CS621 : Artificial Intelligence Pushpak Bhattacharyya CSE Dept., IIT Bombay Lecture 12- Completeness Proof; Self References and Paradoxes 16 th August, 2010

Soundness, Completeness & Consistency Syntactic World Theorems, Proofs Semantic World Valuation, Tautology Soundness Completeness * *

An example to illustrate the completeness proof pqp  (p V q) TFT TTT FTT FFT

Running the completeness proof For every row of the truth table set up a proof: 1. p, ~q |- p  (p V q) 2. p, q |- p  (p V q) 3. ~p, q |- p  (p V q) 4. ~p, ~q |- p  (p V q)

Completeness Proof

P 1 P 2 P 3... P n A F F F... F T F F F... T T. T T T... T T We have a truth table with 2 n rows

We should show P 1 ’, P 2 ’, …, P n ’ |- A’ For every row where P i ’ = P i if V(P i ) = T = ~P i if V(P i ) = F And A’ = A if V(A) = T = ~A if V(A) = F

Completeness of Propositional Calculus Statement If V(A) = T for all V, then |--A i.e. A is a theorem. Lemma: If A consists of propositions P 1, P 2, …, P n then P’ 1, P’ 2, …, P’ n |-- A’, where A’ = A if V(A) = true = ~Aotherwise Similarly for each P’ i

Proof for Lemma Proof by induction on the number of ‘→’ symbols in A Basis: Number of ‘→’ symbols is zero. A is ℱ or P. This is true as, |-- (A → A) i.e. A → A is a theorem. Hypothesis: Let the lemma be true for number of ‘→’ symbols ≤ n. Induction: Let A which is B → C, contain n+1 ‘→’

Induction: By hypothesis, P’ 1, P’ 2, …, P’ n |-- B’ P’ 1, P’ 2, …, P’ n |-- C’ If we show that B’, C’ |-- A’ (A is B → C), then the proof is complete. For this we have to show: B, C |-- B → C True as B, C, B |-- C B, ~C |-- ~(B → C) True since B, ~C, B → C |-- ℱ ~B, C |-- B → C True since ~B, C, B |-- C ~B, ~C |-- B → C True since ~B, ~C, B, C → ℱ |-- ℱ Hence the lemma is proved. Proof of Lemma (contd.)

Proof of Theorem A is a tautology. There are 2 n models corresponding to P 1, P 2, …, P n propositions. Consider, P 1, P 2, …, P n |--A and P 1, P 2, …, ~P n |--A P 1, P 2, …, P n-1 |-- P n → A and P 1, P 2, …, P n-1 |-- ~P n → A RHS can be written as: |--((P n → A) → ((~P n → A) → A)) |--(~P n → A) → A |--A Thus dropping the propositions progressively we show |-- A

Self Reference and Paradoxes

Paradox -1 “This statement is false” The truth of this cannot be decided

Paradox -2 (Russell Paradox or Barber Paradox) “In a city, a barber B shaves all and only those who do not shave themselves” Question: Does the barber shave himself? Cannot be answered

Paradox -3 (Richardian Paradox) Order the statements about properties of number in same order. E.g., 1. “A prime no. is one that is divisible by itself of 1.” 2. “ A square no. is one that is product of 2 identical numbers.”.

Paradox -3 (Richardian Paradox) Definition: A number is called Richardian if it does not have the property that it indexes. For example, in the above arrangement 2 is Richardian because it is not a square no.

Paradox -3 (Richardian Paradox) Now, suppose in this arrangement M is the number for the definition of Richardian M : “A no. is called Richardian … “ Question : is M Richardian? Cannot be answered.

Self Reference: source of paradoxes All these paradoxes came because of 1. Self reference 2. Confusion between what is inside a system and what is outside