Unit 6 Answers

1.

1. Addition
2. Simplification
3. Modus ponens
4. Modus tollens
5. Disjunctive syllogism

2.

1. The argument is translated as follows:
   T S
   T C
   S
   ----------------------------------------------------
   C
   
   The inference rules used are:
   From ( T S )   and S
   by modus tollens we deduce T
   From T and ( T C )
   by disjunctive syllogism we conclude C.



2. The argument is translated as follows:
   ( S W )     ( R C )
   -----------------------------------------------------
   ( W C )
   
   The inference rules used are:
   ( S W ) is equivalent to ( S W )
   by implication and De Morgan
   Also ( R C ) is equivalent to ( R C ) by De Morgan.
   Hence ( S W ) ( R C ) is equivalent to ( S W ) ( R C ), which is equivalent to
   ( S R ) ( W C )
   Hence by simplification ( W C )



3. The argument is translated as follows:
   C S
   O L
   O C
   -----------------------------------------------------
   S L
   
   The inference rules used are:
   S C
   by Contrapositive of C S
   C O
   by Contrapositive of O C
   By hypothetical syllogism from the last two
   S O
   By another hypothetical syllogism from this and O L,
   S L
   is obtained.



4. The argument is translated as follows:
   G F
   Z M
   M F
   --------------------------------------------
   Z G
   
   The inference rules used are:
   >From Z M and
   M F
   by hypothetical syllogism
   Z F is obtained.
   Then from G F
   by taking contrapositive M F is obtained.
   >From this and Z F
   by hypothetical syllogism Z G is obtained.



5. The argument is translated as follows:
   R ( H L )
   H ( L R )
   --------------------------------------------
   R
   
   The inference rules used are:
   From H ( L R ) by distributive law
   ( H L ) ( H R )
   From this by simplification
   ( H L )
   From this by De Morgan ( H L )
   With this and R ( H L )
   by disjunctive syllogism R is concluded.