## The Difference That Makes A Difference That Peirce Makes : 15

Re: Peirce List Discussion • JAGF

One could hardly dispute the importance of logical implication relations like $A \Rightarrow B.$  Their set-theoretic analogues are subset relations like $A \subseteq B,$ which are almost the canonical way of expressing constraint, determination, information, and so on.  There is moreover a deep analogy or isomorphism between propositions like $A \Rightarrow B$ and functional types like $A \to B$ of considerable importance in the theory of computation.  That is probably enough to earn implications a primary and fundamental status but there are several reasons we might stop short of claiming these order relations are exclusively primary and fundamental.

For one thing, implication in existential graphs is expressed in a compound form, as $\texttt{(} A \texttt{(} B \texttt{))},$ “not A without B”.  For another, there is Peirce’s own discovery of the amphecks, the logical connectives expressed by “not both” and “both not”, respectively, which appear to have a primary and fundamental status all their own.  Lastly, implicational inferences are in general information-losing while the fundamental operations in Peirce’s logical graphs, either entitative or existential, give us the option of equational rules of inference, that is, information-preserving steps.

Just a few things to think about …

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