Advanced Symbolic Logic, PL 330

Homework set # 3, partial answer key

 

Page 52, # 2

p↔ qr . → pq DOES NOT imply p → qr. ↔ pq

p↔ qr . → pq : → : p → qr. ↔ pq

p

qr.

pq:

→:

p

qr.

pq

p

qr.

pq:

→:

p

qr.

pq

T

qr.

Tq:

→:

T

qr.

Tq

^

qr.

^q

:→:

^

qr.

^q

 

 

qr

q

.→.

qr

q.

 

 

 

 

-(qr)

^.

→.

 

T

^

 

Tr

T

..

Tr

T

 

 

 

 

 

 

 

 

qr

^

 

 

 

 

 

 

T

r

 

 

 

 

 

 

 

 

 

 

 

-(qr)

 

 

 

 

 

 

 

 

T

 

^

 

 

 

 

 

 

 

-(Tr)

 

 

 

 

 

 

 

-(^r)

 

 

 

 

 

 

 

 

 

 

 

 

 

-r

 

 

 

 

 

 

 

T

 

 

 

 

 

 

 

 

 

 

 

 

^

T

 

 

 

 

 

 

 

 

 

 

BUT p → qr. ↔ pq does imply p↔ qr . → pq

p → qr. ↔ pq :: p↔ qr . → pq

p

qr.

pq

:→:

p

qr.

pq

p

qr.

pq

:→:

p

qr.

pq

T

qr.

Tq

:→:

T

qr.

Tq

^

qr.

^q

:→:

^

qr.

^q

 

 

qr

q

.→.

qr

q.

 

 

 

 

T

^

:→:

qr

 

 

 

 

 

 

 

 

 

T

 

 

 

 

 

 

 

 

^

qr

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

T

 

 

 

 

 

 

 

Page 52, # 5

p implies  as shown by fell swoop

p..

TT.. ^^

               T

 

p implies pqq as shown by fell swoop

p..pqq

TT..Tq^

             T

 

 implies pqq as shown by fell swoop

..pqq

T^ ..^q^

    ^         ^

             T

 

 does not imply p

 

 .. p

^q..T                                 Tq..^

                                                                                             q^

                                                       T                                                

                                                                                                          ^ T

 

pqq does not imply  as shown by fell swoop

pqq..

TqT..^T

   T        ^

           ^

 

pqq does not imply p

pqq.. p

TqT..T^

    T    ^

          ^

 

Page 52 # 7

p ↔ qr DOES NOT IMPLY q → pr. →

p ↔ qr :: q → pr. →

T

qr

:→:

q

Tr.

^

^

qr

→:

q

^r.

T

 

 

qr

:→:

q

r.

q

 

 

 

 

-(qr)

→:

.

 

 

 

 

Patently valid

 

 

 

 

-(Tr)

→:

^.

^

 

 

 

 

 

 

 

 

 

 

 

 

 

 

^

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

r

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

T

 

^

 

 

 

 

 

 

But q → pr. →  DOES IMPLY p ↔ qr

q → pr. →  :→: p ↔ qr

T

pr.

^

:→:

p

Tr

^

pr.

T

:→:

p

^r

 

 

pr.

T

 

 

:→:

p

r

 

T.

 

 

 

.→

 

 

 

 

Patently valid

 

 

 

 

 

 

.→

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Patently valid