html) he wrote:
> "FOL (through its practically driven variants) is the only appropriate
> vehicle for the foundations of mathematics. I doubt if many people would
> disagree with this. The only issue is how to go about carefully formulating
> just what this means."
> Obviously, no much progress has been made on this front in the last 
> 7 years. Anyway, this statement was made as a part of an exchange between us
> (with contributions of others) that started within a thread that was called 
> "First-order arithmetical truth", and continued with various postings under 
> the titles: "Are first-order languages adequate for mathematics?", 
> "First Order Logic/status", and "Concerning Ancestral Logic". I do not
> have much to add to the arguments I made in this exchange 7 years ago
> (and neither does Harvey, or so it seems to me). So I do not
> see much point in repeating what I wrote then in several postings.
> I'll just repeat the words  started that debate (in my posting from Fri 
> Oct 20 19:32:16 EDT 2006): 
> "For many years I maintain that the appropriate language
> for formalizing logic and mathematics is neither the first-order 
> language nor the second-order one. The first is too weak for 
> expressing what we all understand, the second involves too strong 
> ontological commitments. The adequate language
> is something  in the middle: what is called "ancestral logic"
> in Shapiro's book "Foundations without Foundationalism". This logic 
> is equivalent to weak second-order logic (as is shown
> in Shapiro's book, as well as in  his chapter in Vol. 1 of the 
> 2nd ed. of the Handbook of Philosophical logic). However, I prefer 
> the "ancestral logic" version, because the notion of "ancestor" 
> is part of everybody's logic, 100% understood also by 
> non-mathematicians." 
>> In summary, whereas it can be interesting and important to give second
>> order formulations of various theories, ultimately in each case you are
>> compelled to provide a clarifying first order associate.
> So again: the right choice is neither first-order logic
> nor higher-order logic, but something in the middle: ancestral
> logic. This logic is in fact 
> absolutely necessary even to define what first-order logic
> is, because all the basic notions of first-order logic,
> like formulas and formal proofs, are defined inductively.
> Arnon Avron
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