80-211: Logic and Mathematical Inquiry
(Spring 2012)

Course Information  [  Logistics  |  Course Links  |  Calendar of Classes  |  Coursework Calendar  ]



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Logistics

Lectures: Mo,We   14:30 - 15:50 (room 1190)

Class Webpage:   http://qatar.cmu.edu/cs/80211


Instructors: Wilfried Sieg
Office hours:  by appointment
Office:  CMU-Q 2040
Email:  sieg@cmu.edu
Iliano Cervesato
Office hours:  by appointment (check schedule)
Office:  CMU-Q 1008
Email:  iliano@cmu.edu

Course Links

Calendar of Classes

Click on a class day to go to that particular lecture or recitation. Due dates for homeworks are set in bold. The due date of the next homework blinks.

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Coursework Calendar

Test
Percentage
(learning obj.)		Hw1
6.25%		Hw2
6.25%		Hw3
6.25%		Hw4
6.25%		Exam
15%		Hw5
6.25%		Hw6
6.25%		Hw7
6.25%		Hw8
6.25%		Hw9
6.25%		Hw10
6.25%		Final
30%
Posted25 Jan1 Feb8 Feb15 Feb29 Feb22 Feb14 Mar21 Mar28 Mar4 Apr11 Apr3 May
(8:30-11:30)
[2152]
Due
(23:59)		29 Jan5 Feb12 Feb19 Feb11 Mar18 Mar25 Mar1 Apr8 Apr15 Apr
Corrected1 Feb8 Feb15 Feb22 Feb4 Mar14 Mar21 Mar28 Mar4 Apr11 Apr18 Apr6 May

About this course  [  Description  |  Prerequisites  |  Feedback  |  Readings  |  Grading  |  Assessment  ]

Description

This course has the purpose of introducing first and second year students to elements of formal logic as well as to the historical context in which this discipline developed. As many courses in the undergraduate curriculum rely on students having mastered basic logical notions and skills, it will test and enhance your preparation, thereby putting you in a better position to succeed in your program. It will also help you understand and appreciate how aspect of modern mathematics and computer science came about since these disciplines grew out of logic.

This course is divided into four parts:
  1. Analysis of Informal Arguments
  2. Sentential Logic: Boolean operations, semantics, and derivations
  3. Predicate Logic with equality: quantifiers, semantics and derivations
  4. Formalization of mathematics: elementary set theory, Cantor's theorem; incompleteness; Turing machines and undecidability of predicate logic
This course is a gateway to the fascinating and multifaceted world of computational logic. If you find it enjoyable, you may also like 15-312 Foundations of Programming Languages (a systematic, logic-based investigation to programming languages), 15-317 Constructive Logic (the use of logic to express algorithms and computation), 15-414 Bug Catching: Automated Program Verification and Testing (the use of a logical technique known as model-checking to verify complex systems), and 80-311 Computability and Incompleteness (a study of the logical foundations of Computer Science).

Prerequisites

There are no prerequisites for this course.

Readings

      

Feedback

It is our goal to make this course successful, stimulating and enjoyable. If at any time you feel that the course is not meeting your expectations or you want to provide feedback on how the course is progressing for you, please contact us. If you would like to provide anonymous comments, please use the feedback form on the course home page or slide a note under my door. Comments of general interest will be answered on the course discussion board.

Grading

This is a 9 unit course. For Computer Science majors, it counts as a free CS elective.

Tasks and Percentages

Evaluation Criteria

Academic Integrity

You are expected to comply with the University Policy on Academic Integrity and Plagiarism.

Collaboration is regulated by the whiteboard policy: you can bounce ideas about an assignment, but when it comes to typing it down for submission, you are on your own — no notes, snapshots, etc., you can at most reconstruct the reasoning from memory.

Class etiquette

Assessment

Course Objectives

This course seeks to develop students who:

  1. can understand and formulate logical representations to problems
  2. can think abstractly, i.e., for form, analyze, and use abstractions
  3. can understand and articulate technical ideas
  4. can follow and form cogent arguments
  5. have a historical perspective of logic and Computer Science

Learning Outcomes

Upon successful completion of this course, students will be able to:

  1. express problems symbolically using propositional and predicate logic
  2. describe the importance and limitations of predicate logic
  3. gain deeper knowledge about a topic by researching it using a variety of sources and tools
  4. describe technical concepts clearly, so as to be readily understood by their peers within the discipline and by those outside the discipline
  5. follow and summarize arguments presented in texts and papers used as supporting materials for their courses or consulted to research a topic
  6. form a cogent, logical argument asserting and reiterating technical concepts that lie within the bounds of any researched topic
  7. make clear, well-articulated, well-organized and creative presentations about topics of their choice

Schedule of Classes

At a glance ...

January 2012
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March 2012
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April 2012
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May 2012
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Mon 16 Jan
Lecture 1
Welcome and Course Introduction
We provide an overview over the course, covering topics, approach, and learning goals. We also outline administrative issues, such as assignments, exams, course policies, and grading.
Key concepts:
  • What is logic?
Wed 18 Jan
Lecture 2
Analysis of Informal Arguments
In this lecture, we reflect on how we reason and introduce the fundamental notions of inference. We observe that reasoning is language-independent.
Key concepts:
  • Inferences
  • Formal reasoning
  • Logic and language
Readings:

Mon 23 Jan
Lecture 3
The Syntax of Sentential Logic: Connectives
In this lecture, we examine some of the most elementary linguistic patterns used in reasoning, sentential logic and introduce symbols to express them.
Key concepts:
  • Statements
  • Propositional connectives
Readings:
Wed 25 Jan
Lecture 4
The Syntax of Sentential Logic: Formulas
In this lecture, we combine statements using connectives into logical formulas and practice translating English sentences in symbolic logic.
Key concepts:
  • Propositional formulas
  • Symbolization
  • Valid inferences
Readings:

Mon 30 Jan
Lecture 5
The Semantics of Sentential Logic: Truth Tables
In this lecture, we introduce the semantics of sentential logic, exploiting the syntactic structure of formulae: we provide a truth-definition for all formulae and define the central logical concepts (logical truth and consequence) and the truth table method for deciding related questions.
Key concepts:
  • Truth value assignments
  • Truth tables
  • Validity
Readings:
Wed 1 Feb
Lecture 6
The Semantics of Sentential Logic: Truth Trees
In this lecture, we introduce truth trees, a second method for determining the validity of a formula or of an inference.
Key concepts:
  • Satisfiability
  • Truth trees
Readings:

Mon 6 Feb
Lecture 7
Derivations in Sentential Logic: Introduction Rules
In this lecture, we introduce the concept of derivation and examine the introduction rules for the logical connectives.
Key concepts:
  • Inference rules
  • Derivations
  • Introduction rules
Readings:
Wed 8 Feb
Lecture 8
Derivations in Sentential Logic: Elimination Rules
In this lecture, we examine the elimination rules for the logical connectives.
Key concepts:
  • Elimination rules
  • Reasoning by cases
Readings:

Mon 13 Feb
Lecture 9
Derivations in Sentential Logic: Negation
In this lecture, we examine the inference rules relating to negation and falsehood.
Key concepts:
  • Rules for falsehood
  • Rules for negation
  • Reasoning by contradiction
Readings:
Wed 15 Feb
Lecture 10
Derivations in Sentential Logic: Double Negation
In this lecture, we introduce additional inference rules regarding negation and show how they can be used to derive additional formulas.
Key concepts:
  • Double negation rules
  • Classical vs. intuitionistic logic
Readings:

Mon 20 Feb
Lecture 11
Sentential Logic Equivalences: Conjunction and Disjunction
In this lecture, we examine a number of logical equivalences regarding conjunction and disjunction and show how they can be used as derived rules in proofs.
Key concepts:
  • Logical equivalence
  • Derived rules
  • Cut
  • De Morgan's Law
Readings:
Wed 22 Feb
Lecture 12
Sentential Logic Equivalences: Implication
In this lecture, we introduce logical equivalences concerning implication and the related derived rules.
Key concepts:
  • Modus tollens
  • Transitivity
  • Negated conditionals
Readings:

Mon 27 Feb
Lecture 13
Elementary Meta-mathematics in Sentential Logic
In this lecture, we add a new connective the language, the biconditional, and prove the completeness of the derivation system, i.e., that every tautology can be derived.
Key concepts:
  • Biconditional
  • Normal forms
  • Completeness
Readings:
Wed 29 Feb
Exam
Midterm

Mon 5 Mar
No class (Spring Break)
Wed 7 Mar

Mon 12 Mar
Lecture 14
Terms in Predicate Logic: Syntax
In this lecture, the logical language is extended to incorporate the notions of predicates and terms, allowing a new dimension of logical structure to be represented.
Key concepts:
  • Structured atoms
  • Predicates
  • Singular terms
Readings:
Wed 14 Mar
Lecture 15
Terms in Predicate Logic: Semantics
In this lecture, we give a meaning to predicative formulas by intepreting constants relative to a domain of discourse and predicates as relations in this domain of discourse.
Key concepts:
  • Domain of discourse
  • Intepretations
Readings:

Mon 19 Mar
Lecture 16
Quantifiers in Predicate Logic: Syntax
In this lecture, we extend the logical language further by introducing the notion of quantification, giving us the full language of predicate logic.
Key concepts:
  • Variables
  • Quantifiers
Readings:
Wed 21 Mar
Lecture 17
Quantifiers in Predicate Logic: Semantics
In this lecture, we revise the semantics of the quantifier-free predicate logic and extend the truth tree method to work on them.
Key concepts:
  • Interpretations revisited
  • More truth trees
Readings:

Mon 26 Mar
Lecture 18
Derivations in Predicate Logic
In this lecture, we extend the derivation system from sentential logic for use with predicate arguments by adding introduction and elimination rules for the quantifiers.
Key concepts:
  • Introduction rules
  • Elimination rules
Readings:
Wed 28 Mar
Lecture 19
Equivalences in Predicate Logic
In this lecture, we present a number of derived rules, which capture the interaction of quantifiers and sentential connectives. These fundamental connections allow us to establish a normal form theorem for the formulae of predicate logic.
Key concepts:
  • Quantifiers and bound variables
  • Predicative equivalences
  • Cantor's Theorem
Readings:

Mon 2 Apr
Lecture 20
Equality in Predicate Logic: Identity of Constants
In this lecture, we extend predicate logic with equality, which allows us to identify two terms.
Key concepts:
  • Identity
  • Inference rules
  • Functional predicates
Readings:
Wed 4 Apr
Lecture 21
Equality in Predicate Logic: Function Symbols
In this lecture, we introduce composite terms in the language by means of function symbols.
Key concepts:
  • Composite terms
  • Interpretation
  • Derivations
Readings:

Mon 9 Apr
Lecture 22
Elementary Set Theory: Axiomatization
In this lecture, we define a system of axioms that captures the notion of set and its operations.
Key concepts:
  • Zermelo-Fraenkel axioms
  • Interpretation of set theory
Readings:
  • The Universal Computer (ch. 6: Gödel)
Wed 11 Apr
Lecture 23
Elementary Set Theory: Functions
In this lecture, we characterize functions as sets, give logical expressions for fundamental classes of functions and relate them.
Key concepts:
  • Set-theoretic functions
  • Injection, surjection, bijection
  • Cantor-Bernstein theorem
Readings:
  • The Universal Computer (ch. 6: Gödel)

Mon 16 Apr
Lecture 24
Incompleteness of Set Theory
In this lecture, we show that the objects of predicate logic (terms, formulas, derivations) can be represented as sets.
Key concepts:
  • Syntactic objects as sets
  • Representability of syntax
Readings:
  • The Universal Computer (ch. 6: Gödel)
Wed 18 Apr
Lecture 25
Incompleteness of Set Theory
In this lecture, we build a the representation of a sentence that talks about itself, and with it follow the steps of Kurt Gödel in showing that there are true formulas that are not derivable in set theory.
Key concepts:
  • Self-referential sentences
  • Gödel's first incompleteness theorem
Readings:
  • The Universal Computer (ch. 7: Turing)

Mon 23 Apr
Lecture 26
Undecidability of predicate logic
In this lecture, we define Turing machines, a device to capture the notion of computation, and use it to show that there are problems whose solution cannot be computed.
Key concepts:
  • Turing machines
  • Halting problem
Readings:
  • The Universal Computer (ch. 7: Turing)
Wed 25 Apr
Lecture 27
Undecidability of predicate logic
In this lecture, we build on undecidability of the Halting problem and show that predicate logic is itself undecidable.
Key concepts:
  • Reductions
  • Turing's theorem
Readings:
  • The Universal Computer (ch. 9: Beyond)

Thu 3 May
(8:30-11:30)
[2152]
final
Final
© 2012 Iliano Cervesato iliano@cmu.edu