CSE390D Notes for Wednesday, 10/2/24
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what is an argument? (premises & conclusion)
what makes an argument valid (premises -> conclusion is a tautology)
An argument in propositional logic is a sequence of propositions. All
but the final proposition in the argument are called premises and the
final proposition is called the conclusion. An argument is valid if
the truth of all its premises implies that the conclusion is true.
an argument p1, p2, ..., pn, q is valid when p1 ^ p2 ^ ... ^ pn -> q
what if you have p -> q and one other thing (two valid, two invalid):
p -> q p -> q p -> q p -> q
p ~q ~p q
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q (modus ponens) ~p ~q p
modus ponens modus tollens fallacy of fallacy of affirming
denying the the conclusion
hypothesis
for example: p = you are a student, q = you have an adviser
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Example
Show that the premises "It is not sunny this afternoon and it is colder than
yesterday," "We will go swimming only if it is sunny," "If we do not go
swimming, then we will take a canoe trip," and "If we take a canoe trip, then
we will be home by "sunset" lead to the conclusion "We will be home by sunset."
p = it is sunny this afternoon
q = it is colder than yesterday
r = we will go swimming
s = we will take a canoe trip
t = we will be home by sunset
premises: ~p ^ q, r -> p, ~r -> s, s -> t
conclusion: t
step reason
1. ~p ^ q premise
2. ~p simplification using (1)
3. r -> p premise
4. ~r modus tollens using (2) and (3)
5. ~r -> s premise
6. s modus ponens using (4) and (5)
7. s -> t premise
8. t modus ponens using (6) and (7)
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Example
Show that the premises "If you send me an email message, then I will finish
writing the program," "If you do not send me an email message, then I will go
to sleep early," and "If I go to sleep early, then I will wake up feeling
refreshed" lead to the conclusion "If I do not finish writing the program, then
I will wake up feeling refreshed."
p = you send me an email message
q = I will finish writing the program
r = I will go to sleep early
s = I will wake up feeling refreshed
premises: p -> q, ~p -> r, r -> s
conclusion: ~q -> s
step reason
1. p -> q premise
2. ~q -> ~p contrapositive of (1)
3. ~p -> r premise
4. ~q -> r hypothetical syllogism using (2) and (3)
5. r -> s premise
6. ~q -> s hypothetical syllogism using (4) and (5)
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Quantifier rules of inference:
universal instantiation:
(all x) P(x)
---------------
P(c)
universal generalization:
P(c) for an arbitrary c
---------------
(all x) P(x)
existential instantiation:
(exists x) P(x)
---------------
P(c) for some c
existential generalization:
P(c) for an some c
---------------
(exists x) P(x)
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Example
Show that the premises "Everyone in this discrete math class has taken a course
in computer science" and "Marla is a student in this class" imply the
conclusion "Marla has taken a course in computer science."
D(x) x is in this discrete math class
C(x) x has taken a course in computer science
premises: (all x)(D(x) -> C(x)), D(Marla)
conclusion: C(Marla)
step reason
1. (all x) (D(x) -> C(x)) premise
2. D(Marla) -> C(Marla) universal instantiation from (1)
3. D(Marla) premise
4. C(Marla) modus ponens from (2) and (3)
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Show that the premises "A student in this class has not read the book," and
"Everyone in this class passed the first exam" imply the conclusion "Someone
who passed the first exam has not read the book."
C(x) x is in this class
B(x) x has read the book
P(x) x passed the first exam
premises: (exists x)(C(x) ^ ~B(x)), (all x)(C(x) -> P(x))
conclusion: (exists x)(P(x) ^ ~B(x))
step reason
1. (exists x)(C(x) ^ ~B(x)) premise
2. C(a) ^ ~B(a) existential instantiation from (1)
3. C(a) simplification from (2)
4. (all x)(C(x) -> P(x)) premise
5. C(a) -> P(a) universal instantiation from (4)
6. P(a) modus ponens from (3) and (5)
7. ~B(a) simplification from (2)
8. P(a) ^ ~B(a) conjunction from (6) and (7)
9. (exists x)(P(x) ^ ~B(x)) existential generalization from (8)
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Rules of inference can be combined with quantifiers, as in:
universal modus ponens
(all x) (P(x) -> Q(x))
P(a)
----------------------
Q(a)
universal modus tollens
(all x) (P(x) -> Q(x))
~Q(a)
----------------------
~P(a)
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Stuart Reges
Last modified: Wed Oct 2 15:34:10 PDT 2024