PROPOSITIONAL LOGIC IN-CLASS EXERCISES

PROPOSITIONAL STATEMENTS (modified from Nolt)

Using the abbreviations below, put the following sentences into standard propositional form.

p=Paul is happy.

q=Quigley is happy.

r=Robin is happy.

s=Sally is happy.

(1) Example

Robin is happy or Sally is happy.

(2) Paul is happy, although Quigley is not.

(3) Sally is not happy or Sally is happy.

(4) Paul is happy if Sally is.

(5) Robin is happy if and only if Sally is.

(6) Neither Paul nor Sally is happy.

(7) Robin is not happy if Sally is.

(8) Sally is not happy, but Robin and Quigley are happy.

(9) If Sally is happy, then both Paul and Quigley are happy.

(10) Sally and Quigley are happy, but Paul and Robin are not happy.

(11) Robin is happy only if Paul and Quigley are not happy.

(12) If Sally is happy, then either Quigley or Paul is happy, and if Sally is not happy, then both Paul and Robin are happy.

TRUTH TABLES

(13) Example

Construct a truth table for the following wff:

p & q

p          q          p & q

T          T          T

T          F          F

F          T          F

F          F          F

(14) Construct a truth table for the following wff:

p v q

(15) Seven separate wwfs are lined up on the top of the following table, separated by commas. Beneath each wff, fill in its truth values. Indicate which of these wffs is a tautology, inconsistency, or contingency.

p          q          p → q, p ↔ q, ~p,       ~q,       p & ~p,            p & ~q,            p ↔ p, p ↔ ~q

T          T

T          F

F          T

F          F

p & ~p is an inconsistency, p ↔ p is a tautology, the rest are contingencies

(16) Construct a truth table for the following wff (hint: remember to first write down the truth value of the sub-wff contained in parentheses):

p v (q →  r)

(17) Six separate wwfs are lined up on the top of the following table, separated by commas. Beneath each wff, fill in its truth values.

p          q          r           ~p,       ~q,       (~p v r),           (~q & r),          (~p v r) → (~q &  r)

T          T          T

T          T          F

T          F          T

T          F          F

F          T          T

F          T          F

F          F          T

F          F          F

(18) Construct a truth table for the following argument and indicate why it is or is not valid:

p → q

q

:. p

(2) p & ~q

(3) ~s v s

(4) s → p

(5) r ↔ s

(6) ~p v ~s

(7) s → ~r

(8) ~s & (r & q)

(9) s → (p & q)

(10) (s & r) and ~(p and r)

(11) r → ~(p & q)

(12) [s → (q v p)] & [~s → (~p & ~ r)]

(14)

p          q          p v q

T          T          T

T          F          T

F          T          T

F          F          F

(15)

p          q          p → q, p ↔ q, ~p,       ~q,       p & ~p,            p & ~q,            p ↔ p, p ↔ ~q

T          T          T          T                      F          F          F                      F                      T          F

T          F          F          F                      F          T          F                      T                      T          T

F          T          T          F                      T          F          F                      F                      T          T

F          F          T          T                      T          T          F                      F                      T          F

p & ~p is an inconsistency, p ↔ p is a tautology, the rest are contingencies

(16)

p          q          r           (q → r), p v (q →  r)

T          T          T          T          T

T          T          F          F          T

T          F          T          T          T

T          F          F          T          T

F          T          T          T          T

F          T          F          F          F

F          F          T          T          T

F          F          F          T          T

(17)

p          q          r           ~p,       ~q,       (~p v r),           (~q & r),          (~p v r) → (~q &  r)

T          T          T          F          F          T                      F                                  F

T          T          F          F          F          F                      F                                  T

T          F          T          F          T          T                      T                                  T

T          F          F          F          T          F                      F                                  T

F          T          T          T          F          T                      F                                  F

F          T          F          T          F          T                      F                                  F

F          F          T          T          T          T                      T                                  T

F          F          F          T          T          T                      F                                  F

(18)

p          q          p →q,  q          ˫p

T          T          T         T         T

T          F          F          F          T

F          T          T         T         F

F          F          T          F          F

Invalid: row 3 had true premises but a false conclusion