Unit 2950: Mathematical Logic

1. Introduction to Mathematical Logic

• Overview of mathematical logic
• Importance in mathematics and computer science
• Distinction between propositional and predicate logic
• Applications across disciplines

2. Propositional Logic

2.1 Syntax and Semantics

• Propositions and propositional variables
• Logical connectives: AND, OR, NOT, IMPLIES, IFF

2.2 Truth Tables

• Constructing truth tables
• Evaluating logical expressions

2.3 Logical Equivalence

• Definition and examples
• Tautologies and contradictions
• Logical implications and entailment

3. Predicate Logic

3.1 Language of Predicate Logic

• Predicates, functions, constants, and variables
• Terms and atomic formulas

3.2 Quantifiers

• Universal quantifier (∀)
• Existential quantifier (∃)
• Scope and binding

3.3 Relations and Functions

• Expressing relations and functions using predicates
• Examples and interpretations

4. Inference Rules and Proofs

4.1 Deductive Reasoning

• Concept of logical implication
• Validity and soundness

4.2 Rules of Inference

• Modus Ponens
• Modus Tollens
• Hypothetical syllogism
• Disjunctive syllogism
• Addition, simplification, conjunction

4.3 Methods of Proof

• Direct proof
• Proof by contradiction
• Mathematical induction

5. Logical Equivalences

• Commutative laws
• Associative laws
• Distributive laws
• Idempotent laws
• De Morgan’s laws
• Application in simplifying logical expressions

6. Formal Proofs

• Structure of formal proofs
• Use of axioms and definitions
• Deriving theorems from prior results
• Examples of formal proof construction

7. Set Theory and Logic

7.1 Basic Set Concepts

• Sets, subsets, and membership
• Set operations: union, intersection, difference, complement

7.2 Cardinality

• Finite and infinite sets
• Comparing sizes of sets

7.3 Paradoxes

• Russell’s paradox
• Implications for naive set theory

7.4 Logic in Set Definitions

• Using logic to define sets
• Logical characterization of set operations

8. Applications of Mathematical Logic

• Role in computer science (algorithms, programming languages)
• Artificial intelligence (knowledge representation, reasoning)
• Linguistics (formal semantics)
• Philosophy (foundations of mathematics, reasoning)

9. Gödel’s Incompleteness Theorems

• Historical background
• Statement of the first incompleteness theorem
• Statement of the second incompleteness theorem
• Implications for formal systems and mathematics
• Overview of proof ideas (non-technical)