On 3/16/2017 8:44 AM, Jon Awbrey wrote:
This is the question of “convergence”, a question that mathematicians,
physicists, systems theorists, etc. have investigated in great detail.
As a rule we find that some methods of procedure, of stepping through
a sequence of states, will eventually converge on a settled or stable
state while others will not.
To formalize the idea of convergence, I combined a Lindenbaum
lattice with methods of belief (or theory) revision. The lattice
contains all possible theories expressible within a given logic,
and the AGM operators for belief revision give a measure of how
close one theory is to another.
I discuss this measure and relate it to Peirce and some critics
(including Quine) in the signproc.pdf article. (Excerpt below)
John
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From page 33 of http://www.jfsowa.com/pubs/signproc.pdf
Peirce’s definition of truth and his logic of pragmatism, which supports
that definition, are an elegant generalization of the practices of
working scientists. Yet many philosophers who seized upon one brief
quotation have failed to appreciate the full ramifications. In a survey
of various theories of truth, Kirkham (1992) said
Peirce’s theory of truth is plausible only because it is parasitic
on another, hidden theory of truth: truth as correspondence with
reality. So why doesn’t Peirce simply offer the latter as his theory
of truth? (p. 83)
If he had read more of Peirce’s writings, Kirkham might have found the
answer to his question:
That truth is the correspondence of a representation with its object
is, as Kant says [1787, A58, B82], merely the nominal definition of it.
Truth belongs exclusively to propositions. A proposition has a subject
(or set of subjects) and a predicate. The subject is a sign; the
predicate is a sign; and the proposition is a sign that the predicate
is a sign of that of which the subject is a sign. If it be so, it is
true. But what does this correspondence, or reference of the sign to
its object, consist in? The pragmaticist answers this question as
follows... if we can find out the right method of thinking and can
follow it out, — the right method of transforming signs, — then truth
can be nothing more nor less than the last result to which the
following out of this method would ultimately carry us. (EP 2.379-380)
Quine (1960) is more subtle, but he hadn’t read much more of Peirce’s
writings than Kirkham:
But there is a lot wrong with Peirce’s notion, besides its assumption
of a final organon of scientific method and its appeal to an infinite
process. There is a faulty use of numerical analogy in speaking of a
limit of theories, since the notion of limit depends on that of “nearer
than,” which is defined for numbers and not for theories. And even if
we by-pass such troubles by identifying truth somewhat fancifully with
the ideal result of applying scientific method outright to the whole
future totality of surface irritations, still there is trouble in the
imputation of uniqueness (“the ideal result“).... It seems likelier, if
only on account of symmetries or dualities, that countless alternative
theories would be tied for first place. (p. 23)
Quine’s objection has three parts, each of which requires a separate answer:
1. Peirce made no “assumption of a final organon of scientific
method,” other than the repeated and unfettered cycles of induction,
abduction, deduction, and testing illustrated in Figure 7. In rejecting
Kant’s claim that there is anything that could be inherently unknowable,
Peirce maintained that for any question that science might ask, there
exists a discoverable theory that could answer it. He admitted that
discovering such a theory might take an indefinitely long time, but the
existence of a theory in the infinite lattice does not depend on the
method of search, its duration, or the nature of the minds that do the
search.
2. The lattice of all possible theories provides a notion of “nearer
than”: a theory T1 is nearer to a theory T2 than it is to T3 iff fewer
belief revision steps (contraction, expansion, and analogy) are needed
to convert T1 to T2 than to convert T1 to T3.
3. Peirce was well aware of the infinite number of symmetries,
dualities, and other transformations that can change a statement’s form
without making any change in its implications. They can all be
accommodated by grouping theories in equivalence classes (Sowa 2000).
The ultimate goal of science is not a particular statement of a theory,
but any statement within the equivalence class.
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