Logic
For Beginners
1. Introduction to Mathematical Logic:
1.1. What is Mathematical Logic?
Mathematical logic is a branch of mathematics that explores the principles and structure of valid reasoning. It provides a framework for analyzing and understanding the nature of mathematical reasoning and proofs.
1.2. Propositions and Statements:
- Propositions: Statements that are either true or false but not both. For example, "The sky is blue" is a proposition.
- Compound Propositions: Formed by combining simple propositions using logical operators.
1.3. Logical Connectives:
- Negation (¬): Represents "not." If p is true, ¬p is false, and vice versa.
- Conjunction ( ∧): Represents "and." p ∧ q is true only when both p and q are true.
- Disjunction ( ∨): Represents "or." p ∨ q is true if at least one of p or q is true.
- Implication ( →): Represents "if... then." p → q is false only when p is true and q is false.
- Biconditional ( ↔): Represents "if and only if." p ↔ q is true if both are true or both are false.
2. Truth Tables:
2.1. Constructing Truth Tables:
- Truth Values: Assigning true or false values to propositions and evaluating compound propositions.
2.2. Logical Equivalence:
- Two propositions are logically equivalent if they have the same truth values under all possible circumstances.
3. Predicate Logic:
3.1. Introduction to Predicates:
- Predicates: Statements containing variables. For example, P(x) could be "x is even."
3.2. Quantifiers:
- Universal Quantifier (∀): Represents "for all" or "for every." ∀x P(x) means "P(x) is true for every x."
- Existential Quantifier (∃): Represents "there exists." ∃x P(x) means "There exists an x for which P(x) is true."
4. Rules of Inference:
4.1. Modus Ponens:
- If p → q is true and p is true, then q is true.
4.2. Modus Tollens:
- If p → q is true and q is false, then ¬p is true.
4.3. Hypothetical Syllogism:
- If p → q is true and q → r is true, then p → r is true.
5. Mathematical Proof:
5.1. Direct Proof:
- A direct proof establishes the truth of a proposition by a series of logical implications.
5.2. Proof by Contraposition:
- Proving a proposition by proving the contrapositive.
5.3. Proof by Contradiction:
- Assuming the negation of what you want to prove and showing that this assumption leads to a contradiction.
6. Set Theory and Logic:
6.1. Sets and Operations:
- Union (∪): A ∪ B includes all elements in A or B or both.
- Intersection (∩): A ∩ B includes only elements common to both A and B.
- Complement ('): A' includes elements not in A.
6.2. De Morgan's Laws:
- ¬(A ∪ B) = ¬A ∩ ¬B
- ¬(A ∩ B) = ¬A ∪ ¬B
7. Applications of Mathematical Logic:
7.1. Computer Science:
- Logical operators form the basis of Boolean algebra, essential in computer science and digital circuit design.
7.2. Linguistics:
- Logical structure is crucial in analyzing and understanding the syntax of natural languages.
7.3. Philosophy:
- The study of logic is fundamental to philosophical reasoning and argumentation.
8. Conclusion:
Mathematical logic is a powerful tool for constructing rigorous arguments and understanding the structure of reasoning. As a beginner, start with the basics of propositions, logical connectives, and truth tables. Gradually delve into predicate logic, rules of inference, and proof techniques.
Remember, practice is key in mastering mathematical logic. As you become more familiar with the concepts, you'll gain a deeper understanding of logical reasoning and its applications in various fields.
Enjoy your exploration into the fascinating world of mathematical logic!