QUESTION 7 Please arrange in a correct order steps of the given argument in logical form. "Everyone in New Jersey lives within 50 miles of the ocean. Someone in New Jersey has never seen the ocean. Therefore, someone who lives within 50 miles of the ocean has never seen the ocean." + (+) 3x(P(x) ^ Q(x)) Existential generalization of (%) + (%) P(c) ^ Q(c) Conjunction from ($), (&) + (&) There is x=c in the domain such that P() Universal instantiation (*) + (*) VxP(x) Hypothesis Let P(x) = "x lives within 50 miles of the ocean", and Q(x) = " x has never seen the ocean" with the domain of the variable consisting of all people who live in New Jersey. The argument in the logical form: "Everyone in New Jersey lives within 50 miles of the ocean"= VxP(x) "Someone in New Jersey has never seen the ocean" = 3xQ(x) Someone who lives within 50 miles of the ocean has never seen the ocean = 3x(P(x) A Q(x)) : (#) 3xQ(x) + ($) There is x=c in the domain for which Q(c) Existential instantiation (#)
QUESTION 7 Please arrange in a correct order steps of the given argument in logical form. "Everyone in New Jersey lives within 50 miles of the ocean. Someone in New Jersey has never seen the ocean. Therefore, someone who lives within 50 miles of the ocean has never seen the ocean." + (+) 3x(P(x) ^ Q(x)) Existential generalization of (%) + (%) P(c) ^ Q(c) Conjunction from ($), (&) + (&) There is x=c in the domain such that P() Universal instantiation (*) + (*) VxP(x) Hypothesis Let P(x) = "x lives within 50 miles of the ocean", and Q(x) = " x has never seen the ocean" with the domain of the variable consisting of all people who live in New Jersey. The argument in the logical form: "Everyone in New Jersey lives within 50 miles of the ocean"= VxP(x) "Someone in New Jersey has never seen the ocean" = 3xQ(x) Someone who lives within 50 miles of the ocean has never seen the ocean = 3x(P(x) A Q(x)) : (#) 3xQ(x) + ($) There is x=c in the domain for which Q(c) Existential instantiation (#)
Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:QUESTION 7
Please arrange in a correct order steps of the given argument in logical form.
"Everyone in New Jersey lives within 50 miles of the ocean. Someone in New Jersey has never seen the ocean. Therefore,
someone who lives within 50 miles of the ocean has never seen the ocean."
+ (+) 3x(P(x) ^ Q(x)) Existential generalization of (%)
+ (%) P(c) ^ Q(c) Conjunction from ($), (&)
+ (&) There is x=c in the domain such that p(c)
Universal instantiation (*)
- : (*) VxP(x) Hypothesis
Let P(x) = "x lives within 50 miles of the ocean", and Q(x) = "x has never seen the ocean"
with the domain of the variable consisting of all people who live in New Jersey.
%3D
%3D
The argument in the logical form:
"Everyone in New Jersey lives within 50 miles of the ocean"= XP(x)
"Someone in New Jersey has never seen the ocean" = 3xQ(x)
Someone who lives within 50 miles of the ocean has never seen the ocean =
3x(P(x) ^ Q(x))
+ (#) 3xQ(x)
+ ($) There is x=c in the domain for which Q(c)
Existential instantiation (#)
Transcribed Image Text:QUESTION 8
Steps below show that the premises "Linda, a student in this class, owns a red convertible" and "Everyone who owns a red
convertible has gotten at least one speeding ticket" imply conclusion "Someone in this class has gotten a speeding ticket."
Fill in the blanks.
Let R(x) = "x owns a red convertible", S(x) = "x has gotten at least one speeding ticket", and C(x) = "x is a student in our class",
where the domain of x consists of all people in the world.
Premises in logical form, therefore, will be C(linda), C(Linda) a R(Linda) and Vx(R(x)
→ S(x)).
(1) Vx(R(x)→ S(x))
(2) R(Linda) → S(Linda)
instantiation from (1)
(3) C(Linda) A R(Linda) Premise
(4) C(Linda)
from (3)
(5) R(Linda)
from (3)
(6) S(Linda) Modus ponens from
and (5)
(7) C(Linda) A S(Linda)
from (4) and (6)
(8) 3x(C(x) A S(x)) Existential
from
Step (8) expressed in words is "Therefore, someone in the class has gotten at least one speeding ticket".
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