popper-logic-scientific-discovery

zero
dimensional
classes 1
one
dimensional
classes
two
dimensional
classes
three
dimensional
classes
four
dimensional
classes
— — straight line circle parabola
—
straight line
through two
given points
circle
through
three given
points
straight line
through one
given point
circle
through two
given points
parabola
through
three given
points
circle
through one
given point
parabola
through two
given points
conic
through
three given
points
degrees of testability 117
parabola
through one
given point
conic
through two
given points
conic
through one
given point
of ellipses with given ratio of the axes is four-dimensional (as is that of
parabolas), and so is the set of ellipses with given numerical eccentricity.
The transition from the ellipse to the circle, of course, is equivalent
to specifying an eccentricity (the eccentricity 0) or a particular
ratio of the axes (unity).
As we are interested in assessing degrees of falsifiability of theories
we will now ask whether the various methods of reducing the number
of dimensions are equivalent for our purposes, or whether we should
examine more closely their relative merits. Now the stipulation that a
curve should pass through a certain singular point (or small region) will
often be linked up with, or correspond to, the acceptance of a certain
singular statement, i.e. of an initial condition. On the other hand, the
transition from, say, an ellipse-hypothesis to a circle-hypothesis, will
obviously correspond to a reduction of the dimension of the theory itself.
—
— —
1 We could also, of course, begin with the empty (over-determined) minus-onedimensional
class.