[FOM] First Order Logic

Richard Heck richard_heck at brown.edu
Sat Sep 7 10:49:05 EDT 2013

On 09/07/2013 06:46 AM, Arnon Avron wrote:
> 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."

For what it's worth, there is a very natural generalization of ancestral 
logic, discussed in my paper "A Logic for Frege's Theorem" [1], that is 
equivalent to the \Pi_1^1 fragment of second-order logic. The idea is 
just to take the mechanism through which the ancestral is characterized 
in ancestral logic, and then generalize it to cover any relation that is 
defined through \Pi_1^1 comprehension. The suggestion is then that the 
very same resources needed if one is to understand the ancestral give 
one access to a much stronger logic. It's arguable, moreover, that the 
resulting logic is still first-order, in the sense in which non-standard 
quantifiers such as "most" are still first-order.

Indeed, it is arguable, and more or less has been argued by Aldo 
Antonelli, that the alleged "uniqueness" of first-order logic has much 
less to do with first-order-ness than it has to do with the restriction 
to the simple quantifiers "every" and "some" [2], a restriction whose 
motivations are not exactly obvious.

Richard Heck

[1] http://rgheck.frege.org/pdf/published/LogicOfFregesTheorem.pdf
[2] http://aldo-antonelli.org/Papers/genint.pdf

Richard G Heck Jr
Romeo Elton Professor of Natural Theology
Brown University

Website:   http://rgheck.frege.org/
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