Introduction: Mathematical
Paradoxes
Intuitive approach. Until recently, till the end
of the 19th century, mathematical theories
used to be built in an intuitive or axiomatic
way.
The historical development of mathematics
has shown that it is not sucient
CSE371
Q8
Fall 2014
(20pts )
NAME
ID:
Math/CS
Here are 10 YES/NO questions
1. CIRCLE proper ANSWER - 1point
2. WRITE detailed JUSTIFICATION - 1point
10 QUESTIONS
1.
xR (x2 < 0) xR (x 0)
JUSTIFY:
y
y
n
y
2.
n
n
y
n
For any predicate formulas A(x), B(x), th
CSE/MAT371
QUIZ 1 SOLUTIONS Fall 2015
(20pts)
PART 1: DEFINITIONS
D1. Write denition of a LOGICAL PARADOX
Logical Paradoxes, also called Logical Antinomies are paradoxes concerning the notion of a set
2. Give an example (by name ) of a logical paradox
Her
CSE371 Q3 Fall 2014
Solutions
QUESTION 1 Given a proof system:
S = (Lcfw_, , F,
1.
LA = cfw_(A A), (A (A B),
(r)
(A B)
)
(B (A B)
Prove that S is sound under classical semantics.
Denition: System S is sound if and only if
(i) Axioms are tautologies and
(i
CSE371 Q3 Fall 2014
(20pts + 5 extra pts)
NAME
ID:
Math/CS
QUESTION 1 (10pts) Given a proof system:
S = (Lcfw_, , F,
LA = cfw_(A A), (A (A B),
(r)
1.
Prove that S is sound under classical semantics.
2.
Write a formal proof of your choice in S with 2 appli
CSE371 Q2 Fall 2014
(20pts + 10 extra pts)
NAME
ID:
CS/MAT
PART 1: Denitions
Q1 (2pt)
Write a denition of a propositional language L
Q2 (3pts)
Write the denition of the set of formulas of a language Lcfw_,
Q3 (5pts)
Given the truth assignment (in classic
CSE371
Q4
Fall 2014
(20pts)
NAME
ID:
Math/CS
QUESTION 1 (12pts)
Consider a system RS obtained from RS by changing the sequence into in all of the rules of inference
of RS
Axioms are the same
1. Construct THREE dierent decomposition trees one for RS and TW
CSE371 MIDTERM Fall 2014
(100pts + 25 extra pts)
NAME
ID:
Math/CS
PART 1: DEFINITIONS 20pts
D1 5pts
Given a language Lcfw_, and a formula A of this language
Write a denition of |= A
D2 5pts
Given two languages:
L1 = LCON1 and L2 = LCON2 , for CON1 = CON2
CSE371
Q2 Fall 2014
Solutions
Problem 1a
Given a formula
A = (a (b (b c)
and its restricted model
vA : cfw_a, b, c cfw_T, F ,
vA (a) = T, vA (b) = T, vA (c) = F
Extend vA to the set of all propositional variables VAR to obtain 2 dierent, non restricted m
CSE371
Q5 Extra Credit
(20pts)
NAME
Fall 2014
ID:
Denition 1
Dene a set F of all formulas of a propositional language Lcfw_,
1
Math/CS
Denition 2
Given the truth assignment (in classical semantics)
v : V AR cfw_T, F
Dene its extension v to the set F of a
Chapter 13: Predicate Languages
Predicate Languages are also called First Order Languages. The same applies to the
use of terms Propositional and Predicate
Logic; they are often called zero Order and
First Order Logics and we will use both
terms equally.
Chapter 12:
Gentzen Sequent Calculus for
Intuitionistic Logic
Part 1: LI System
The proof system LI was published by Gentzen
in 1935 as a particular case of his proof
system LK for the classical logic.
We discussed a version of the original Gentzens
syste
Chapter 11: Automated Proof
Systems
SYSTEM RS OVERVIEW
Hilbert style systems are easy to dene and
admit a simple proof of the Completeness
Theorem but they are dicult to use.
Automated systems are less intuitive then
the Hilbert-style systems, but they wi
Chapter 2: Introduction to
Propositional Logic
PART ONE: History and Motivation
Origins: Stoic school of philosophy (3rd century B.C.), with the most eminent representative was Chryssipus.
Modern Origins: Mid-19th century - English
mathematician G. Boole,
Chapter 3: Propositional
Languages
We dene here a general notion of a propositional language.
We show how to obtain, as specic cases, various languages for propositional classical
logic and some non-classical logics.
We assume the following :
All proposit
Chapter 4: Classical Propositional
Semantics
Language :
Lcfw_,.
Classical Semantics assumptions:
TWO VALUES: there are only two logical
values: truth (T) and false (F), and
EXTENSIONALITY: the logical value of a
formula depends only on a main connective
a
Chapter 5: Some Extensional
Many Valued Semantics
First many valued logic (dened semantically
only) was formulated by Lukasiewicz in 1920.
We present here ve 3-valued logics semantics that are named after their authors:
Lukasiewicz, Kleene, Heyting, and B
Chapter 6
Propositional Tautologies, Logical
Equivalences, Denability of
Connectives and Equivalence of
Languages
Propositional Tautologies for Implication
Modus Ponens known to the Stoics (3rd century B.C)
|= (A (A B ) B )
Detachment
|= (A (A B ) B )
|=
Chapter 7: Proof Systems:
Soundness and Completeness
Proof systems are built to prove statements.
Proof systems are an inference machine with
special statements, called provable statements being its nal products.
The starting points of the inference are c
Chapter 8
Hilbert Systems, Deduction
Theorem
Hilbert Systems The Hilbert proof systems
are based on a language with implication
and contain a Modus Ponens rule as a rule
of inference.
Modus Ponens is the oldest of all known rules
of inference as it was al
Chapter 9
Completeness Theorem Proofs
We consider a sound proof system (under classical semantics)
S = ( Lcfw_,,
AL,
M P ),
such that the formulas listed below are provable in S .
1. (A (B A),
2. (A (B C ) (A B ) (A C ),
3. (B A) (B A) B ),
1
4. (A A),
5.
Chapter 10: Introduction to
Intuitionistic Logic
PART 1: INTRODUCTION
The intuitionistic logic has developed as a
result of certain philosophical views on the
foundation of mathematics, known as intuitionism.
Intuitionism was originated by L. E. J. Brouwe
CSE371 Q7
Fall 2014
(20pts + 10 extra)
NAME
ID:
Math/CS
QUESTION 1
Given a Mathematical Statement written with logical symbols
xR nN (x + n > 0 mN (m = x + n)
1. Translate it into a LOGICAL FORMULA with restricted domain quantiers
2. Translate your restri
CSE371
Q6
Fall 2014
(20pts)
NAME
ID:
QUESTION 1
Let GL be the Gentzen style proof system for classical logic.
Prove, by constructing a proper decomposition tree that
1. Prove, by constructing a proper decomposition tree that
GL (a
b) b) (b (a b)
1
Math/C
cse371/ mat371
LOGIC
Professor Anita Wasilewska
Fall 2014
LECTURE 18
Chapter 13
Part 1: Predicate Languages
Predicate Languages
Predicate Languages are also called First Order
Languages
The same applies to the use of terms for Propositional and
Predicate
cse371/ mat371
LOGIC
Professor Anita Wasilewska
Fall 2014
LECTURE 14
Chapter 11
Introduction to Intuitionistic Logic
Short History
Intuitionistic logic has developed as a result of certain
philosophical views on the foundation of mathematics, known
as int
Introduc)on to
Predicate Logic
Part 1
Lecture 16
cse371/ math371
Professor Anita Wasilewska
Predicate Logic Language
Symbols:
1. P, Q, R predicates symbols, denote rela6ons
in real life, countably inn
cse371/ mat371
LOGIC
Professor Anita Wasilewska
Fall 2014
LECTURE 12
Chapter 10
CLASSICAL AUTOMATED PROOF SYSTEMS
PART 1: RS SYSTEM
PART 2: RS1, RS2, RS3 SYSTEMS
PART 3: GENTZEN SYSTEMS
CLASSICAL AUTOMATED PROOF SYSTEMS
Hilbert style systems are easy to d
cse371/mat371
LOGIC
Professor Anita Wasilewska
Fall 2015
DEFINITIONS and FACTS for QUIZ 1
Denition
Logical Paradoxes, also called Logical Antinomies are
paradoxes concerning the notion of a set
FACT
Russell Paradox
Consider the set A of all those sets X s