- LOGIC, PART 11: THE PREDICATE CALCULUS Chapter 3
Exercises
- Given a propositional function p(w, x, y, z) over a domain of discourse,
U = U, x U, x U, x U,, let a, be a specific element of U, for each i = 1,2, 3,4.
Identify each of the following as either a "proposition" or a "propositional function
in n variables." In the latter, determine n. - Translate into an English sentence each of the following symbolized statements
involving an arbitrary predicate p(x, y) or p(x, y, z):
3. Write a symbolized statement equivalent to the negation of each of parts (a)
through (h) of Exercise 2, in which the negation connective does not occur as a
main connective (i.e., the negation connective should modify the predicate only). - Let U be the set of all people living in the year 1987. Define a propositional
function f in two variables over U (i.e., the domain off equals U x U) by f(x, y):
x is a friend of y. Translate into a good English sentence.
5. (Continuation of 4) (a) Express in symbols the negation of each of parts (a)
through (h) of Exercise 4. (As usual, do not use "not" as a main connective in
your final answer to any part).
(b) Translate each of your symbolized statements in (a) into a good English sen-
tence. Compare each of these translations with its corresponding translation
in Exercise 4.
(c) Write an argument similar to that given in Example 4 to justify (b) of
Theorem 3. - (a) A statement of the form (Vx)(3y)p(x, y) asserts, when true, that to every x
there corresponds at le&t one y for which p(x, y) is true. Express in your own
words the "extra" property that must hold if the stronger (by Theorem 1) state-
ment (3y)(Vx)p(x, y) is also to be true.
(b) In each of parts (i) through (v), use the remark preceding Theorem 2 to deter-
mine which of the two given symbolized statements is stronger: