Bool introduced integer "values" for logical values.

He defined True to be 1 and False to be 0.

Then, Bode found the relations:

X⋅(1-Y)=0 All X are Y
X⋅Y=0 No X are Y
v=X⋅Y Some X are Y
v=X⋅(1-Y) Some X are not Y
X⋅(1-X)=1 Contradiction

See also Curry-Howard Isomorphism (1980).

See also Isabelle Pro (Paulson, 1993).