Differential Logic • Discussion 5

Posted on June 17, 2021 by Jon Awbrey

Re: Differential Logic • Discussion 4
Re: Laws of Form • Lyle Anderson

JA:
The differential proposition \mathrm{d}A is one we use to describe a change of state
(or a state of change) from A to \texttt{(} A \texttt{)} or the reverse.
LA:
Does this mean that if A is the proposition “The sky is blue”, then \mathrm{d}A would be the statement “The sky is not blue”?  Don’t you already have a notation for this in A and \texttt{(} A \texttt{)} \, ?  From where does “state” and “change of state” come in relation to a proposition?

Dear Lyle,

The differential variable \mathrm{d}A : X \to \mathbb{B} is a derivative variable, a qualitative analogue of a velocity vector in the quantitative realm.

Let’s say x \in \mathbb{R} is a real value giving the membrane potential in a particular segment of a nerve cell’s axon and A : \mathbb{R} \to \mathbb{B} is a categorical variable predicating whether the site is in the activated state, A(x) = 1, or not, A(x) = 0.  We observe the site at discrete intervals, a few milliseconds apart, and obtain the following data.

  • At time t_1 the site is in a resting state, A(x) = 0.
  • At time t_2 the site is in an active state, A(x) = 1.
  • At time t_3 the site is in a resting state, A(x) = 0.

On current information we have no way of predicting the state at time t_2 from the state at time t_1 but we know action potentials are inherently transient so we can fairly well guess the state of change at time t_2 is \mathrm{d}A = 1, in other words, about to be changing from A to \texttt{(} A \texttt{)}.  The site’s qualitative “position” and “velocity” at time t_2 can now be described by means of the compound proposition A ~ \mathrm{d}A.

Resources

cc: Cybernetics • Ontolog • Peirce (1) (2) (3) (4) • Structural Modeling • Systems Science
cc: FB | Differential Logic • Laws of Form

This entry was posted in Amphecks, Animata, Boolean Algebra, Boolean Functions, C.S. Peirce, Cactus Graphs, Category Theory, Change, Cybernetics, Differential Analytic Turing Automata, Differential Calculus, Differential Logic, Discrete Dynamics, Equational Inference, Frankl Conjecture, Functional Logic, Gradient Descent, Graph Theory, Hologrammautomaton, Indicator Functions, Inquiry Driven Systems, Leibniz, Logic, Logical Graphs, Mathematics, Minimal Negation Operators, Painted Cacti, Peirce, Propositional Calculus, Surveys, Time, Topology, Visualization, Zeroth Order Logic and tagged , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , . Bookmark the permalink.

1 Response to Differential Logic • Discussion 5

  1. Pingback: Survey of Differential Logic • 3 | Inquiry Into Inquiry

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