Logic

Logic Curriculum

This curriculum provides a structured overview of the study of logic, from its foundational principles to advanced topics.

Core Concepts

Basic Reasoning/Informal Logic: (Foundation)

• Identifying Arguments (Premises and Conclusions)
• Recognizing Common Fallacies (e.g., Ad Hominem, Straw Man)
• Inductive vs. Deductive Reasoning
• Evaluating the Strength of Arguments
• Understanding Implicit Assumptions
• Basic use of words like "all", "some", "none" in everyday arguments.

Symbolic Representation (Introduction): (Foundation)

• The need for formalizing arguments.
• Representing Simple Statements with Letters (e.g., P, Q, R)
• Using Symbols for Basic Connectives (e.g., ¬, ∧, ∨)
• The Concept of a Formal Language
• The relationship between a natural language sentance and its symbolic representation.

Propositional Logic: (Core Logic)

• Atomic Propositions
• Compound Propositions
• Logical Connectives (¬, ∧, ∨, →, ↔)
• Truth Tables
• Tautologies, Contradictions, and Contingencies
• Logical Equivalence
• Rules of Inference (e.g., Modus Ponens, Modus Tollens)
• Formal Proofs (e.g., Natural Deduction)

Predicate Logic (First-Order Logic): (Core Logic)

• Predicates and Subjects
• Quantifiers (Universal, Existential)
• Variables and Constants
• Formulas and Sentences
• Bound and Free Variables
• Interpretations and Models
• Rules of Inference for Quantifiers
• Formal Proofs in Predicate Logic

Quantifiers (Universal, Existential): (Part of Predicate Logic)

• Universal Quantifier (∀) - "for all"
• Existential Quantifier (∃) - "there exists"
• Scope of Quantifiers
• Negation of Quantified Statements
• Nested Quantifiers
• Quantifiers and Domains of Discourse

Logical Connectives (AND, OR, NOT, IMPLICATION, BICONDITIONAL): (Part of Propositional/Predicate Logic)

• Negation (¬)
• Conjunction (∧) - "and"
• Disjunction (∨) - "or"
• Implication (→) - "if...then"
• Biconditional (↔) - "if and only if"
• Truth Tables for Each Connective
• Precedence of Connectives

Truth Tables: (Tool within Propositional Logic)

• Constructing Truth Tables for Compound Propositions
• Using Truth Tables to Determine Validity
• Using Truth Tables to test for logical equivalence.
• Understanding the relationship between true and false.

Logical Equivalence: (Concept within Propositional/Predicate Logic)

• Definition of Logical Equivalence
• Common Logical Equivalences (e.g., De Morgan's Laws)
• Using Truth Tables to Prove Equivalence
• Using Rules of Inference to Prove Equivalence
• Substitution of Equivalents

Rules of Inference (Modus Ponens, Modus Tollens, etc.): (Core Logic)

• Modus Ponens
• Modus Tollens
• Hypothetical Syllogism
• Disjunctive Syllogism
• Introduction and Elimination Rules
• Formal Derivations

Formal Proofs: (Core Logic)

• Natural Deduction
• Axiomatic Systems
• Proof Strategies
• Constructing Valid Derivations
• Conditional Proofs
• Indirect Proofs (Proof by Contradiction)

Soundness and Completeness: (Metalogical Concepts)

• Definition of Soundness
• Definition of Completeness
• The Relationship Between Soundness and Completeness
• Gödel's Completeness Theorem (for First-Order Logic)
• Limitations of Soundness and Completeness

Gödel's Incompleteness Theorems: (Subject of Mathematical Logic/Philosophy of Mathematics)

• Gödel's First Incompleteness Theorem
• Gödel's Second Incompleteness Theorem
• The Concept of Undecidability
• Implications for Formal Systems
• The Significance for the Foundations of Mathematics
• Arithmetization of syntax.

Related but Separate Subjects

Set Theory

• Basic Set Operations
• Types of Sets
• Relations and Functions
• Cardinality
• Axiomatic Set Theory

Model Theory

• First-Order Structures
• Interpretations
• Elementary Equivalence
• Completeness Theorem
• Compactness Theorem

Proof Theory

• Formal Systems
• Natural Deduction
• Sequent Calculus
• Proof Normalization
• Constructive Proofs

Boolean Algebra

• Boolean Operations
• Boolean Expressions
• Truth Tables
• Logic Gates
• Applications in Computer Science

Higher-Order Logic

• Quantification over predicates
• Lambda calculus
• Type theory
• Applications in mathematics

Modal Logic

• Necessity and Possibility
• Kripke Semantics
• Axiomatic Systems
• Applications in Philosophy and Computer Science

Intuitionistic Logic

• Constructive Proofs
• Heyting Algebra
• Brouwer-Heyting-Kolmogorov interpretation
• Relevance to constructive mathematics

Temporal Logic

• Reasoning about time
• Linear Temporal Logic (LTL)
• Computation Tree Logic (CTL)
• Applications in computer science

Fuzzy Logic

• Degrees of truth
• Fuzzy sets
• Fuzzy operations
• Applications in control systems and AI

Non-Classical Logics

• Many-Valued Logics
• Paraconsistent Logic
• Substructural Logics
• Relevance Logic

Computability Theory

• Turing Machines
• Decidability and Undecidability
• Church-Turing Thesis
• Computational Complexity

Category Theory

• Categories and Functors
• Universal Properties
• Adjunctions
• связь с логикой и основаниями математики