Korea Advanced Institute of Science and Technology
Logic
HSS Hss105

Fall 2009
MATHEMATICAL INDUCTION
Weak Mathematical Induction
The principle of weak mathematical induction (weak induction, for short) builds upon our most fundamental intuition about the structure of the natural numbers: crudely put, that they are the result of sta
Korea Advanced Institute of Science and Technology
Logic
HSS Hss105

Fall 2009
Other axiom systems M u
1. 2.
Other axiom systems Systems F and M
Final exam: Monday (12/14) 7pm,
Room 412
Covers Chs.1321 (except for Ch.19)
Typos in Ch.21
p.251: (3) From S1 and S2 and F6 p.254, last line: if X is deducible from Y p.258, line 5: ~X
Korea Advanced Institute of Science and Technology
Logic
HSS Hss105

Fall 2009
Completeness of axiom # systems
1. 2. 3.
Axiom system U0 Completeness of U0 Other axiom systems
Final exam: Monday (12/14) 7pm. No paper assignment: focus on the
exam!
HW #10 (last HW set due
Wednesday, 12/9)
1. There is no ambiguity involved in the
fo
Korea Advanced Institute of Science and Technology
Logic
HSS Hss105

Fall 2009
axiom systems IF
1. 2. 3.
SkolemLwenheims theorem What is an axiom system? Euclids case Axiom systems in logic
Last time,
We proved the completeness theorem
for 1storder logic by constructing a systematic tableau.
If a formula X is valid (or FX is not
Korea Advanced Institute of Science and Technology
Logic
HSS Hss105

Fall 2009
Completeness and other ZB results
1. 2. 3.
Completeness for firstorder logic Lwenheims theorem Compactness
Reading for the next week Ch.20
(skip Ch.19) HW #9 (due Monday, 11/30)
1. By applying a procedure similar to the
propositional case, show the cor
Korea Advanced Institute of Science and Technology
Logic
HSS Hss105

Fall 2009
Completeness for firsts order logic
1. 2. 3.
Truth set Compactness Completeness for firstorder logic
Reading for the next week Ch.20
(skip Ch.19) HW #9 (due Monday, 11/30)
1. By applying a procedure similar to the
propositional case, show the correctne
Korea Advanced Institute of Science and Technology
Logic
HSS Hss105

Fall 2009
Completeness for prob positional logic
1. 2. 3.
Completeness Truth set Compactness
Read Ch.18 for the next week HW #8 (due Monday, 11/23)
1. Let A be a formula in the propositional
language (no quantifiers). Let cA be the number of places at which a bin
Korea Advanced Institute of Science and Technology
Logic
HSS Hss105

Fall 2009
Correctness for proposifS tional logic
1. 2. 3.
HW solutions Correctness Completeness
HW #8 (due Monday, 11/23)
1. Let A be a formula in the propositional
language (no quantifiers). Let cA be the number of places at which a binary connective (one of v,
Korea Advanced Institute of Science and Technology
Logic
HSS Hss105

Fall 2009
More difficult results T
1. 2.
Knigs lemma Compactness
HW #8 will be assigned on 11/16. Read Ch.17 for the next week. In HW #6, Q2,
Tx(Kx x(Sx v~Kx) replace all Sx v~Kx),T(Ka free occurrences of x with any parameter a. Then apply the tableau rule for 6
Korea Advanced Institute of Science and Technology
Logic
HSS Hss105

Fall 2009
Generalized induction 0
1. 2.
Generalized induction principle Wellfoundedness
Generalized induction
First we define a component relation
between elements of an arbitrary set.
C(x, y): x is a component of y. Ex1. The x<y Ex2. The x+1 =y. Ex3. The C(X, Y
Korea Advanced Institute of Science and Technology
Logic
HSS Hss105

Fall 2009
Generalized induction n4
1. 2. 3.
The least number principle Generalized induction Descending chain
HW #7 (due 11/11) 1. By using one of the induction principles, show that every natural number greater than 1 is a product of prime numbers.
2. Show why Th
Korea Advanced Institute of Science and Technology
Logic
HSS Hss105

Fall 2009
Ma athematical induction
1. 2.
Cantors theorem Mathematical induction
HW #6 (due 11/4)
1. Problem 13.12 (prove by a tableau) 2. Exercise 13.7 (p.129, in the first for
mula Kx should be ~Kx) 3. Show that the set of all rational numbers (of the form a/b
Korea Advanced Institute of Science and Technology
Logic
HSS Hss105

Fall 2009
Infinity ZE
1. 2. 3.
Subsets Sizes of sets Cantors theorem
Midterm graded
Any questions?
HW #6 (due 11/4)
1. Problem 13.12 (prove by a tableau) 2. Exercise 13.7 3. Show that the set of all rational num
bers (of the form a/b where a and b are integers
Korea Advanced Institute of Science and Technology
Logic
HSS Hss105

Fall 2009
Tableaux for firstorder ]= logic
1. 2. 3.
Validity Tableau rules for and Some examples
Midterm today, 9pm Room 403
(bring your student id for entrance to the building)
13.9
Find a formula that is valid in all non
empty domains (hence simply valid) bu
Korea Advanced Institute of Science and Technology
Logic
HSS Hss105

Fall 2009
Firstorder logic jq
1. 2. 3.
All and some Firstorder formulas Validity
Midterm
Monday 19th 9pm, Room 403 (this build
ing) Covers Smullyan, pp.1 119 (up to todays lecture) Format
Truefalse questions Problemsolving questions
F
Similar to Problems i
Korea Advanced Institute of Science and Technology
Logic
HSS Hss105

Fall 2009
all5 and some
1. 2. 3.
HW #4 Puzzles involving all and some The Barber Paradox
Midterm
Monday 19th 9pm Covers Smullyan, pp.1 119 (up to
Wednesdays lecture)
HW #5 due Wednesday
1. 11.1(j) on p.90 2. 11.3(b) on p.94 3. For synthetic tableaux, show: if
Korea Advanced Institute of Science and Technology
Logic
HSS Hss105

Fall 2009
The tableau method :
1. 2. 3.
Two types of tableau rules Correct and complete Synthetic tableau
Midterm
Monday 19th 9pm
HW #5
1. 11.1(j) on p.90 2. 11.3(b) on p.94 3. For synthetic tableaux, show: if X=>Y
and ~X=>Y are provable, so is Y.
HW #3, Q1
k
Korea Advanced Institute of Science and Technology
Logic
HSS Hss105

Fall 2009
The tableau method )
1. 2. 3.
Interpretation The tableau method Two types of tableau rules
A tight schedule
This week: HW #4 due Wed., cover the
rest of Ch.11 Next week: Ch.12 and first half of Ch.13 (HW #5 due) On Oct. 19th cover the rest of Ch.13
Nee
Korea Advanced Institute of Science and Technology
Logic
HSS Hss105

Fall 2009
Relationships between Ug Connectives
1. 2. 3.
Joint denial, Sheffer stroke Binary connectives Relationships between connectives
Errata on p.76 Problem 10.15 => 16
definable from
and =/>
and =/
>
Reading for the next week
Ch.11
HW #4 due Oct. 7th in
Korea Advanced Institute of Science and Technology
Logic
HSS Hss105

Fall 2009
Connectives and varix~ able liars
1. 2. 3.
HW #2 The concept of definability Boolean island
Serious errata in Ch.10
Between problem 10.1 and Problem
10.2:
(should be Problem 10.2) Next a lady from Venus claims to understand the meaning of ~
Therefore,
Korea Advanced Institute of Science and Technology
Logic
HSS Hss105

Fall 2009
Knights and knaves w again
1. 2. 3.
Tautologies, contradictions Application of the symbolic method NGP revisited
Errata in Ch.8
p. 56, 15 lines from bottom, Problem 8.5
> 1.5 p. 57, three lines below NGP Revisited: of > if p. 59, 12 lines from bottom
Korea Advanced Institute of Science and Technology
Logic
HSS Hss105

Fall 2009
Propositional logic
1. 2. 3.
Symbolic method Logical connectives Truth tables
Symbolic method
The strength of modern logic con
sists mainly in the use of symbols (modeled upon math). Ex. A puzzle attributed to Euclid
A mule and a donkey were talking.
Korea Advanced Institute of Science and Technology
Logic
HSS Hss105

Fall 2009
Unifying NGP
w
1. 2. 3.
HW solutions Crazy island An allround NGP
HW #1 graded (3/3/4 points) HW #2 due 9/21 => 9/23(next Wed
nesday)
3.5 (HW solution)
Will you flash a red card in answer to
this question? C: (flashes black)
Will you say yes to this
Korea Advanced Institute of Science and Technology
Logic
HSS Hss105

Fall 2009
Doubling difficulties
1. 2.
Mad or Sane Puzzles Knight/knave, sane/mad combined
Corrections
Typo on the last line of p.34: a knave
=> mad Pronunciation of knave [neiv]
Unlike knack [nk] Trough anyone?
Ch.4 puzzles
If one believes p, then one honestly
s
Korea Advanced Institute of Science and Technology
Logic
HSS Hss105

Fall 2009
Logic of Lying and Truthtelling (2) A/
1. 2. 3.
Metapuzzle Male or female Silent knights and knaves
1.19 (application of NGP)
Three natives a knight, a knave, and a
spy (neither a knight or a knave) Ask two questions to find out who the spy is.
Again th
Korea Advanced Institute of Science and Technology
Logic
HSS Hss105

Fall 2009
Logic of Lying and Truthtelling
1. 2.
Puzzles Nelson Goodman Principle (NGP)
Check out the course web board:
http:/web.kaist.ac.kr/~philist (up and
running)
Typo on p.15
About 10 lines from the top Two claims couldnt both be false => true
The syllab
Korea Advanced Institute of Science and Technology
Logic
HSS Hss105

Fall 2009
HSS105 Logic
firstorder Logic ]P
1. 2. 3.
Administrative information Smullyan and his Logical Labyrinths Content introduction If you havent registered yet, please check out with me for my signature after this session.
Who am I?
Jeongmin Lee, KAIST class