Module 3: Proof Techniques
Theme 1: Rule of Inference
Let us consider the following example.
Example 1
: Read the following “obvious” statements:
All Greeks are philosophers.
Socrates is a Greek.
Therefore, Socrates is a philosopher.
This conclusion seems to be perfectly correct, and quite obvious to us. However, we cannot justify
it rigorously since we do not have any
rule of inference
.
When the chain of implications is more
complicated, as in the example below, a formal method of inference is very useful.
Example 2
: Consider the following hypothesis:
1. It is not sunny this afternoon and it is colder than yesterday.
2. We will go swimming only if it is sunny.
3. If we do not go swimming, then we will take a canoe trip.
4. If we take a canoe trip, then we will be home by sunset.
From this hypothesis, we should conclude:
We will be home by sunset.
We shall come back to it in Example 5.
The above conclusions are examples of a
syllogisms
defined as a “deductive scheme of a formal
argument consisting of a major and a minor premise and a conclusion”. Here what
encyclope
dia.com
is to say about syllogisms:
Syllogism
, a mode of argument that forms the core of the body of Western logical thought.
Aristotle defined syllogistic logic, and his formulations were thought to be the final word
in logic; they underwent only minor revisions in the subsequent 2,200 years.
We shall now discuss
rules of inference
for propositional logic. They are listed in Table 1. These
rules provide justifications for steps that lead logically to a conclusion from a set of hypotheses.
Because the emphasis is on
correctness
of arguments, these rules, when written as a proposition, are
tautologies
. Recall that a tautology is a proposition that is
always
true.
1
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Table 1: Rules of Inference
Rule of Inference
Tautology
Name
Explanation
Addition
If the hypothesis is true,
then the disjunction is true.
Simplification
If a conjunction of hypotheses is
true, then the conclusion is true.
Conjunction
If both hypotheses are true,
then the conjunction of them is true.
Modus ponens
If both hypotheses are true,
then the conclusion is true.
Modus tollens
If a hypothesis is not true
and an implication is true,
then the other proposition cannot be true.
Hypothetical syllogism
If both implications are true,
then the resulting implication is true.
Disjunctive syllogism
If a disjunction is true, and
one proposition is not true, then
the other proposition must be true.
Let us start with the following proposition (cf. Table 1)
The table below shows that it is a tautology.
T
T
T
T
T
T
F
F
F
T
F
T
T
F
T
F
F
T
F
T
This tautology is the basis of the rule of inference called
modus ponens
or
law of detachment
that
we actually used in Example 1 to infer the above conclusion. Such a rule is often written as follows:
.
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 Fall '08
 W.Szpankowski
 Logic, canoe trip

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