hurley truth tree answers full

Answers to Truth Tree Exercises in the Hurley Truth Tree
Supplement
Exercise Sets T-1, T-2, T-3
N.B. The answers contained herein use the typewriter symbols for the SL operators.


EXERCISE T-2
“Use truth trees to determine if each of the following groups of statements
is consistent or inconsistent. If consistent, identify one set of truth values
that makes the group of statements true. Then, test your answer by
entering these truth values into the statements of that group and proceed
to prove that each statement turns out true.”

Set T-2, # 1
«1. M  ~D, K v D, M  K

Conversion: { ( M > ~ D ), ( K v D ), ( M & K ) }

(M>~D)&(KvD)&(M&K)
M > ~ D
K v D
M & K
M
K
/ \
~M ~D
X / \
K D
open X

One open branch: Open Truth Tree: Wffs are consistent: K = T, D = F, M = T




1

,Set T-2, # 2
2. S  R, S  G, ~ R v ~ G

Conversion: { ( S & R ), ( S = G ), ( ~ R v ~ G ) }

( S & R ) & ( S = G ) & ( ~ R v ~ G )
S & R
S = G
~ R v ~ G
S
R
/ \
~R ~G
X / \
S ~S
G ~G
X X


All branches are closed: Closed Truth Tree: This set of wffs is inconsistent.


Set T-2, # 3
3. C  (N  ~ H) , N  ~ C, C v H

Conversion: { [ C > ( N & ~ H )], ( N > ~ C ), ( C v H ) }


[ C > (N & ~ H) ] & ( N > ~ C ) & ( C v H )
C > (N & ~ H)
N > ~ C
C v H
/ \
~C N & ~ H
/ \ N
/ \ ~H
/ \ / \
~N ~C ~N ~C
/ \ / \ X / \
C H C H C H
X open1 X open2 X X


Two branches are open: Open Truth Tree: This set of wffs is consistent: open
branch 1: H = T, C =F, N =F; open branch 2: H = T, C = F, N = T/F.


2

,Set T-2, # 4
«4. (P  ~ B)  (Q v D), ~ (P v Q), B  D

Conversion: { [ ( P & ~ B ) = ( Q v D ) ], ~ ( P v Q ), ( B > D )}


1. [ (P & ~ B ) = ( Q v D ) ] & ~ ( P v Q ) & ( B > D )
2. (P & ~ B ) = ( Q v D ) 
3. ~ (P v Q) 
4. (B > D)
5. ~P
6. ~ Q
/ \
7. P & ~ B  ~ (P & ~ B) 
8. QvD ~ (Q v D) 
9. P ~Q
10. ~B ~D
11. X / \
12. ~P ~~ B
13. / \ B
14. ~B D / \
15. open X ~B D
16. X X


One open branch: Open truth tree: The set of wffs is consistent.

Commentary: There is one open branch in line 15. Note that in line 11, that branch was
closed even though the wff (Q v D) was not decomposed. This is permissible because once
one can close a branch, one should close it, even if there are remaining wffs that may be
decomposed. The one open branch indicates that this set of wffs is consistent when “B” is
false (F), “P” is false (F), “D” is false (F), “Q” is false (F)




3

, Set T-2, # 5
5. (S  C)  (E v G), S  C, E  G, S  ~ G

Conversion: { [ (S & C ) = ( E v G ) ], ( S > C ), (E > G ), ( S & ~ G ) }


[ (S & C ) = ( E v G ) ] & ( S > C ) & (E > G ) & ( S & ~ G ) 
(S & C ) = ( E v G ) 
S>C
E>G
S&~G
S
~G
/ \
~S C
X / \
~E G
/ \ X

S&C  ~ ( S & C )
EvG ~(EvG)
S ~E
C ~G
/ \ / \
E G ~S ~C
X X X X


All branches are closed: Closed Truth Tree: This set of wffs is inconsistent.




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