1.3. Rules of inference

1.3.1 Definition

By an “argument” we mean a sequence of statements that end with a conclusion. By “valid”, we mean that the conclusion must follow from the preceding statements “the premises” of the argument.That is an argument is valid if and only if it is impossible for all premises to be true and the conclusion to be false.

1.3.2 Remark

We can always use a truth table to prove that an argument is valid. However, this can be a tedious approach. For example, when an argument involves 10 different propositional variable, to prove the validity of this argument using a truth table we require 2¹⁰ = 1024 different rows. Instead, we can first establish the validity of some relatively simple argument forms, called “RULES OF INFERENCE”

1.3.3 Definition (modus ponens). The tautology [p˄(p→q)]→q] is the basis of an important rule of inference which is well known as “modus ponens”. This argument is usually written as:

1.3 Rules of inference

where the symbol “∴” denotes “therefore”. This means that if p and p→q are true then q is true.

1.3.4 Example

Consider the following argument:

p→q: (If you have a current password, then you can log onto the network)
p: (you have a current password)
Therefore,
q: (you can log onto the network)
Suppose that p and p→q are true then using the modus ponens we write this argument as


Obviously the given argument is “valid” since q is also true.

1.3 Rules of inference

1.3.5 Example

Suppose that : p: “It snows today ” is true,
p→q: “If it snows today then we will go skiing” is true

Then obviously q: (We will go skiing) is true.
We can express the validity of this argument as (using modus ponens).



1.3.6 Remark

There are many rules of inferences for propositional logic which are listed in the following table:

1.3 Rules of inference

1.3 Rules of inference

1.3 Rules of inference

1.3.7 Example

State which rule of inference is the basis for the following arguments

(i) “It is below freezing now, therefore, it is either below freezing now or it is raining now”
(ii) “It is below freezing now and raining now, therefore it is below freezing”

Solution If p: (It is below freezing)
                  q: (It is raining)
                  then argument (i) has the form


Thus, this argument uses the simplification rule

1.3 Rules of inference

1.3.8 Example

Let p be the proposition (It is raining today), q be the proposition (We will not have a barbecue today), and r be the proposition (We will have a barbecue tomorrow).

Consider now the argument:

“If it rains today, then we will not have a barbecue today. If we don’t have a barbecue today then we will have a barbecue tomorrow. Therefore, it is raining today then we will have a barbecue tomorrow”. Obviously, this argument is of the form: