lec1.md

Logic

• Valid inference:
  • $\frac{A_1, A_2, A_3, ... , A_n}{C}$
  • An inference is valid if and only if every time all the premises are true, the conclusion is also true.

e.g.: if x integer, then $x^3 - x$ is multiple of 3 3 cases: x = 3k, or x = 3k+1, or x = 3k+2 case x = 3k+2: $x^3 - x = $

Propositional language

The language LP is a set of formulas satisfying:

1. All the basic propositions are in LP: p ∈ LP, q ∈ LP, r ∈ LP, . . .
2. If ϕ ∈ LP and ψ ∈ LP, then
   • (ϕ ∧ ψ) ∈ LP,
   • (ϕ → ψ) ∈ LP,
   • (ϕ ∨ ψ) ∈ LP,
   • (ϕ ↔ ψ) ∈ LP
   • Nothing else is in Lp

Constructing formulas

(¬(p ∨ q) → r) can be seen as building a tree

name	symbol
not	¬
and / conjuction	∧
or / disjunction	∨
"if...then" / implication	→
"if and only if" / equivalence	↔

• Never true (contradiction)
• Can be true (satisfiable)
• Always true (valid, tautology)
• Two formulas ϕ and ψ are logically equivalent (ϕ ≡ ψ) if and only if ϕ |= ψ and ψ |= ϕ

Symbolic inference

• A proof is a finite sequence of formulas where each formula is either an axiom or else it has been infered from previous formulas by using an inference rule.
• A formula is a theorem if it occurs in a proof
• A set of axioms and rules is called an axiom system or an axiomatization.
  • An axiom system is sound for a logic if every theorem is valid in the logic.
  • An axiom system is complete if every valid formula of the logic is a theorems.