Proposition Logic

Proposition

<aside> 💡 Proposition: a sentence (or statement) that have a truth value (either True/False)

• Anything that can be put in an if(…) is a proposition </aside>

$\forall$ : forall

$\exists$: there exists

Variable

<aside> 💡 Variable can be aused to represent a proposition

</aside>

• Examples:
  • $p:\text{It is raining} \\ q: \text{I am sick}$

Logial Operators

<aside> 💡 Logical operators are used to combine variables into a proposition

</aside>

• Negation
  • $\neg \ p$ denotes the proposition “not p”
• Conjuction
  • $p \wedge q$ denotes the proposition “p and q”
• Disjunction
  • $p\vee q$ denotes the proposition “p or q”
• Implication
  • ‣
  • $p\to q$ denotes the proposition “if p, then q”
  • $p$ is called premise, $q$ is called the consequence
  • Explanation:
    • Think of $p\to q$ as a “promise”: I promise $q$ if $p$ happen
    • The whole statement
• Contrapositive
  • ‣
  • $\neg p\to \neg q$ is the contrapositive of $q\to p$
  • $\neg p\to \neg q$ have same truth value as $q \to p$
• Biconditional
  • ‣
  • $p\leftrightarrow q$ only true when $p,q$ have same values
    • either both p and q are true, or both are false
  • $p\leftrightarrow q \equiv (p\to q)\wedge (q\to p)$
    • $p\leftrightarrow q$ is essentially a “mutual promise” from both sides
    • Promise 1: if $p$ then $q$ ($p\to q)$
    • Promise 2: if $q$ then $p$ ($q\to p$)

Proposition classification

<aside> 💡 Propositon (with variables) called

Tautology: when proposition always true

Contradiction: when proposition always false

</aside>

Predicates and Quantifiers

• Example: “___ is blue” or “___ is an integer”
  • ___ is the variable
  • the latter is the predicate