Pat, John & All: (01)
Pat Hayes wrote:
> On Jul 17, 2007, at 8:50 AM, Kathryn Blackmond Laskey wrote:
>
>
>>At 5:44 PM -0500 7/16/07, Pat Hayes wrote:
>>
>>>On Jul 16, 2007, at 5:20 PM, Gary Berg-Cross wrote:
>>>
>>>
>>>> Pat, Barry
>>>>
>>>> Most authors who write biology text books don't know about, or
>>>> aren't thinking about Taskian models.
>>>
>>>They may not know about them, but they can still be thinking about
>>>them. Just as someone who knows nothing of botany can think about
>>>plants.
>>
>>I don't think it's correct to say they "really are" thinking about
>>Tarskian models even if they don't know about them. However, it is
>>the case that their thought processes are logically consistent to the
>>degree that they could be formalized as Tarskian models of a set of
>>axioms.
>
>
> Those seem exactly the same to me. Suppose I know nothing about set
> theory, and am thinking about six pennies on a table in front of me
> (say, trying to solve a puzzle). Is it correct to say that I am
> thinking about members of a set containing six pennies? IMO, yes, it
> is. Because the members of the set are the pennies, and I am thinking
> about those pennies. (02)
Seems a good place to plug back into the Peirce manuscript that started
this discussion. Somehow we branched into model theory exclusively and
there's a nice connection between cognition and model theory to be made
here. The Peirce manuscript describes quality as the first conception in
passing from being to substance. So, the pennies have certain
qualities. For example color, shape, size, etc. that allow us to infer
the individual objects make up a set. In this case, the observer asserts
a proposition based on the similarity of the observed qualities. When
the interpretation of the proposition is true, we have a model. Unless
the observer has prior knowledge, the observer can explain the model
with a theory: every object with those qualities is from the set of pennies. (03)
Peirce claims that we first use similarity (quality), second comparison
(correlate) and third mediation (interpretant) in the formation of a
proposition which has a truth value. Even a simple proposition as "the
snow is white" allows us to connect cognition with model theory.
Although model theory provides a richer exposition of truth and/or
satisfaction through sentences, interpretation, structures, signatures,
etc., Peirce's early description of semiotics complements model theory. (04)
So, maybe a composition of formalized specification is in order, but way
to much work to define explicitly here. Notionally, we could define the
specification morphism of Peirce and Tarski algebraic specifications as
follows: (05)
Peirce Specification from "On a New List of Categories"(early semiotics)
signature (structure)
sorts: conception, impression ...
ops: proposition, predication ...
axioms (sentences) sorts + ops (06)
Tarski Model Theory
signature (structure)
sorts: model, theory ...
ops: interpretation ...
axioms (sentences) sorts + ops (07)
Composition (Peirce + Tarski specification morphism)
signature
sorts: P + T
ops: P + T
axioms: P + T (08)
Anyway, that would be a lot of work, but it gets the idea across that we
could formalize what's in the Peirce manuscript about cognition and
what's in model theory to enhance the discussion. Other list members who
are experts in algebraic specifications may want to comment on this. I
think there has already been some work done in this area called
algebraic semiotics ... (09)
> Now, suppose I am a scientist thinking about some scientific topic,
> and putting my thoughts into reasonably exact words. I might be
> thinking about, say, the energy levels of a photon being emitted from
> an organic molecule which I have synthesized. If my thoughts/
> assertions are veridical, then they relate to the reality they
> describe in a certain way. That relationship is what Tarskian truth
> theory describes. So if my thoughts are veridical, then they relate
> to some reality in a Tarskian way: in other words, that (piece of )
> reality is a Tarskian model of my thoughts/assertions. That is all I
> meant. Now, this does not mean that I am thinking about Tarskian
> theory or even need to have heard of Tarskian theory, any more than
> in the first example I need to know set theory to think about coins.
> Nevertheless, the objects of my thought in the first case are the
> elements of a set, and in the second case are a Tarskian model of my
> thoughts. (010)
I understand your point that a model is a true interpretation of reality
based on knowledge, but despite their best efforts at veridicality,
scientists often work on belief and normal science often operates in a
soon to be defunct paradigm re: Thomas Kuhn's Structure of Scientific
Revolutions. (011)
Has the literature on model theory addressed this issue ? (012)
> Pat
>
>
>
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> (013)
--
Best wishes, (014)
Rick (015)
email: rick@xxxxxxxxxxxxxx
web: http://www.rickmurphy.org
blog: http://spout.rickmurphy.org
cell: 703-201-9129 (016)
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