Computability and Logic

PROBLEMS 269
if it is preceded by ∼. Show that if A implies C, and neither ∼A nor C is valid,
then there is another such sentence B such that: (i) A implies B; (ii) B implies
C; (iii) any predicate occurs positively in B only if it occurs positively in both
A and C, and occurs negatively in B if and only if it occurs negatively in both
A and C.
20.2 Give an example to show that Lyndon’s theorem does not hold if constants are
present.
20.3 (Kant’s theorem on the indefinability of chirality). For points in the plane, we
say y is between x and z if the three points lie on a straight line and y is between
x and z on that line. We say w and x and y and z are equidistant if the distance
from w to x and the distance from y to z are the same. We say x and y and z form
a right-handed triple if no two distances between different pairs of them are the
same, and traversing the shortest side, then the middle side, then the longest side
of the triangle having them as vertices takes one around the triangle clockwise,
as on the right in Figure 20-1.
Figure 20-1. Right and left handed triangles.
Show that right-handedness cannot be defined in terms of betweenness and
equidistance. (More formally, consider the language with a three-place predicate
P, a four-place predicate Q, and a three-place predicate R; consider the
interpretation whose domain is the set of the points in the plane and that assigns
betweenness and equidistance and right-handedness as the denotations of P and
Q and R; and finally consider the theory T whose theorems are all the sentences
of the language that come out true under this interpretation. Show that R is not
definable in terms of P and Q in this theory.)