Propositional Logic

What's a proposition?


a statement or assertion that expresses a judgment or opinion. (



a declarative sentence (that is, a sentence that declares a fact) that is either true or false, but not both.


Some propositions

  • George is a lecturer
  • Honolulu is the capital of Hawaii
  • The temperature outside is 90 degrees

What isn't a proposition?

  • Watch this screencast
  • How much time is left in this screencast?
  • x > 5
  • This statement is false

Propositional Variables

  • We typically use the letters p, q, r, s ...
  • Each value can either be true or false (T/F)
  • We'll see what this has to do with computers later

Compound Propositions

  • Each of the letters are "atoms"
  • Make more "interesting" propositions with atoms and logical operators.

Logical Operators



If p is a proposition, then ¬p is the opposite of the truth value of p.


Let p and q be propositions. The disjunction p v q is equivalent to logical OR. The disjunction is false if both and are false and true otherwise.


Let p and q be propositions. The conjunction p Λ q is equivalent to logical AND. The conjunction is true if both and are true and false otherwise.

Exclusive Or:

Let p and q be propositions. The Exclusive Or of and q is p  q and is equivalent to logical XOR. Exclusive or is true if either p or are true but not both.



Let p and q be propositions. The conditional statement p → is the proposition "if then q". The conditional is false when p is true and is false and true otherwise.


Let p and q be propositions. The converse of the proposition → q is q p.


Let p and q be propositions. The inverse of the proposition → q is ¬p ¬q.


Let p and q be propositions. The contrapositive of the proposition → q is ¬q ¬p.


A conditional statement is equivalent to its contrapositive. The converse of a conditional statement is equivalent to the inverse.


Let p and q be propositions. The biconditional statement p ↔ is the proposition "p if and only if q". The conditional is true when p and have the same truth value and false otherwise.

Truth Tables

// One Truth Variable


// Two truth variables

p | q
T | T
T | F
F | T
F | F

// Three variables

p | q | r
T | T | T
T | T | F
T | F | T
T | F | F
F | T | T
F | T | F
F | F | T
F | F | F

The number of rows is 2^n

Logic and Bit Operations

Computers use binary

  • 1's and 0's (T and F)
  • A chain of 1's and 0's are a bitstring
  • Bitstrings are typically split into groups of 4 (Hexadecimal)
  • Read 1.2 to see some applications


By George Lee


A presentation on logics.

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