Inference: Addition, Conjunction, and Simplification

Addition

If we know that $q_1$ is true, we can consistently conclude that, $\left(q_1\lor q_2\right)$ is true. Here $q_2$ is an arbitrary proposition. This rule of inference is called addition. The following statement is always true.

“If $q_1$ is true then $q_1 \lor q_2$ is also true.”

We can write this tautology as follows:

$$q_1 \Rightarrow \left(q_1\lor q_2\right).$$

Examples

Let’s look at a few examples to see this rule in action.

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For the next example, consider the following two propositions.

• $L$: $\sqrt{2} < 2.$
• $E$: $\sqrt{2} = 2.$

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As $L$ is true, we know that the following proposition is true by the addition rule.

$$L \lor E : \sqrt{2} \le 2.$$

Conjunction

If we know that $q_1$ is true and we also know independently that $q_2$ is true, we can conclude that $\left(q_1\land q_2\right)$ is true.

Examples

Let’s look at some examples.

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Assume that the following two propositions are true:

• $C_T$: A car has four tires.
• $C_{W}$: A car has a steering wheel.

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We can conclude that the following proposition is true by applying the rule of conjunction.

• $C_T \land C_{W}$: A car has four tires and a steering wheel.

Let’s look at the next example.

Simplification

If we know that $\left(q_1\land q_2\right)$ is true, we can conclude that $q_1$ is true. We can write it as the following tautology:

$$\left(q_1 \land q_2\right)\Rightarrow q_1.$$

Examples

For further elaboration, let’s look at a few examples.

From the fact above, we can conclude that the following proposition is true.

• $N_{GH}$: Nutritious food is good for health.

Indeed, we can also conclude that $E_{GH}$ is true.

Quiz