Discrete Mathematics - Rules of Inference

To deduce new statements from the statements whose truth that we already know, Rules of Inference are used.

What are Rules of Inference for?

Mathematical logic is often used for logical proofs. Proofs are vapd arguments that determine the truth values of mathematical statements.

An argument is a sequence of statements. The last statement is the conclusion and all its preceding statements are called premises (or hypothesis). The symbol “$ herefore$”, (read therefore) is placed before the conclusion. A vapd argument is one where the conclusion follows from the truth values of the premises.

Rules of Inference provide the templates or guidepnes for constructing vapd arguments from the statements that we already have.

Table of Rules of Inference

Addition

If P is a premise, we can use Addition rule to derive $ P lor Q $.

$$egin{matrix} P \ hpne herefore P lor Q end{matrix}$$

Example

Let P be the proposition, “He studies very hard” is true

Therefore − "Either he studies very hard Or he is a very bad student." Here Q is the proposition “he is a very bad student”.

Conjunction

If P and Q are two premises, we can use Conjunction rule to derive $ P land Q $.

$$egin{matrix} P \ Q \ hpne herefore P land Q end{matrix}$$

Example

Let P − “He studies very hard”

Let Q − “He is the best boy in the class”

Therefore − "He studies very hard and he is the best boy in the class"

Simppfication

If $P land Q$ is a premise, we can use Simppfication rule to derive P.

$$egin{matrix} P land Q\ hpne herefore P end{matrix}$$

Example

"He studies very hard and he is the best boy in the class", $P land Q$

Therefore − "He studies very hard"

Modus Ponens

If P and $P ightarrow Q$ are two premises, we can use Modus Ponens to derive Q.

$$egin{matrix} P ightarrow Q \ P \ hpne herefore Q end{matrix}$$

Example

"If you have a password, then you can log on to facebook", $P ightarrow Q$

"You have a password", P

Therefore − "You can log on to facebook"

Modus Tollens

If $P ightarrow Q$ and $lnot Q$ are two premises, we can use Modus Tollens to derive $lnot P$.

$$egin{matrix} P ightarrow Q \ lnot Q \ hpne herefore lnot P end{matrix}$$

Example

"If you have a password, then you can log on to facebook", $P ightarrow Q$

"You cannot log on to facebook", $lnot Q$

Therefore − "You do not have a password "

Disjunctive Syllogism

If $lnot P$ and $P lor Q$ are two premises, we can use Disjunctive Syllogism to derive Q.

$$egin{matrix} lnot P \ P lor Q \ hpne herefore Q end{matrix}$$

Example

"The ice cream is not vanilla flavored", $lnot P$

"The ice cream is either vanilla flavored or chocolate flavored", $P lor Q$

Therefore − "The ice cream is chocolate flavored”

Hypothetical Syllogism

If $P ightarrow Q$ and $Q ightarrow R$ are two premises, we can use Hypothetical Syllogism to derive $P ightarrow R$

$$egin{matrix} P ightarrow Q \ Q ightarrow R \ hpne herefore P ightarrow R end{matrix}$$

Example

"If it rains, I shall not go to school”, $P ightarrow Q$

"If I don t go to school, I won t need to do homework", $Q ightarrow R$

Therefore − "If it rains, I won t need to do homework"

Constructive Dilemma

If $( P ightarrow Q ) land (R ightarrow S)$ and $P lor R$ are two premises, we can use constructive dilemma to derive $Q lor S$.

$$egin{matrix} ( P ightarrow Q ) land (R ightarrow S) \ P lor R \ hpne herefore Q lor S end{matrix}$$

Example

“If it rains, I will take a leave”, $( P ightarrow Q )$

“If it is hot outside, I will go for a shower”, $(R ightarrow S)$

“Either it will rain or it is hot outside”, $P lor R$

Therefore − "I will take a leave or I will go for a shower"

Destructive Dilemma

If $(P ightarrow Q) land (R ightarrow S)$ and $ lnot Q lor lnot S $ are two premises, we can use destructive dilemma to derive $lnot P lor lnot R$.

$$egin{matrix} (P ightarrow Q) land (R ightarrow S) \ lnot Q lor lnot S \ hpne herefore lnot P lor lnot R end{matrix}$$

Example

“If it rains, I will take a leave”, $(P ightarrow Q )$

“If it is hot outside, I will go for a shower”, $(R ightarrow S)$

“Either I will not take a leave or I will not go for a shower”, $lnot Q lor lnot S$

Therefore − "Either it does not rain or it is not hot outside"