Jerry, We already have a universal foundation for logic. It's called "Peirce's semiotic".
JLRC
the mathematics of the continuous can not be the same as the mathematics of the discrete. Nor can the mathematics of the discrete become the mathematics of the continuous.
They are all subsets of what mathematicians say in natural languages. In Wittgenstein's terms, they are "language games" that mathematicians play with a subset of NL semantics. It's irrelevant whether they use special symbols or words like 'set', 'integral', 'derivative' ... For that matter, chess, go, and bridge are just as mathematical as any other branch of mathematics. They have different language games, but nobody worries about unifying them with algebra or topology. I believe that Richard Montague was half right: RM, Universal Grammar (1970).
There is in my opinion no important theoretical difference between natural languages and the artificial languages of logicians; indeed, I consider it possible to comprehend the syntax and semantics of both kinds of languages within a single natural and mathematically precise theory.
But Peirce would say that NL semantics is a more general version of semiotic. Every version of formal logic is a disciplined subset of NL (ie, one of Wittgenstein's language games). JLRC
I am simply saying that the thought processes of the scientific community (and my thought processes) did not stop on April 19, 1914.
Peirce would certainly agree. He said that building on the foundations he laid "would be a labor for generations of analysts, not for one" (MS 478). The 20th c logicians who ignored Peirce were on the wrong track. Many of them haven't yet reached the 14th c. Peirce was far ahead of the 20th c because he did his homework. JLRC
For a review of recent advances in logic, see http://www.jyb-logic.org/Universallogic13-bsl-sept.pdf, 13 QUESTIONS ABOUT UNIVERSAL LOGIC.
Thanks for the reference. On page 134, Béziau makes the following point, and Peirce would agree:
Universal logic is not a logic but a general theory of different logics. This general theory is no more a logic itself than is meteorology a cloud.
JYB, p. 137
we argue against any reduction of logic to algebra, since logical structures are differing from algebraic ones and cannot be reduced to them. Universal logic is not universal algebra.
Peirce would agree. JYB, 138
Universal logic takes the notion of structure as a starting point; but what is a structure?
Peirce's answer: a diagram. Mathematics is necessary reasoning, and all necessary reasoning involves (1) constructing a diagram (the creative part) and (2) examining the diagram (observation supplemented with some routine computation). What is a diagram? Answer: an icon that has some structural similarity (homomorphism) to the subject matter. JYB, 138
structuralism as we understand it is something still larger that includes linguistics, mathematics, psychology, and so on... what concerns us are not so much historical and sociological considerations about the development of structuralism, but rather the issue of the ultimate view of structuralism as underlying mathematical structuralism and universal logic.
If you replace 'structuralism' with 'diagrammatic reasoning', Peirce would agree. JYB, 145
Some wanted to go further and out of the formal framework, namely those working in informal logic or the theory of argumentation. The trouble is that one runs the risk of being tied up again in natural language.
See my comment above about Montague, Wittgenstein, and Peirce. Universal logic (diagrammatic reasoning) is *independent of* any language or notation. The differences between the many variants are the result of drawing different kinds of diagrams for sets, continua, quantum mechanics, etc. (Note Feynman diagrams.) Whatever the reasoning stuff may be, it would support NL-like reasoning as a more general version of the 20th c kinds of logic. I develop these points further in the following lecture on Peirce's natural logic: http://www.jfsowa.com/talks/natlogP.pdf See also "Five questions on epistemic logic" and the references cited there: http://www.jfsowa.com/pubs/5qelogic.pdf John
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