Howard, Cathy, list,
Howard, at first I thought you were making a point that I had made in a
previous thread on the subject, when I said that Peirce disbelieved that
the seeming meaning of the ordinary language was captured by the formal
logic, and I started talking about veiled constants, modal logic, and
branching quantifiers, as ways rendering the ordinary-language sense.
However, I see that you're onto something different. The difference
between AND gates and NAND gates is a good example. This also relates to
Peirce's distinction between corollarial and theorematic reasoning
http://en.wikipedia.org/wiki/Corollary#Peirce_on_corollarial_and_theorematic_reasonings
. In a way that certainly seems related, a mathematical deduction's
conclusion may be equivalent to, or is at least entailed by, the set of
premisses, yet be different in ways that make us call it a nontrivial or
surprising result. Then you go on to another idea, that of
equisatisfiability, and I followed your link, and went on to
Skolemization, but unfortunately I'm just not well-grounded enough in
these things to get it, but it certainly looks interesting and to the point.
Best, Ben
On 2/24/2015 8:21 PM, Howard Pattee wrote:
Ben, Catherine and list,
At 04:29 PM 2/24/2015, Catherine Legg wrote:,
I'm confused though about Peirce's big announcement about now being
able to give a meaning to graphs which cross a cut.
[snip]
I once tried to prove Peirce's famous two statements about the
suiciding wife and the man who fails in business equivalent in
regular FOL, but couldn't do it. Are people sure they're equivalent
in FOL, as in the beta graphs?
HP: I'm still not sure. Ben's link to Peirce
<http://www.existentialgraphs.com/peirceoneg/prolegomena.htm#Paragraph569>
convinced me that the logical equivalences is clear. But on second
thought, the natural language statement is not clear. Here is Peirce’s
natural language statement: “The logical Principle is that to say that
there is some one individual of which one or other of two predicates
is true is no more than to say that there either is some individual of
which one is true or else there is some individual of which the other
is true.”
But notice that Peirce also recognizes the strangeness of the true
natural language consequent when he says, “This evidently goes far
beyond saying that if every married man fails in business some married
woman will commit suicide.”
It seems to me that expression in formal logic and natural language
need not be equivalent because their domains (universes of discourse)
exist at different levels of abstraction. Bringing live humans and
natural language images to illustrate an abstract formal logical
principle is confusing two levels of abstraction.
What first bothered me was that at the linguist level Peirce’s two
clauses do not appear to be equivalent because his first clause
applies only to one individual while his second clause must refer to
two or more individuals. (NB. If the second clause were interpreted as
referring to only “some one individual” then it would be
indistinguishable from the first clause.) In other words, the two
clauses have two linguistically different /conceptual models/ even
though they are abstractly logically equivalent. For logicians this
level of abstraction is fine. For wives who are considering suicide
this is too abstract.
Here is an analogy. For many years I taught discrete mathematics for
computer scientists. Those were the days when abstract Boolean logic
and hardware gates were not separated by as many hierarchical levels
of abstraction as they are today, and programmers had to actually do
logic in their brains! Now computer programs do most of the formal
logic. In teaching, I found it important to always distinguish the
several logic levels of abstraction from the several physical levels,
otherwise the students were easily confused.
As a simple example, Peirce discovered that two-to-four NAND gates can
formally execute any Boolean function. One AND gate can be logically
executed by two NAND gates in series. At the formal logic level of
abstraction they are equivalent. At the circuit design level of
abstraction they are obviously not equivalent. For the logician this
equivalence is essential. For the circuit or chip designer the
difference is essential.
Incidentally, the first Texas Instruments 4 NAND chips
<http://en.wikipedia.org/wiki/7400_series> came out during my
teaching years (/ca/. 1968). I think Peirce would have been
fascinated by abstract Turing Theory, but probably irritated by the
logically redundant hardware proliferation (See TI 7400 Series
<http://en.wikipedia.org/wiki/List_of_7400_series_integrated_circuits>
list)
My thinking is that at the logician’s level of abstraction formal
equivalence may be clear, but /at the same time/ the less abstract
material view, like computer circuit design, must see it as only a
/functional/ equivalence. I think some logicians call this
equisatisfiability <http://en.wikipedia.org/wiki/Equisatisfiability> –
two formal logically equivalent statements that have different
conceptual or formal models. I don't know where Peirce discusses
levels of abstraction, but he mostly focused on the logic level.
Howard
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