PROPOSITIONAL LOGIC HOMEWORK

 

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

If Paul is happy, then so is Quigley.

Answer: p → q

 

(2) If Robin is happy if and only if Robin is happy.

 

(3) Sally is happy only when Paul and Quigly are unhappy.

 

(4) Robin is happy only if Quigley is.

 

(5) Robin and Quigley are not both happy.

 

(6) Either Robin is not happy or Sally is not happy.

 

(7) Either Paul is happy, or Robin and Quigley are happy.

 

(8) If neither Sally nor Quigley is happy, then Robin is happy.

 

(9) If Sally is happy, then if Paul is not happy, Robin is not happy.

 

(10) Robin or Sally is happy only if Paul and Quigley are happy.

 

(11) If either Paul or Quigley is happy, then Robin is happy and Sally is not happy.

 

(12) If either Quigley or Paul is happy then sally is not happy, and if Sally is not happy, then Robin is happy or Paul is not 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          T

F          F          T

 

(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 v ~q, ~p v ~q,           p v ~p  ~p →  q,          p ↔  ~q

T          T         

T          F         

F          T         

F          F         

 

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

~p

:. ~q