## Differential Logic • Discussion 5

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 $A ~ \mathrm{d}A.$

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