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Jon AwbreyDifferential Logic and Dynamic Systems • Review and Transition 1 • <a href="https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_1#Review_and_Transition" rel="nofollow noopener noreferrer" target="_blank">https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Part_1#Review_and_Transition</a>This note continues a previous discussion on the problem of dealing with change and diversity in logic-based intelligent systems. It is useful to begin by summarizing essential material from previous reports.Table 1 outlines a notation for propositional calculus based on two types of logical connectives, both of variable \(k\)-ary scope.• A bracketed list of propositional expressions in the form \(\texttt{(} e_1 \texttt{,} e_2 \texttt{,} \ldots \texttt{,} e_{k-1} \texttt{,} e_k \texttt{)}\) indicates that exactly one of the propositions \(e_1, e_2, \ldots, e_{k-1}, e_k\) is false.• A concatenation of propositional expressions in the form \(e_1 ~ e_2 ~ \ldots ~ e_{k-1} ~ e_k\) indicates that all of the propositions \(e_1, e_2, \ldots, e_{k-1}, e_k\) are true, in other words, that their logical conjunction is true.All other propositional connectives can be obtained in a very efficient style of representation through combinations of these two forms. Strictly speaking, the concatenation form is dispensable in light of the bracketed form but it is convenient to maintain it as an abbreviation of more complicated bracket expressions.<a href="https://mathstodon.xyz/tags/Peirce" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#Peirce</a> <a href="https://mathstodon.xyz/tags/Logic" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#Logic</a> <a href="https://mathstodon.xyz/tags/LogicalGraphs" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#LogicalGraphs</a> <a href="https://mathstodon.xyz/tags/DifferentialLogic" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#DifferentialLogic</a> <a href="https://mathstodon.xyz/tags/DynamicSystems" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#DynamicSystems</a> <a href="https://mathstodon.xyz/tags/BooleanFunctions" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#BooleanFunctions</a> <a href="https://mathstodon.xyz/tags/BooleanDifferenceCalculus" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#BooleanDifferenceCalculus</a> <a href="https://mathstodon.xyz/tags/QualitativeChange" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#QualitativeChange</a> <a href="https://mathstodon.xyz/tags/MinimalNegationOperators" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#MinimalNegationOperators</a> <a href="https://mathstodon.xyz/tags/NeuralNetworkSystems" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#NeuralNetworkSystems</a> <a href="https://mathstodon.xyz/tags/Semiotics" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#Semiotics</a>

Jon AwbreyDifferential Logic and Dynamic Systems • Overview • <a href="https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Overview" rel="nofollow noopener noreferrer" target="_blank">https://oeis.org/wiki/Differential_Logic_and_Dynamic_Systems_%E2%80%A2_Overview</a>❝Stand and unfold yourself.❞ — Hamlet • Francisco • 1.1.2In modeling intelligent systems, whether we are trying to understand a natural system or engineer an artificial system, there has long been a tension or trade-off between dynamic paradigms and symbolic paradigms. Dynamic models take their cue from physics, using quantitative measures and differential equations to model the evolution of a system’s state through time. Symbolic models use logical methods to describe systems and their agents in qualitative terms, deriving logical consequences of a system’s description or an agent’s state of information. Logic-based systems have tended to be static in character, largely because we have lacked a proper logical analogue of differential calculus. The work laid out in this report is intended to address that lack.This article develops a differential extension of propositional calculus and applies it to the analysis of dynamic systems whose states are described in qualitative logical terms.The work pursued here is coordinated with a parallel application focusing on neural network systems but the dependencies are arranged to make the present article the main and the more self-contained work, to serve as a conceptual frame and a technical background for the network project.<a href="https://mathstodon.xyz/tags/Peirce" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#Peirce</a> <a href="https://mathstodon.xyz/tags/Logic" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#Logic</a> <a href="https://mathstodon.xyz/tags/LogicalGraphs" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#LogicalGraphs</a> <a href="https://mathstodon.xyz/tags/DifferentialLogic" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#DifferentialLogic</a> <a href="https://mathstodon.xyz/tags/DynamicSystems" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#DynamicSystems</a> <a href="https://mathstodon.xyz/tags/BooleanFunctions" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#BooleanFunctions</a> <a href="https://mathstodon.xyz/tags/BooleanDifferenceCalculus" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#BooleanDifferenceCalculus</a> <a href="https://mathstodon.xyz/tags/QualitativeChange" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#QualitativeChange</a> <a href="https://mathstodon.xyz/tags/MinimalNegationOperators" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#MinimalNegationOperators</a> <a href="https://mathstodon.xyz/tags/NeuralNetworkSystems" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#NeuralNetworkSystems</a> <a href="https://mathstodon.xyz/tags/Semiotics" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#Semiotics</a>

Jon Awbrey<a href="https://mathstodon.xyz/@bblfish" class="u-url mention" rel="nofollow noopener noreferrer" target="_blank">@bblfish</a> <a href="https://fosstodon.org/@josd" class="u-url mention" rel="nofollow noopener noreferrer" target="_blank">@josd</a> <a href="https://social.logilab.org/@semwebpro" class="u-url mention" rel="nofollow noopener noreferrer" target="_blank">@semwebpro</a> <a href="https://scholar.social/@hochstenbach" class="u-url mention" rel="nofollow noopener noreferrer" target="_blank">@hochstenbach</a> One thing I found out early on is how critical it is to get <a href="https://mathstodon.xyz/tags/AlphaGraphs" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#AlphaGraphs</a> (<a href="https://mathstodon.xyz/tags/BooleanFunctions" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#BooleanFunctions</a>, <a href="https://mathstodon.xyz/tags/PropositionalCalculus" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#PropositionalCalculus</a>, <a href="https://mathstodon.xyz/tags/ZerothOrderLogic" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#ZerothOrderLogic</a>) down tight. If you do that it changes how you view <a href="https://mathstodon.xyz/tags/FOL" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#FOL</a> (<a href="https://mathstodon.xyz/tags/PredicateCalculus" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#PredicateCalculus</a>, <a href="https://mathstodon.xyz/tags/QuantificationalLogic" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#QuantificationalLogic</a>). That tends to rub people who view FOL as <a href="https://mathstodon.xyz/tags/GOL" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#GOL</a> (<a href="https://mathstodon.xyz/tags/GodsOwnLogic" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#GodsOwnLogic</a>) the wrong way so you have watch out for that if you go down this road.Here's a primer on \(\alpha\) <a href="https://mathstodon.xyz/tags/LogicalGraphs" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#LogicalGraphs</a> as I see them — • <a href="https://oeis.org/w/index.php?title=Logical_Graphs&stable=0&redirect=no" rel="nofollow noopener noreferrer" target="_blank">https://oeis.org/w/index.php?title=Logical_Graphs&stable=0&redirect=no</a>

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