1. 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 Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page

2. 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 Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page

3. 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 Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page

4. 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 Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page

5. 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 Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page

6. 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 Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page

7. 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 Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page

8. 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 Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page

9. 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 Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page

10. 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 Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page

11. Lecture 8 - CS 1813 Discrete Math, University of Oklahoma
4/20/2019
End of Lecture
CS Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page

12. 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 Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page

13. 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 Discrete Mathematics, Univ Oklahoma
Copyright © 2000 by Rex Page