Now, John, I'm sure you must know, if you think about it, that
many of the statements that you make below are not exactly so.
Peirce's interpretation of basic logical symbols was far more
abstract, formal, or general than our conventional readings of
boolean syntax, as evidenced among many other facts by the deep
duality between the entitative interpretation and the existential
interpretation of logical graphs. It is also evident in the fact
that Peirce gave axioms for boolean algebra in dual symmetric form,
even writing them in the fashion that was customary to present dual
axiomsets for projective geometry. This leads to a very distinctive
way of regarding the interpretation of logical calculi, where symbols
that we once regarded as "logical constants" take on variable meanings. (02)
In the sense that this variability of interpretation was always implicit
from the very beginning of our logical languages, one could say that what
semantics they actually had was no different than it ever was, but what is
marked in the use of manifestly symmetric syntax and axiomsets is our level
of awareness as to what that semantics actually is. (03)
Jon Awbrey (04)
John F. Sowa wrote:
> There is a very *profound* reason why logic is different:
> > SB: We suffer from identical syntax and allegedly identical
> > semantics all the time in data exchange (ISO 10303), so
> > I don't see why logic should be any different.
> Classical Boolean logic was defined in the mid 19th century, and
> *every* version since then has had absolutely identical semantics.
> The model-theoretic foundation of Boolean logic, which is called
> "truth tables", is a subset of the model-theoretic foundation for
> full first-order logic (FOL).
> Full FOL was independently discovered and published in 1879 by
> Gottlob Frege and in a totally different notation by C. S. Peirce
> in 1880 and 1885. Despite the fact that their notations were
> radically different in appearance and they were working on
> different continents with no communication between them,
> their semantics was *identical*.
> Furthermore, it is a superset of Boole's semantics, which Peirce
> was consciously following, while Frege was consciously trying
> to avoid. Yet they converged on *identical* semantics.
> During the 20th century, many different versions of what is
> called classical FOL were invented and written in many different
> notations, yet they all have *identical semantics*.
> The model-theoretic foundation defined in the ISO standard for
> Common Logic is a superset of *all* the previous versions plus
> many others. It includes the semantics by Boole, Frege, Peirce,
> Tarski, Gentzen, etc., etc., as subsets. It also includes the
> semantics of RDF(S), OWL, and many other notations as subsets.
> No two programming systems defined by behavioral semantics can
> come anywhere close to that result -- even when one is called
> an upward compatible version of the other.
> SB> Its only when we ground the semantics of the data in the
> > behaviour of the application/organization that we have any hope
> > of success, and then only after a long and painful process of
> > testing. Even in the relatively well specified area of geometry,
> > it has taken many years of effort and continuous testing to get
> > a reasonably reliable exchange (though not 100%), and there is
> > still a considerable amount to do.
> As E. W. Dijstra pointed out over 40 years ago, testing the
> behavior of programs can only prove the presence of errors,
> it can never prove their absence.
> Logic can prove both.
> John Sowa (05)
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