Advanced Symbolic Logic, PL 330                               Homework # 2 Solution

Page 4, problem # 1 First part, symbolizing statement # 2 on p 32

pq q r. st  Now apply the appropriate truth values as supplied on p 40

T^q^.    ^T  and resolve

^ q ^. ^

^ ^

T

Page 45, # 1, upper left

p q. q. q p

p q. q. q p

Make p true   Make p false
p q. q. q

p

  p q. q. q

p

T q. q. q T   ^ q. q. q ^
    q q   T       T   q    
      T               T      

Since both sides of the truth value analysis resolve to T, the schema is valid

 

upper right hand part

p q qr q. . q

 

p q qr q. . q

T

q

q

r

q.

^.

q

 

^

q

q

r

q.

T.

q

 

T

 

 

q

^

 

 

 

 

 

 

q

r

q.

q

 

 

 

 

 

T

 

 

 

 

 

Tr

q.

^

q

^r

q.

T

q 

 

 

 

 

 

 

 

 

 

 

r

q

 

 

^

q

T

 

 

 

 

 

 

 

 

 

 

 

 

T

 

 

 

 

T

 

 

All branches resolve to T, so the schema is valid

 

lower right hand part

p

q

.q.

q

r

.q.

p

r

 

 

 

 

 

 

 

 

 

 

 

 

 

p

q

.q.

q

r

.q.

p

r

 

 

 

p

q

.q.

q

r

.q.

p

r

T

q

.q.

q

r

.q.

T

r

 

 

 

^

q

.q.

q

r

.q.

^

r

 

 

q

.q.

q

r

.q.

r

 

 

 

 

 

 

 

.q.

q

r

.q.

 

T

q.

T

r

.q

r

^

q.

^

r.

q

r

^

.q.

T↔r

.q.

T

.q.

^↔r

.q.

 

T

 

 

 

 

 

 

 

 

 

q

r

 

 

r

 

T

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

T

 

 

 

 

T

 

 

 

 

 

 

All branches resolve to true, the schema is valid

 

 

 

 

 

 

 

 

 

 

Page 42, # 3

Yes, a schema that is consistent but not valid can, by appropriate substitutions be transformed into both a valid schema and an inconsistent schema.  Since alternation and negation together are sufficient to express any truth functional schema, first transform the original consistent but not valid schema into an alternation of literals and negations of literals.  To create a valid schema by substitution, replace one altern with a patently valid one.  To create an inconsistent schema by substitution, replace each altern with an inconsistent one.

 

Truth Value Analysis for  -(pq):qr..  v

-(pq):qr..  v

 

 

-(Tq):

q

r.

↔.

^

 

 

-(^q):

q

r.

↔.

T

 

 

:

q

r.

↔.

 

 

 

 

T:

q

r.

↔.

T

 

 

^:

T

r.

T:

^

r.

↔.

 

 

 

q

r

 

 

^

T→ r

 

^ → r

 

 

 

 

 

 

 

 

^

T

 

 

r

 

 

T

 

 

 

 

 

 

 

 

 

 

 

 

 

 

T

^

 

 

 

 

 

Consistent but not valid