On 12/28/2010 4:54 AM, Christopher Spottiswoode wrote:
> You seem (to me...) to be saying that every tool for a subset
> of CL or IKL can support full CL. (01)
No. I was elaborating the following point by Chris Menzel: (02)
CM:
> Common Logic supports most every logic-based framework, in
> the sense that such frameworks are all Common Logic dialects.
> All that is needed in order for a framework to "support"
> Common Logic is an explicit characterization of it as such
> a dialect. It is a mistake to think of Common Logic as some
> sort of competitor to OWL, RDF, etc. One of the central goals
> of Common Logic's design was to serve as an overarching abstract
> logical framework in which the underlying commonalities and
> differences between concrete frameworks could be easily identified. (03)
For any version of logic L, there is no such thing as a tool that
supports "full" L. No tool could or should support every possible
method for using L in deduction, induction, abduction, analogy,
question answering, learning, data mining, knowledge discovery... (04)
CS:
> It seems rather at odds with the usual implications of subsetting
> languages, where tools for subsets need to be expanded to handle
> full languages. I am intrigued by what seem to be subtle
> redefinitions of terms such as "subset/superset" or "include ...
> tools". (05)
Instead of saying "redefinition", I would claim that the words
'support' and 'full' have never been adequately defined for any
version of logic. That's one theme of the following article: (06)
http://www.jfsowa.com/pubs/fflogic.pdf
Fads and fallacies about logic (07)
A programming language, such as Java or C, has a specific purpose
that determines the meaning of words like 'support' and 'full':
Each language was designed to represent algorithms that are
translated to an executable form, such as Java byte codes for
Java or machine instructions (plus supporting data) for C. (08)
To say that tool X "supports" the "full" programming language L
means that X translates every syntactically correct statement
in L to an executable statement in the target language that
has whatever semantic effect is specified by the documentation. (09)
But logics are declarative languages that have no single "purpose"
or "goal" or "use". There are as many different meanings of the
words 'support' and 'full' as there are ways of using logic --
and that's probably countably infinite. (See footnote.) (010)
Some logic-based languages, such as Prolog or OWL, were designed
for a specific method of use. For them, the programming-language
definition could be stretched to cover that method of use. (011)
When mathematicians talk about logic, they think "theorem proving".
But look at the Thousands of Problems for Theorem Provers (tptp.org).
The tools that score the best overall are hybrids composed of many
specialized tools optimized for various tasks: translating source
notations to some internal form or forms; analyzing problem statements
(axioms + goal) to determine what strategy or strategies to use;
breaking down the top-level goal to subgoals processed by different
methods for special cases; storing and reusing previous results... (012)
Cyc could be used to prove theorems in "full" CL, but Doug Lenat
said that CycL is equivalent to CL + the IKL extensions. The IKRIS
project showed that IKL can be used as the interchange language for
interoperability of CycL and other logics used in AI. But Cyc itself
is actually a hybrid that uses techniques similar to the TPTP tools. (013)
In fact, Cyc and many other AI systems go beyond theorem proving
to implement some subset of the infinite variety of uses for logic.
Lenat likes to use the term 'commonsense reasoning', but I consider
that a catch-all term with no fixed definition. In fact, common sense
is a moving target that changes with every innovation in science and
even every political and/or religious motivation. (And by politics
and religion, I include academic and commercial politics and faiths.) (014)
John
_____________________________________________________________________ (015)
Footnote: Excerpt from http://www.jfsowa.com/pubs/rolelog.pdf (016)
5. Steps Toward Formalization (017)
In his Philosophical Remarks from the transitional period of 1929-30,
Wittgenstein analyzed some “minor” inconsistencies in the Tractatus. His
analysis led to innovations that form a bridge between his early system
and the far more flexible language games. Shanker (1987) noted two new
terms that are key to Wittgenstein’s transition: (018)
1. Satzsystem: a system of sentences or propositions stated
in a given syntax and vocabulary. (019)
2. Beweissystem: a proof system that defines a logic for a
Satzsystem. (020)
Formally, the combination of a Satzsystem with a Beweissystem
corresponds to what logicians call a theory — the deductive closure of a
set of axioms. Informally, Wittgenstein’s remarks about Satzsysteme are
compatible with his later discussions of language games. In
conversations reported by Waismann (1979:48), Wittgenstein said that
outside a Satzsystem, a word is like “a wheel turning idly.” Instead of
a separate mapping of each proposition to reality, as in the Tractatus,
the Satzsystem is mapped as a complete structure: “The Satzsystem is
like a ruler (Maßstab) laid against reality. An entire system of
propositions is now compared to reality, not a single proposition.”
(Wittgenstein 1964, §82). (021)
For a given logic (Beweissystem), each Satzsystem can be formalized as a
theory that defines the ontology of a narrow subject. The multiplicity
of Satzsysteme implies that any word that is used in more than one
system will have a different sense in each. For natural languages, that
principle is far more realistic than the monolithic logic and ontology
of the Tractatus. Yet Wittgenstein illustrated his Philosophical Remarks
primarily with mathematical examples. That turning point, as Shanker
called it, implies that the goal of a unified foundation for all of
mathematics, as stated in the Principia Mathematica, is impossible. That
implication alarmed Russell, who observed “The theories contained in
this new work of Wittgenstein’s are novel, very original, and
indubitably important. Whether they are true, I do not know. As a
logician who likes simplicity, I should wish to think that they are not.” (022)
From the mid 1930s to the end of his life, Wittgenstein focused on
language games as a more general basis for a theory of meaning. But he
continued to teach and write on mathematical topics, and he compared
language games to the multiple ways of using words such as number in
mathematics: “We can get a rough picture of [the variety of language
games] from the changes in mathematics.” These remarks imply that
Satzsysteme can be considered specialized language games. The crucial
addition for natural language is the intimate integration of language
games with social activity and even the “form of life.” As Wittgenstein
said in his notebooks, language is an “extension of primitive behavior.
(For our language game is behavior.)” (Zettel, §545) The meaning of a
word, a chess piece, or a mathematical symbol is its use in a game — a
Sprachspiel or a Beweissystem. (023)
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