Lecture 8 CS 1813 – Discrete Mathematics Lecture 8 - CS 1813 Discrete Math, University of Oklahoma 4/20/2019 Lecture 8 CS 1813 – Discrete Mathematics Deriving Absurdity and The Big Surprise CS 1813 Discrete Mathematics, Univ Oklahoma Copyright © 2000 by Rex Page
Deriving Absurdity Theorem (absurdity): (a b) (a (b)) = a (p q) (p (q)) = ((p) q) (p (q)) {imp} a, b = ((p) q) ((p) (q)) {imp} a, b = (p) (q (q)) { dist/} a, b, c = (p) False { complement} = p { identity} a QED CS 1813 Discrete Mathematics, Univ Oklahoma Copyright © 2000 by Rex Page
CS 1813 Discrete Mathematics, Univ Oklahoma Two Implications for the Price of One (a b) c = (a c) (b c) { imp} equations {rule} substitution [formula in eqn / variable in rule] (p q) r = (p q) r {imp} [p q /a] [r /b] = ((p) (q)) r {DeMorgan } [p /a] [q /b] = r ((p) (q)) { comm} [(p) (q) /a] [r /b] = (r (p)) (r (q)) { dist/} [r /a] [p /b] [q /c] = ((p) r) (r (q)) { comm} [r /a] [p/b] = ((p) r) ((q) r) { comm} [r /a] [q/b] = (p r) ((q) r) {imp} [p /a] [r /b] = (p r) (q r) {imp} [q /a] [r /b] QED CS 1813 Discrete Mathematics, Univ Oklahoma Copyright © 2000 by Rex Page
True, Incognito True = a ((a) b) equations {rule} substitution p ((p) q) = (p) ((p) q) {imp} [p /a] [(p) q /b] = (p) (((p)) q) {imp} [p /a] [q /b] = (p) (p q) {dbl neg} [p /a] = ((p) p) q { assoc} [p /a] [p /b] [q /c] = (p (p)) q { comm} [p /a] [p /b] = True q { comp} [p /a] = q True { comm} [True /a] [q /b] = True { null} [q /a] QED CS 1813 Discrete Mathematics, Univ Oklahoma Copyright © 2000 by Rex Page
Boolean Equations, Theorems, and Tautologies Tautology WFF that is true for all combinations of values of its atomic consituents Let p, q stand for arbitrary WFFs Suppose this is a theorem: p |– q Then p q is a tautology Suppose this is a theorem: |– p q Suppose Boolean laws prove p = q Then the following WFFs are tautologies p q q p p q The equation: p = q has the same meaning as the formula: p q CS 1813 Discrete Mathematics, Univ Oklahoma Copyright © 2000 by Rex Page
Inference and Tautologies Let p, q, … stand for arbitrary WFFs Suppose rules of inference prove: p |– q Then p q is a tautology — you already know this Suppose rules of inference prove: |– p q Then p q is a tautology — you know this, too Suppose rules of inference prove p, q, r |– s Then (p q r) s is a tautology Suppose rules of inference prove |– p Then p is a tautology CS 1813 Discrete Mathematics, Univ Oklahoma Copyright © 2000 by Rex Page
Why You Must Accept the Truth Table of p q You have no objection to P True False ¬P Q (¬P) Q False False False False True True True False True True True True P Q False True True True From the homework, you know: p q |– (¬p) q So, (p q) ((¬p) q) is a tautology Next slide will prove: (¬p) q |– p q So, ((¬p) q) (p q) is a tautology That means, ((¬p) q) (p q) is a tautology In other words, ((¬p) q) = (p q) Therefore, their truth tables must be the same CS 1813 Discrete Mathematics, Univ Oklahoma Copyright © 2000 by Rex Page
CS 1813 Discrete Mathematics, Univ Oklahoma Theorem: (¬p) q |– p q q p ¬p { +&- } False {CTR} {I} p q Lemma 1: ¬p |– p q p q p q {ER} {I} p q {I} Lemma 2: q |– p q ¬p {Lemma 1} p q q {Lemma 2} p q (¬p) q {E} p q CS 1813 Discrete Mathematics, Univ Oklahoma Copyright © 2000 by Rex Page
CS 1813 Discrete Mathematics, Univ Oklahoma What It All Means Notions of Consistency in Formal Systems If p |– q, then p |= q (p, q arbitrary WFFs) if q is provable from p, then q is true whenever p is true Another way to say it |– p and |– p means the system is not consistent Completeness in Formal Systems If p |= q, then p |– q (p, q arbitrary WFFs) If q is true whenever p is, there is a proof of p |– q Propositional Logic: consistent and complete Consistency — propositional logic is consistent Inference preserves tautologies Inconsistency would make all WFFs tautologies Some WFFs aren’t tautologies — QED (consistency) Completeness — propositional logic is complete if p q is a tautology, then p |– q Informally: all true statements are provable — No surprise CS 1813 Discrete Mathematics, Univ Oklahoma Copyright © 2000 by Rex Page
Kurt Gödel — the first computer scientist The recognition of Kurt Gödel as the first computer scientist is an insight of John Allen, a present-day logician and computer scientist, author of Anatomy of Lisp. Arithmetic is consistent Comforting, but not surprising Arithmetic is not complete Some arithmetic tautologies cannot be proved A humongous surprise for mathematicians in the 1930s Consistency and Completeness Impossible goal No formal system can be both Except stripped-down systems (less powerful than arithmetic) In essence, this has nothing to do with numbers It has to do with the limits of formal systems (computers) Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I — Gödel, 1931 46 definitions + several dozen lemmas precede main point Translation by Meltzer — $6.25, Amazon.com CS 1813 Discrete Mathematics, Univ Oklahoma Copyright © 2000 by Rex Page
Lecture 8 - CS 1813 Discrete Math, University of Oklahoma 4/20/2019 End of Lecture CS 1813 Discrete Mathematics, Univ Oklahoma Copyright © 2000 by Rex Page
Or Elimination — the Boolean way ((a b) (a c) (b c)) c Note: Theorem statement not fully grouped with parens Doesn’t affect meaning because of { assoc} law equations {rule} substitution (p q) ((p r) (q r)) convenient grouping chosen = (p q) ((p q) r)) { imp} [p /a] [q /b] [r /c] r {Mod Pon} [p q /a] [r /b] Modus Ponens gives implication, not equality QED CS 1813 Discrete Mathematics, Univ Oklahoma Copyright © 2000 by Rex Page
Contradiction ala Boole from False, everything follows False a equations {rule} substitution False = p (p) { comp} [p /a] p {conj imp} [p /a] [p /b] conjunctive implication gives implication, not equality QED CS 1813 Discrete Mathematics, Univ Oklahoma Copyright © 2000 by Rex Page