(1) There
are three people in the story: The sister, her nephew and Doc Brady
(2) Someone
in the story sent the letter to the sister
(3) Anyone
who sent the letter to the sister had a motive
(4) But the
nephew didn’t have a motive—and neither did Doc Brady
:. The
sister send the letter to herself
Let: s = the sister n = the nephew d = Doc
Brady
Sx = x is in the story
Exy = x sent the letter to y
Mx = x had a motive
Represent
the argument and prove the conclusion.
(1) (Ss &
Sn & Sd) & (∀x)(Sx → (x=s v x=n v x=d)
(2) (∃x)(Sx
& Exs)
(3) (∀x)[(Exs
→ Mx]
(4) ~Mn &
~Md
:. Ess
(5) Sa* &
Ea*s 2,
∃-Elimination
(6) Sa* 5,
&-Elimination
(7) (∀x)(Sx →
(x=s v x=n v x=d) 1,
&-Elimination
(8) Sa* →
(a*=s v a*=n v a*=d) 7,
∀-Elimination
(9) (a*=s v
a*=n v a*=d) 6, 8
Modus Ponens
(10) a*=n Assumption
for RAA
(11) Ea*s 5,
&-Elimination
(12) Ens 10, 11
=Elimination
(13) Ens
→ Mn 3, ∀-Elimination
(14) Mn 12,
13 Modus Ponens
(15) ~Mn 4,
&-Elimination
(16) Mn
& ~Mn 14,
15 &-Introduction
(17) ~(a*=n) 10-16
RAA
(18) (a*=s v
a*=d) 9, 17 Disjunctive Syllogism
(19) a*
= d Assumption
for RAA
(20) Ea*s 5,
&-Elimination
(21) Eds 19, 20
=Elimination
(22) Eds
→ Md 3, ∀-Elimination
(23) Md 21,
22 Modus Ponens
(24) ~Md 4,
&-Elimination
(25) Md
& ~Md 23,
24 &-Introduction
(26) ~(a*=d) 19-25
RAA
(27) a*=s 18,
26 Disjunctive Syllogism
(28) Ess 11,
27 =Elimination
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