Proofs with quantifiers
There are four rules of inference for the quantifier symbols, which are the following:
Universal elimination (instantiation)
Universal introduction (generalization)
Existential introduction (generalization)
Existential elimination (instantiation)
The first rule listed for each quantifier is relatively simple to use, while the other two rules are
considerably more complicated.
Let’s first look at the rules of universal elimination and existential introduction.
Universal elimination
Suppose we are given the following set of premises:
∀
x Cube(x)
Large(a)
What might we conclude about object
a
?
The rule of
universal elimination
states that for any universal statement, we can remove the universal
quantifier and consistently replace the variables with the name of some object in the domain of
discourse.
Consider the following argument.
1.
∀
x (Cube(x)
∨
Dodec(x))
2.
∀
y (Cube(y)
→
Small(y))
3.
∀
z (Dodec(z)
→
Large(z))
4.
Cube(a)
∨
Dodec(a)
5.
Cube(a)
∨
Dodec(b)
6.
Cube(b)
→
Small(b)
7.
∀
y (Cube(b)
→
Small(y))
8.
Dodec(a)
→
Large(a)
9.
Dodec(z)
→
Large(z)
10.
∀
x (Dodec(x)
→
Large(x))
Which of the conclusions drawn is a valid application of the rule of universal elimination?
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View Full DocumentUniversal introduction
There are two ways to apply the rule of
universal introduction
.
The first way is called
general
conditional proof
, while the second way is called
universal generalization
.
The method you will need
to use will, in large part, depend on the type of conclusion you want to draw.
Suppose we want to give an informal proof for the following argument:
∀
x (Large(x)
→
FrontOf(x, f)
∀
x (Dodec(x)
→
Large(x))
∀
x (Dodec(x)
→
FrontOf(x, f))
Because our conclusion is a universal statement that contains a conditional statement, we will use the
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 Fall '06
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 Logic, Predicate logic, Dodec, Existential Introduction, universal elimination

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