Hi Jon, about Lobe Connective and Node Connective and their consequences, I have a question? You say that genus species are evaluated by the proposition (a,(b),(c)). the following preposition would no longer be appropriate: a (b,c). And another question about differential calculus: when we talk about A and dA we talk about A and (A) or is it more similar to A and B? thanks in advance Mauro
On Tue, 15 Jun 2021 at 22:56, Jon Awbrey <[email protected]> wrote: > Cf: Differential Logic • 2 > https://inquiryintoinquiry.com/2020/03/23/differential-logic-2/ > > Cactus Language for Propositional Logic > ======================================= > > https://oeis.org/wiki/Differential_Logic_%E2%80%A2_Part_1#Cactus_Language_for_Propositional_Logic > > The development of differential logic is facilitated by having a moderately > efficient calculus in place at the level of boolean-valued functions and > elementary logical propositions. One very efficient calculus on both > conceptual and computational grounds is based on just two types of > logical connectives, both of variable k-ary scope. The syntactic > formulas of this calculus map into a family of graph-theoretic > structures called “painted and rooted cacti” which lend visual > representation to the functional structures of propositions > and smooth the path to efficient computation. > > The first kind of connective takes the form of a parenthesized sequence > of propositional expressions, written (e₁, e₂, …, eₖ) and meaning exactly > one of the propositions e₁, e₂, …, eₖ is false, in short, their “minimal > negation” is true. An expression of this form maps into a cactus structure > called a “lobe”, in this case, “painted” with the colors e₁, e₂, …, eₖ as > shown below. > > Figure 1. Lobe Connective > > https://inquiryintoinquiry.files.wordpress.com/2020/03/cactus-ej-lobe-connective.jpg > > The second kind of connective is a concatenated sequence of propositional > expressions, > written e₁ e₂ … eₖ and meaning all of the propositions e₁, e₂, …, eₖ are > true, in short, > their logical conjunction is true. An expression of this form maps into a > cactus structure > called a “node”, in this case, “painted” with the colors e_1, e_2, ..., > e_k as shown below. > > Figure 2. Node Connective > > https://inquiryintoinquiry.files.wordpress.com/2020/03/cactus-ej-node-connective.jpg > > All other propositional connectives can be obtained through combinations > of these two forms. As it happens, the parenthesized form is sufficient > to define the concatenated form, making the latter formally dispensable, > but it's convenient to maintain it as a concise way of expressing more > complicated combinations of parenthesized forms. While working with > expressions solely in propositional calculus, it's easiest to use > plain parentheses for logical connectives. In contexts where > ordinary parentheses are needed for other purposes an alternate > typeface (...) may be used for the logical operators. > > References > [1] https://oeis.org/wiki/Boolean-valued_function > [2] https://oeis.org/wiki/Minimal_negation_operator > [3] https://oeis.org/wiki/Logical_conjunction > _ _ _ _ _ _ _ _ _ _ > ► PEIRCE-L subscribers: Click on "Reply List" or "Reply All" to REPLY ON > PEIRCE-L to this message. PEIRCE-L posts should go to > [email protected] . > ► To UNSUBSCRIBE, send a message NOT to PEIRCE-L but to > [email protected] with UNSUBSCRIBE PEIRCE-L in the SUBJECT LINE of the > message and nothing in the body. More at > https://list.iupui.edu/sympa/help/user-signoff.html . > ► PEIRCE-L is owned by THE PEIRCE GROUP; moderated by Gary Richmond; and > co-managed by him and Ben Udell. > -- "[..] events are primarily linguistic or cognitive in nature. That is, the world does not really contain events. Rather, events are the way by which agents classify certain useful and relevant patterns of change." Allen and Fergusson "No, no. *History of Eternity*. At first I wanted to find every single one of the buyers to apologize because of the book and also to thank them for what they had done. There is an explanation for that. If you think of thirty-seven people—those people are real, I mean every one of them has a face of his own, a family, he lives on his own particular street. Why, if you sell, say two thousand copies, it is the same thing as if you had sold nothing at all because two thousand is too vast—I mean, for the imagination to grasp. While thirty-seven people—perhaps thirty-seven are too many, perhaps seventeen would have been better or even seven—but still thirty-seven are still within the scope of one's imagination." http://www.theparisreview.org/interviews/4331/the-art-of-fiction-no-39-jorge-luis-borges
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