[FOM] is logic universal?

Vaughan Pratt pratt at cs.stanford.edu
Fri Oct 22 02:30:10 EDT 2010

A common element in all four of the essays in this issue of Logica 
Universalis (of which the first, Gaines's, is far and away the longest) 
is that they assume a local reasoner, one whose sensors and reasoning 
apparatus are all sufficiently near each other to make communication 
time between them a negligible factor.

This assumption comes to a head in decentralized management and 
distributed computing as exemplified by the Internet. (And perhaps 
physics, though I don't have much of a handle on it yet; likewise 
biology, at least at the level of ecosystems if not individual organisms.)

The problem is that a distributed organization, such as an army on the 
battlefield, or a multinational corporation, may sometimes have to 
respond to situations in a time-frame too short to convey the situation 
to headquarters, yet long enough to get it to a responsible nearby 
representative of management for suitable handling.

If we postulate a common goal or objective function for the 
organization, then there ought to be some sort of rationally based 
system by which it optimizes its behavior relative to that goal subject 
to the prevailing limitations and urgencies.

Such a system would surely fit within the scope of a logical system, one 
that reasons logically about appropriate facts, responses, behaviors, 
protocols, etc, but in a distributed way that trades off urgency against 
communication delay.

I bring this up because quite some time ago I found dynamic logic (and 
related modal logics of behavior) suitable only for traditional 
centralized reasoning systems and not appropriate for distributed 
systems.  I therefore moved towards a more physically motivated 
semantics that took delay into consideration.

A result was that the conventional notion of truth disappeared because 
there was no locus of "truth at a point" (or in a world) in the 
traditional sense.   In place of pointwise truth I found myself working 
with natural transformations between functors.  The Curry-Howard 
isomorphism confers on these an analogue of logical structure.  What I 
found hard to convey however is that there is no natural way to convert 
this back to pointwise truth; the truth in a natural transformation, to 
the extent one can even imagine there could be such a thing, is an 
intrinsically distributed notion of truth.

This is not something that has a counterpart in the extant notions of 
"universal logic."

I bring this up more as a question than an answer, namely how to make 
sense of this concept.  So far I don't have a clearly convincing story 
to tell.  I do however hope to be able to develop the idea in more 
detail as more of the structure becomes clear to me, which has been a 
slow process.

The relevant semantics is abstractly that of *-autonomous categories, 
and concretely Chu spaces as a natural generalization of Nielsen, 
Plotkin and Winskel's event structures, see e.g. 
http://boole.stanford.edu/pub/coimbra.pdf and 
http://boole.stanford.edu/pub/seqconc.pdf for my most recent thinking on 
this (which is not all that recent). 
http://boole.stanford.edu/pub/topoalg.pdf *is* more recent but suffers 
from being both cryptic (being just slides for a talk) and tangential. 
I'm currently working on a paper that may tie some of this together better.

Vaughan Pratt

On 10/21/2010 8:47 AM, jean-yves beziau (by way of Martin Davis 
<eipye at pacbell.net>) wrote:
> A special issue of the journal Logica Universalis dedicated to the
> question "is logic universal?" has been released.
> <http://www.springerlink.com/content/120443/?Content+Status=Accepted>http://www.springerlink.com/content/120443/?Content+Status=Accepted
> The authors have tried to answer the following questions:
> 1. Do all human beings have the same capacity of reasoning? Do men,
> women, children, Papuans, yuppies, reason in the same way?
> 2. Does reasoning evolve? Did human beings reason in the same way two
> centuries ago? In the future will human beings reason in the same
> way? Are computers changing our way of reasoning? Is a mathematical
> proof independent of time and culture?
> 3. Do we reason in different ways depending on the situation? Do we
> use the same logic for everyday life, in physics, and in questions to
> do with the economy?
> 4. Do the different systems of logic reflect the diversity of reasoning?
> 5. Is there any absolute true way of reasoning?
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