# Rules of Inference

Each rule of inference is a single step in a logical argument. In the table below, the propositions above the line are the premises (also known as the hypotheses) and the proposition below the line is the conclusion.

Addition Simplification Conjunction Modus Ponens Modus Tollens
p
∴ p ∨ q
p ∧ q
∴ p
p
q
∴ p ∧ q
p ⇒ q
p
∴ q
p ⇒ q
¬q
∴ ¬p

p = "The sky is blue", q = "I can fly"
If we know "The sky is blue" then we can conclude "The sky is blue or I can fly".

Simplification Example:
p = "The sky is blue", q = "The moon is full"
If we know "The sky is blue and the moon is full" then we can conclude "The sky is blue".

Conjunction Example:
p = "The sky is blue", q = "The moon is full"
If we know "The sky is blue" and we know "The moon is full" then we can conclude "The sky is blue and the moon is full".

Modus Ponens Example:
p = "It is raining", q = "it must be cloudy"
Suppose we know that "If it is raining then it must be cloudy" is a true statement and we also know that "it is raining" then we are allowed to conclude "it must be cloudy".

Modus Tollens Example:
p = "It is raining", q = "it must be cloudy"
Suppose we know that "If it is raining then it must be cloudy" is a true statement and we also know that "it is not cloudy" then we are allowed to conclude "it is not raining".

The rules of inference above are quite simple and intuitive. We have 3 more that are slightly more involved.

Hypothetical Syllogism Disjunctive Syllogism Resolution
p ⇒ q
q ⇒ r
∴ p ⇒ r
p ∨ q
¬p
∴ q
p ∨ q
¬p ∨ r
∴ q ∨ r

Hypothetical Syllogism Example:
p = "It is raining", q = "it must be cloudy", r = "the sky is grey"
Suppose we know that "If it is raining then it must be cloudy" is a true statement and we also know that "If it is cloudy then the sky is grey" is a true statement then we are allowed to conclude "If it is raining then the sky must be grey" is a true statement.

Disjunctive Syllogism Example:
p = "The sky is blue", q = "The sky is grey"
Suppose we know that "The sky is blue or the sky is grey" and we also know that "The sky is not grey" then we can conclude that "the sky is blue".

Resolution Example:
p = "It is raining", q = "the sky is blue", r = "the sky is grey"
Suppose we know that "it is raining or the sky is blue" is a true statement and we also know that "it is not raining or the sky is grey" is a true statement then we are allowed to conclude "the sky is blue or the sky is grey".

Most people find the rules of inference quite natural, but be careful that you do not fall for these two common fallacies. These are INCORRECT deductions.

Fallacy: Affirming the Consequent Fallacy: Denying the Hypothesis
p ⇒ q
q
∴ p
p ⇒ q
¬p
∴ ¬q

Affirming the Consequent Example:

p = "It is raining", q = "it must be cloudy"
Suppose we know that "If it is raining then it must be cloudy" is a true statement and we also know that "it is cloudy". We are not allowed to conclude that "it is raining". We might have clouds and not rain.

Denying the Hypothesis Example:
p = "It is raining", q = "it must be cloudy"
Suppose we know that "If it is raining then it must be cloudy" is a true statement and we also know that "it is not raining". We are not allowed to conclude that "it is not cloudy". We might have no rain but have cloudy skies anyway.