Columbo:
You see, we got our poison ivy in the same place. We both touched the slide.
You touched it when you picked up the microscope and hit him. That's when the
slide broke. I got it when I put my hand on the floor and it touched a piece of
glass. I remember because I said: "Fellows, feels like there's broken
glass here." The fingerprint man, he thought the glass came from a
drinking glass.
Viveca:
Wow, very good, Lieutenant.
Let:
Hx = x’s hands are itching
Txy = x touched y
Px = poison ivy is on x
Kx = x is the killer
Ix =
x is investigating the killing
c = Columbo s = the microscope-slide
v = Viveca
Represent
the following argument and prove the conclusion:
(1)
Anyone whose hands are itching touched something on which there was poison ivy.
(2)
Poison ivy was on the microscope-slide—and only the microscope slide.
(3)
Anyone who touched the microscope-slide is either the killer or investigating
the killing.
(4)
Columbo and Viveca’s hands are itching.
(5)
Viveca is not investigating the killing.
∴
Viveca is the killer
Solution:
(1) (∀x)(Hx → (∃y)(Txy & Py)) Premise
(2) Ps & (∀x)(Px → x=s) Premise
(3) (∀x)(Txm → (Kx v Ix)) Premise
(4) Hc & Hv Premise
(5) ~Iv Premise
∴ Kv
(6) Hv 4,
&-Elimination
(7) Hv → (∃y)(Tvy & Py) 1, ∀-Elimination
(8) (∃y)(Tvy & Py) 6, 7 Modus Ponens
(9) Tva* & Pa* 8, ∃-Elimination
(10) Pa* 9, &-Elimination
(11) (∀x)(Px → x=s) 2, &-Elimination
(12) Pa* → a*=s 11,
∀-Elimination
(13) a*=s 10,
12 Modus Ponens
(14) Tva* 9,
&-Elimination
(15) Tvs 13,
14 Indiscernibility of Identicals
(16) Tvs → (Kv v Iv) 3, ∀-Elimination
(17) Kv v Iv 15,
16 Modus Ponens
(18) Kv 5, 17 v-Elimination
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