Chris, Ali, and Gary, (01)
These discussions help clarify many of the issues. In particular,
they show why we need to distinguish the formal operations rather than
vague "intuitions" about "intended meaning". In particular, I would
prefer to avoid using the term 'meaning' except in discussions of how
the formalism relates to informal discussions in natural languages. (02)
I would also like to add some examples that illustrate how these
discussions can be rephrased in terms of the lattice of theories.
And it is also important to show the implications for the finite,
implemented subset, which may be called the *hierarchy* of theories. (03)
CM> To say that an extension T' of a theory T is conservative is
> to say only that you can't prove anything new about the original
> primitives in the language of the original theory... (04)
Your discussion of that point was a very good and clear summary
and illustration in terms of Presburger arithmetic. I'd like to
give another example in terms of Boolean algebra. (05)
Boole's original operators (or relations) were '+', 'x', and '-',
which he interpreted variously as operators among propositions,
monadic predicates, and sets. Although Boole used the same symbols
that were used for ordinary arithmetic, he created a very different
algebra by adding axioms that were false about the integers, such as (06)
p + p = p, for all p. (07)
This is definitely not a conservative extension. In fact, it is not
an extension at all, since it is inconsistent with ordinary arithmetic. (08)
Peirce made a conservative extension to Boolean algebra by introducing
a less-than-or-equal operator, which he wrote as the claw symbol '-<'.
He showed that p-<q could be interpreted as subset in terms of sets
or material implication for propositions and monadic predicates. (09)
Peirce also showed that all 16 dyadic functions of two Boolean variables
could be defined in terms of just two operators, such as + and -,
x and -, or -< and -. He introduced symbols for some operators, such
as the ones we now call 'nand' and 'nor', and he showed that just 'nand'
by itself or 'nor' by itself could be used to define all 16. (010)
(Footnote: Boole's original algebra used '+' to represent exclusive OR
for propositions and disjoint union for sets. De Morgan and Peirce used
the '+' symbol to represent inclusive OR and the modern union. But the
extended algebra with multiple symbols subsumes both versions.) (011)
These innovations were very useful, but they were all conservative
extensions. But they certainly contributed many new insights into
Boolean algebra and its applications to propositions, sets, and
monadic predicates. However, every theorem of the extended algebra
could be translated to a logically equivalent theorem in terms of
the original operators that used only +, x, and -. (012)
Those translations, however, would lose all the intuitive 'meaning'
of the extended algebras. But in talking about the formalism, I
would prefer to avoid using the term 'meaning', except perhaps as
an adjective in terms like 'meaning preserving translation' (MPT).
For MPT, see the following short paper: (013)
http://www.jfsowa.com/logic/proposit.htm (014)
Another significant issue: Boole used Boolean algebra with multiple
interpretations, including the two-valued propositional logic in which
0 represents falsum and 1 represents verum. But he also applied it
to the many-valued sets, in which 0 is the empty set, 1 is the universe,
and variables can represent sets of any size between 0 and 1. (015)
C. I. Lewis took advantage of that option to introduce a version of
modal logic. He observed that one could restrict Boolean algebra
to just the values 0 and 1 by adding the following axiom: (016)
p = (p = 1), for all p. (017)
This says that asserting any proposition p by itself is equivalent
to asserting the proposition that p is true. (018)
Adding this axiom is a nonconservative extension (a specialization)
of Boolean algebra, which restricts the number of possible models.
In particular, it makes it a very poor choice for set theory. (019)
But Lewis also showed that the negation of the above axiom would
also make a very interesting algebra: (020)
-(p = (p = 1)), for at least one p. (021)
As an example, he used a 4-value algebra, with 1 for necessary
and 0 for impossible. The intermediate values are (022)
necessary > true > false > impossible. (023)
Cyc currently uses a 5-value algebra: (024)
true > true by default > unknown > false by default > false. (025)
Both versions are nonconservative extensions (specializations)
of the original Boolean algebra. (026)
In terms of the lattice of theories, a conservative extension creates
a larger theory (more theorems) but each of the new theorems has a
translation to an equivalent theorem that uses only the original
vocabulary. But in terms of MPTs (meaning preserving translations),
the new theorems would be considered distinct from the old ones
because they use different vocabulary. (027)
John (028)
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