Spring 2020 PHIL 279 Test 2 Name _____________________________________

This test is due in Dropbox by Friday, May 29 at 11:00 pm. Please submit a single PDF or Word file.

Part I. All Natural Deductions are to be done using only the 12 rules from the Natural Deductions handout.
Number each line and justify each line. Justify all initial assumptions. Justify all additional assumptions by
indicating the ND rule that you intend to apply to the sub-derivation created by that assumption, e.g., A/ ⸧ I.
(Note: You will get part marks for properly setting up your ND and for making the appropriate additional
assumptions that indicate your strategy.) (15)

i. Provide a Natural Deduction to establish that the following is a valid sentence. This ND can
be done in 9 lines. (3)
[( M   L) & ( L  M)]  ( L ≡  M)

ii. Provide a Natural Deduction to establish that the following argument is valid. This ND can
be done in 13 lines. (3)
(A &  B), ( B  C)  (D   E), D   F,  F ≡  E ⸫ (G ≡ H)   F

iii. Provide a Natural Deduction to establish that the following sentences are equivalent. The NDs
can be done in 22 lines. (3)
A & ( B  C) (A & B)  (A & C)

iv. Provide a Natural Deduction to establish that the following set is inconsistent. This ND can be
done in 18 lines. (3)
{(M  N)  (G &  C), (G  N)  C,  N  M}


v. Provide a Natural Deduction to establish that the following set is inconsistent. This ND can be
done in 16 lines. (3)
{M   N,  M   L, ( M  N) & L}

Part II

i. ND proof procedure claims that if Γ ⊢ P then Γ ⊨ P. Carefully explain what this claim means.
(In your explanation, you’ll need to clarify the difference between ⊢ and ⊨.) (1)

ii. Suppose that there is a proof procedure ND* that had the following alternative disjunction
elimination rule (E*):
m P  Q


P E* m
Provide a natural deduction in ND* that’s two lines long and explain why this two line ND*
would illustrate why with ND* the claim that if Γ ⊢ P then Γ ⊨ P would be false.


iii. Provide an explanation for why, according to the ND proof procedure, does a ND of the
following form establish P to be unsatisfiable? (‘m’ is a line number. Note:  is unsatisfiable.)
(1)
1 P A


m 

iv. Suppose you have a ND of the following form, where P, Q, R, S, T, V, W, Z, X,
Y and Z are sentences of TFL.


P
Q_____
R__
S
T
V
W
X
Y
Z

According to the ND proof procedure, this ND would establish the following four entailment
claims. (Note: S, X, Y, Z in the following entailment claims are from the above ND.)

Γ1 ⊨ S Γ2 ⊨ X Γ3 ⊨ Y Γ4 ⊨ Z

Identify these four sets, e.g., {…}, by identifying the sentences in these sets, e.g., {Y, Z}. (1)

Γ1 =
Γ2 =
Γ3 =
Γ4 =
v. If Q is derived along the main scope line from a set of initial assumptions {P}, then ND
establishes that there is an interpretation in which both P and Q are true. (Indicate whether this
claim is true or false, e.g., circle the correct answer) (.5)
True False

vi. According to the ND proof procedure, every sentence of TFL can be derived along the main
scope line from a set of initial assumptions {}, i.e., the empty set. (Indicate whether this claim is
true or false.) (.5)
True False