Squares of opposition

The traditional Aristotelian square of opposition is like that of first-order logic apart from existential import:

contradictory \begin{bmatrix} \forall x Px & & \forall x \neg Px = \neg \exists xPx\\ & implies & \\ \neg \forall x Px & & \neg \forall x \neg Px = \exists xPx \end{bmatrix} \newline \newline \indent \indent \indent \indent \indent \indent \indent \indent contrary

Or in words:

contradictory \begin{bmatrix} for\; all \; x:P(x) & & for\;no\;x:P(x)\\ & implies & \\ for\;not\; all \; x:P(x) & & for\;some\;x:P(x) \end{bmatrix} \newline \newline \indent \indent \indent \indent \indent \indent \indent \indent \indent \indent \indent \indent contrary

Outer negation is the contradictory, i.e., affirm/deny, and inner negation is the contrary, i.e., all/none. For quantifiers (or other operators) there is a duality square:

outer \; negation \begin{bmatrix} Q & & Q \sim \\ & dual & \\ \ \sim Q & & \sim Q \sim \end{bmatrix} \newline \newline \indent \indent \indent \indent \indent \indent \indent inner \; negation

Outer negation is negation of the whole quantifier; inner negation is negation within the quantifier range. For rates the inverse acts like a negation (with appropriate treatment of zero):

outer \; inverse \begin{bmatrix} \frac{\mathrm{d}x(t) }{\mathrm{d} t}= \left ( \frac{\mathrm{d} t}{\mathrm{d} x(t)} \right )^{-1} & & \frac{\mathrm{d} t}{\mathrm{d} x(t)}=\left ( \frac{\mathrm{d} x(t)}{\mathrm{d} t} \right )^{-1} \\ & dual & \\ \ \frac{\mathrm{d} s}{\mathrm{d} z(s)} =\left ( \frac{\mathrm{d} z(s)}{\mathrm{d} s} \right )^{-1} & & \frac{\mathrm{d}z(s) }{\mathrm{d} s} =\left (\frac{\mathrm{d}s}{\mathrm{d} z(s)} \right )^{-1}\end{bmatrix} \newline \newline \indent \indent \indent \indent \indent \indent \indent \indent \indent \indent inner \;inverse

The outer inverse inverts the dependency, i.e., the independent variable switches between the denominator and the numerator, and the inner inverse inverts the rate, i.e., takes the reciprocal.

For vector rates the inverses are then:

outer \; inverse \begin{bmatrix} \frac{\mathrm{d}\mathbf{x}(t) }{\mathrm{d} t}= \left ( \frac{\mathrm{d} t}{\mathrm{d} \mathbf{x}(t)} \right )^{-1} & & \frac{\mathrm{d} t}{\mathrm{d} \mathbf{x}(t)}=\left ( \frac{\mathrm{d} \mathbf{x}(t)}{\mathrm{d} t} \right )^{-1} \\ & dual & \\ \ \frac{\mathrm{d} s}{\mathrm{d} \mathbf{z}(s)} =\left ( \frac{\mathrm{d} \mathbf{z}(s)}{\mathrm{d} s} \right )^{-1} & & \frac{\mathrm{d}\mathbf{z}(s) }{\mathrm{d} s} =\left (\frac{\mathrm{d}s}{\mathrm{d} \mathbf{z}(s)} \right )^{-1}\end{bmatrix} \newline \newline \indent \indent \indent \indent \indent \indent \indent \indent \indent \indent inner \;inverse

As an example, rates of motion form a square of opposition:

outer \; inverse\begin{bmatrix} speed & & harmonic\; pace \\ & dual & \\ \ harmonic\; speed & & pace \end{bmatrix} \newline \newline \indent \indent \indent \indent \indent \indent \indent \indent \indent \; \; inner \; inverse