Propositional Logic – A Primer

4. Rules of Inference

In propositional logic, arguments are made up of a set of premises and a conclusion. Arguments are proved valid by using logical rules of inference. These are formally defined rules that allow one to infer a conclusion from a set of premises. A valid argument is one in which the conclusion necessarily follows from the premises according to these rules.

There are several logical rules of inference that can be used to prove the validity of an argument. The most common rules you will come across are:

Modus Ponens (MP):

If 'If P then Q' is true and 'P' is true, then 'Q' is true.

If 'P → Q' and 'P' are true, then 'Q' is true.

Example: If it is raining, the ground is wet. It is raining. Therefore, the ground is wet.

Modus Tollens (MT):

If 'If P then Q' is true and 'Q' is false, then 'P' is false.

If 'P → Q' and '¬Q' are true, then '¬P' is true.

Example: If it is raining, the ground is wet. The ground is not wet. Therefore, it is not raining.

Hypothetical Syllogism (HS):

If 'If P then Q' is true and 'If Q then R' is true, then 'If P then R' is true.

If 'P → Q' and 'Q → R' are true, then 'P → R' is true.

Example: If it is raining, the ground is wet. If the ground is wet, the grass is slippery. Therefore, if it is raining, the grass is slippery.

Conjunction (CONJ):

If 'P' is true and 'Q' is true, then 'P and Q' is true.

If 'P' and 'Q' are true, then 'P ∧ Q' is true.

Example: It is raining. The grass is slippery. Therefore, it is raining and the grass is slippery.

Here are some other rules of inference that are used in proofs of validity:

Disjunctive Syllogism (DS):

If 'P or Q' is true and the negation of 'P' is true, then 'Q' is true.

If 'P ∨ Q' and '¬P' are true, then 'Q' is true.

Example: The sky is blue or grey. The sky is not blue. Therefore, the sky is grey.

If 'P' is true, then 'P or Q' is also true.

If 'P' is true, then 'P ∨ Q' is true.

Example: The sky is blue. Therefore, the sky is blue or the grass is green.

Simplification (SIMP):

If 'P and Q' is true, then 'P' and 'Q' are both true.

If 'P ∧ Q' is true, then 'P' and 'Q' are true.

Example: The sun is shining and the sky is blue. Therefore, the sun is shining. The sky is blue.

Constructive Dilemma (CD):

If 'If P then Q' is true and 'If R then S' is true and 'P' or 'R' is true, then 'Q' or 'S' is true.

If 'P → Q' and 'R → S' are true and 'P' or 'R' is true, then 'Q ∨ S' is true.

Example: If it is raining, the ground is wet. If the sun is shining, the sky is blue. It is raining or the sun is shining. Therefore, the ground is wet or the sky is blue.

Double Negation (DN):

If 'P' is true, then the negation of the negation of 'P' is true.

If 'P', then '¬¬P' is true.

Example: The sun is shining. Therefore, it is not the case that the sun is not shining.

Exportation (EXP):

If 'If "P and Q" then R' is true, then 'If P then "If Q then R"' is true.

If '"P ∧ Q" → R' is true, then 'P → "Q → R"' is true.

Example: If it is raining and the tent is uncovered, the tent is wet, then if it is raining, then if the tent is uncovered, the tent is wet.

Material Implication (MI):

If 'If P then Q' is true, then 'P' is false or 'Q' is true.

If 'P → Q', then '¬P ∨ Q' is true.

Example: If it is raining, the ground is wet. Therefore, it is not raining or the ground is wet.

De Morgan's Law (DM):

If 'P and Q' is false, then 'P' is false or 'Q' is false.

If 'P ∧ Q' is false, then '¬P ∨ ¬Q' is true.

Example: The sun is not shining and it is not raining. Therefore, the sun is not shining or it is not raining.

To sum up, rules of inference are used in propositional logic to derive new propositions from existing ones. A rule of inference is a logical principle that allows us to infer the truth of one proposition from the truth of other propositions. In the next session, we will explore how these rules of inference are used to validate arguments.