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I would not be surprised if someone has investigated the closure operator C using a variant of fixed point theory, which typically is stated using the operand rather than the operator as the fixed term in the relation, I.e., Ax=x for all x.
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Jon A, List,
I replaced the subject line with a very specific question. That question is closely related to the question "How can we raise ethical children?" The logical positivists destroyed philosophy by rejecting value judgments. Carnap was the very intelligent, but emotionally stupid positivist whose most damning criticism of any mention of value judgments was "That's poetry!" Wittgenstein refused to attend any meeting of the Vienna Circle if Carnap was present. And I don't blame him.
Form alone cannot go beyond "is"
to "ought". It does not provide a basis for value judgments.
Beauty is the first of the emotional responses that provides a value
judgement. Beautiful actions are good. Truth is both beautiful and
good.
JA 1> my last best attempt to devise a bridge between
Peirce's special sense of "formal" and the more generic brands
we likely know.
Peirce did not have a special sense of 'formal'. His definition of formal logic was identical to De Morgan's, and he applied that term to the logic in Russell's 1903 book. If you need more evidence, look at the 119 occurrences of 'formal logic' in CP. As you examine them, note CP 1.672: "the only hope of salvation lies in formal logic, which demonstrates in the clearest manner that reasoning itself testifies to its own ultimate subordination to sentiment."
That is the antithesis of Carnap: Formal logic is the foundation for exact reasoning, but its goals must be determined by sentiment. You can't derive "ought" from "is" by reason alone. Poetry is essential. So is music. But the best poetry and the best music require elegant mathematical forms to elicit the most moving sentiments.
JA 2> How does a concern with form, or goodness of form, along with the question of what is required to achieve an object, modify our perspective on sign relations in a way that duly marks it as a logical point of view?
Form alone can't do that. As Peirce said, form must be subordinate to sentiment. But that leads to the next question: How can we design our robots in a way that makes their formal reasoning subordinate to good, ethical sentiment?
Many good people haven't been very good at teaching their children to do that. Is there any hope for designing our robots to do that?
John
Jon A, List,
I replaced the subject line with a very specific question.
That question is closely related to the question "How can we raise ethical children? he logical positivists destroyed philosophy by rejecting value judgments. Carnap was a very intelligent, but emotionally stupid positivist whose most damning criticism of any mention of value judgments was "That's poetry!"
Wittgenstein refused to attend any meeting of the Vienna Circle if Carnap was present. An d I don't blame him.
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Jon A, List,
Pure mathematics is the study of pure
form (i.e., diagrammatic reasoning) without any admixture of emotion,
sentiment, or value judgments about Beauty, Goodness. and Truth. Boolean
algebra computes the values 1 and 0. But the assumption that 1 and 0
correspond to what we mean by truth and falsehood belongs to the
*application* of Boolean algebra to an analysis of meaning
(semiotic).
JA 1> I think you are missing Peirce's deeper meanings of ideas like Form (think Platonic Ideas and the way Aristotle compounded Form and Matter).
The Platonic forms are the purest of pure mathematics. Aristotle's theory of form and matter is an application of mathematics to physics. Plato's theory that the existence of the forms is prior to the physical combination is also an applied theory, but with different assumptions about the combination.
JA 2> Peirce stands the normative science of Logic on grounds within the pale of Ethics and fixes the sight of Ethics on the prize Aesthetics picks to steer by.
On this point, we are in complete agreement. For Peirce, formal logic is a branch of pure mathematics. And normative logic is the application of formal logic to semiotic, aesthetics, and ethics. See his 1903 classification of the sciences.
John