Propositional Logic Quick Reference

 ¬ means ''not''
 ∨ means  ''or''
 ∧ means ''and''
 ⇒ means ''implies''
 ⇔ means ''equivalent to''

Let p and q be propositions that may be true or false. You can describe arguments in terms of relationships between p and q. For instance, if statement p is the patient was cut and q is the patient is bleeding you can express the idea that the patient was cut therefore the patient is bleeding concisely as p implies q or symbolically as:

        p ⇒ q

Logical implication

If p is true, you know that q is true as a consequence: p ⇒ q.

Transposition

The implication concept has an alternate form, called a transposition:

• (p ⇒ q) ⇔ (¬ q ⇒ ¬ p)

Using the example above, the transposition reads as the patient is not bleeding therefore the patient was not cut.

Syllogism

A syllogism combines two implications, like so:

• p ⇒ q
• q ⇒ r
• Therefore p ⇒ r.

Whenever the word some comes into play, you are dealing with sets and intersections of sets instead of implications. For instance:

• Some cats are orange.
• Some orange things are carrots.
• False: Therefore some cats are carrots.

Addition

p ⇒ p ∨ q.

Simplification

p ∧ q ⇒ p.

Commutative property

• If you know that p AND q is true then you also know that q AND p is true.
• If you know that p AND q is false then you also know that q AND p is false.
• If you know that p OR q is true then you also know that q OR p is true.
• If you know that p OR q is false then you also know that q OR p is false.

p ∧ q ⇔ q ∧ p. p ∨ q ⇔ q ∨ p.