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Truth Tellers, Liars, and Propositional Logic
Reading: EC 1.3
Peter J. Haas
INFO 150Fall Semester 2019
Lecture 2 1/ 16
Truth Tellers, Liars, and Propositional LogicSmullyan’s IslandPropositional LogicTruth Tables for Formal PropositionsLogical EquivalenceThe Big Honking Theorem
Lecture 2 2/ 16
Smullyan’s Island
You meet two inhabitants of Smullyan’s Island. A says “exactly oneof us is lying”. B says “at least one of us is telling the truth”. Who(if anyone) is telling the truth?
Strategy: Focus on the statements, not on who said them
Lecture 2 3/ 16
Truth Table Analysis
Notation
I p = “A is truthful”
I q = “B is truthful”
Statement 1: Statement 2:p q Exactly one is lying At least one is truthful
T T F TT F T TF T T T
*F F F F
Answer: Both A and B are liars
Lecture 2 4/ 16
Another Smullyan’s Island Example
The statements
I A: “Exactly one of use is telling the truth”
I B : “We are all lying”
I C : “The other two are lying
Statement 1: Statement 2: Statement 3:p q r Exactly one truthful All lying A & B lying
T T T F F FT T F F F FT F T F F F
*T F F T F FF T T F F FF T F T F FF F T T F TF F F F T T
Answer: A is truthful; B and C are liars
Lecture 2 5/ 16
Inconclusive or a Paradox
Statement 1:p I am telling the truth
*T T*F F
Inconclusive
Statement 1:p I am lying
T FF T
A paradox
Lecture 2 6/ 16
Propositional Logic Notation
Definitions
Proposition: A sentence that is unambiguously true or falsePropositional variable: Represents a proposition (= T or F)Formal proposition: Proposition written in formal logic notation
Rules of formal propositions (FPs)
1. Any propositional variable is an FP
2. p and q are FPs ) p ^ q is an FP (p and q are true)
3. p and q are FPs ) p _ q is an FP (p or q or both are true)
4. p is an FP ) ¬p is an FP (not p)
Example: (p _ q) ^ ¬(p _ q) is a formal proposition
Precedence: ¬ highest, then ^, then _ (like �, ⇥, and +)
I Ex: ¬p ^ ¬q _ p =�(¬p) ^ (¬q)
�_ p
Lecture 2 7/ 16
Propositional Logic Notation
Definitions
Proposition: A sentence that is unambiguously true or falsePropositional variable: Represents a proposition (= T or F)Formal proposition: Proposition written in formal logic notation
Rules of formal propositions (FPs)
1. Any propositional variable is an FP
2. p and q are FPs ) p ^ q is an FP (p and q are true)
3. p and q are FPs ) p _ q is an FP (p or q or both are true)
4. p is an FP ) ¬p is an FP (not p)
Example: (p _ q) ^ ¬(p _ q) is a formal proposition
Precedence: ¬ highest, then ^, then _ (like �, ⇥, and +)
I Ex: ¬p ^ ¬q _ p =�(¬p) ^ (¬q)
�_ p
Lecture 2 7/ 16
Propositional Logic Notation
Definitions
Proposition: A sentence that is unambiguously true or falsePropositional variable: Represents a proposition (= T or F)Formal proposition: Proposition written in formal logic notation
Rules of formal propositions (FPs)
1. Any propositional variable is an FP
2. p and q are FPs ) p ^ q is an FP (p and q are true)
3. p and q are FPs ) p _ q is an FP (p or q or both are true)
4. p is an FP ) ¬p is an FP (not p)
Example: (p _ q) ^ ¬(p _ q) is a formal proposition
Precedence: ¬ highest, then ^, then _ (like �, ⇥, and +)
I Ex: ¬p ^ ¬q _ p =�(¬p) ^ (¬q)
�_ p
Lecture 2 7/ 16
Logic Notation: Examples
Example 1: p = “A is truthful” and q = “B is truthful”
I A is lying:
I At least one of us is truthful:
I Either B is lying or A is:
I Exactly one of us is lying (exclusive or):
Example 2: e = “Sue is an English major” and j = “Sue is aJunior”
I Sue is a Junior English major:
I Sue is either an English major or she is a Junior:
I Sue is a Junior, but she is not an English major:
I Sue is exactly one of the following: an English major or a Junior:
Lecture 2 8/ 16
tip
pvq7 p V 7g
( '
pry ) Vpn -
q )
en 's
evjj a Te
Len- j ) vcjn - e)
Truth Tables for Formal Propositions
p q p ^ q
T T TT F FF T FF F F
p q p _ q
T T TT F TF T TF F F
p ¬p
T FF T
Lecture 2 9/ 16
Truth Table Examples: Complex Formulas
Example 1: p ^ ¬q
p q ¬q p ^ ¬q
T T F FT F T TF T F FF F T F
Example 2: (p _ q) ^ ¬(p ^ q)
p q p ^ q ¬(p ^ q) p _ q (p _ q) ^ ¬(p ^ q)
T TT FF TF F
Lecture 2 10/ 16
T f T ff T T
TF T T
TF T F f
Negation and Inequalities
Example
I p = “Tammy has more than two children”
I ¬p =:
I If c = number of children, then, mathematically, p =
Lecture 2 11/ 16
Tammy has two or fewer childrenC > 2
Negation and Logical Equivalence
Definition
Two statements are logically equivalent if they have the same truthvalue for for every row of the truth table
Example: Sue is neither an English major nor a Junior
j e j _ e ¬(j _ e)
T TT FF TF F
j e ¬j ¬e ¬j ^ ¬e
T TT FF TF F
Lecture 2 12/ 16
T f if f f
T f f T fT F T f f
F T T T T
4- the same#
DeMorgan’s Laws and Negation
Proposition (DeMorgan’s Laws)
Let p and q be any propositions. Then
1. ¬(p _ q) is logically equivalent to ¬p ^ ¬q2. ¬(p ^ q) is logically equivalent to ¬p _ ¬q
Proof: Via truth tables
Example 1:
I “Sue is not both a Junior and an English major”: ¬(j ^ e)
I Use DeMorgan’s laws to given an equivalent statement:
Example 2: “John got a B’ on the test” = (g � 80) ^ (g < 90) [where g = Johnsscore]
I Write the negation in math and English:
Lecture 2 13/ 16
7 j v - e
Sue is not a Junior ourSue is not an English major
71cg ? so ) rig ago ))=
t ( g 380 ) V 7 ( g 290 ) = C g e 80 ) V ( g > go )
John got less than B or greater than B
Tautology and Contradiction
Definition
1. A tautology is a proposition where every row of the truth table is true
2. A contradiction is a proposition where every row of the truth table is false
p q ¬p ¬q p _ ¬q ¬p _ q (p _ ¬q) _ (¬p _ q)
T TT FF TF F
p ¬p p ^ ¬p
T FT FF TF T
Lecture 2 14/ 16
+ to
f F T T Tf T T f T
T F F T T
T T T T T
I
ff
f
The Big Honking Theorem (BHT) of Propositions
Theorem
Let p, q and r stand for any propositions. Let t indicate a tautology and c indicate acontradiction. Then:
(a) Commutive p ^ q ⌘ q ^ p p _ q ⌘ q _ p(b) Associative (p ^ q) ^ r ⌘ p ^ (q ^ r) (p _ q) _ r ⌘ p _ (q _ r)(c) Distributive p ^ (q _ r) ⌘ (p ^ q) _ (p ^ r) p _ (q ^ r) ⌘ (p _ q) ^ (p _ r)(d) Identity p ^ t ⌘ p p _ c ⌘ p(e) Negation p _ ¬p ⌘ t p ^ ¬p ⌘ c(f) Double negative ¬(¬p) ⌘ p(g) Idempotent p ^ p ⌘ p p _ p ⌘ p(h) DeMorgan’s laws ¬(p ^ q) ⌘ ¬p _ ¬q ¬(p _ q) ⌘ ¬p ^ ¬q(i) Universal bound p _ t ⌘ t p ^ c ⌘ c(j) Absorption p ^ (p _ q) ⌘ p p _ (p ^ q) ⌘ p(k) Negations of t and c ¬t ⌘ c ¬c ⌘ t
Does this look familiar?
Substitution Rule: You can replace a formula with a logically equivalent one
Lecture 2 15/ 16
similar to algebra,with he X
,V it
,
6 it,
c = o
( Not identical , c. g. ,It venson of distributive law )
Proving Logical Equivalences
Ex: Use BHT plus substitution to prove that p _ (¬p ^ q) ⌘ p _ q
p _ (¬p ^ q) = (p _ ¬p) ^ (p _ q) (c) Distributive
= t ^ (p _ q) (e) Negation
= (p _ q) ^ t (a) Commutative
= p _ q (d) Identity
Lecture 2 16/ 16