Page 94 - Discrete Mathematics and Its Applications

1.6 Rules of Inference 73

EXAMPLE 5 State which rule of inference is used in the argument:

If it rains today, then we will not have a barbecue today. If we do not have a barbecue today,
then we will have a barbecue tomorrow. Therefore, if it rains today, then we will have a
barbecue tomorrow.

Solution: Let p be the proposition “It is raining today,” let q be the proposition “We will not
have a barbecue today,” and let r be the proposition “We will have a barbecue tomorrow.” Then
this argument is of the form

p → q
q → r
∴ p → r

Hence, this argument is a hypothetical syllogism. ▲

Using Rules of Inference to Build Arguments

When there are many premises, several rules of inference are often needed to show that an
argument is valid. This is illustrated by Examples 6 and 7, where the steps of arguments are
displayed on separate lines, with the reason for each step explicitly stated. These examples also
show how arguments in English can be analyzed using rules of inference.

EXAMPLE 6 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.”
Solution: Let p be the proposition “It is sunny this afternoon,” q the proposition “It is colder
than yesterday,” r the proposition “We will go swimming,” s the proposition “We will take a
canoe trip,” and t the proposition “We will be home by sunset.” Then the premises become
¬p ∧ q, r → p, ¬r → s, and s → t. The conclusion is simply t. We need to give a valid
argument with premises ¬p ∧ q, r → p, ¬r → s, and s → t and conclusion t.
We construct an argument to show that our premises lead to the desired conclusion as
follows.
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)

Note that we could have used a truth table to show that whenever each of the four hypotheses
is true, the conclusion is also true. However, because we are working with five propositional
variables, p, q, r, s, and t, such a truth table would have 32 rows. ▲