Common inference rules

Modus Ponens

The inference rule that states if P implies Q (P→Q) and P is true, then Q must also be true.

Example of Modus Ponens

If Nara is in London (P), then she is in England (Q). Nara is in London. Therefore, Nara is in England.

Modus Tollens

The rule stating that if P implies Q (P→Q) and Q is false (¬Q), then P must also be false (¬P).

Example of Modus Tollens

If I studied (P), I’d pass the test (Q). I didn’t pass the test (¬Q). So, I must not have studied (¬P).

Hypothetical Syllogism

The rule that concludes that if P implies Q (P→Q) and Q implies R (Q→R), then P implies R (P→R).

Example of Hypothetical Syllogism

If I wake up early (P), I’ll have time for breakfast (Q). If I have time for breakfast (Q), I’ll be more focused (R). Therefore, if I wake up early (P), I’ll be more focused (R).

Disjunctive Syllogism

The inference rule stating that if either P or Q (P Q∨) is true and P is false (¬P), then Q must be true.

Example of Disjunctive Syllogism

Either it’s raining (P) or snowing (Q). It’s not raining (¬P). Therefore, it must be snowing (Q).

Conjunction

The rule that states if both P and Q are true, then P and Q combined (P Q∧) are also true.

Example of Conjunction

I like pizza (P). I like ice cream (Q). Therefore, I like both pizza and ice cream (P Q∧).

Simplification

The inference that states if P and Q are both true (P Q∧), then P is also true.

Example of Simplification

It’s sunny and warm (P Q∧). Therefore, it’s sunny (P).

Addition

The rule stating if P is true, then either P or Q (P Q∨) is also true.

Example of Addition

I have a dog (P). Therefore, I have either a dog or a cat (P Q∨).