Chapter 6: Symbolic Logic
Introduction to Logic
Symbolic logic formalizes reasoning using symbols and mathematical notation. Logic is the foundation of mathematics, computer science, and philosophy. This chapter covers logical connectives, truth values, logical equivalence, and important logical structures like tautologies and contradictions.
1. Propositions and Truth Values
What is a Proposition?
A proposition is a statement that is either true or false (not both). "2 + 2 = 4" is a proposition (true). "Paris is in England" is a proposition (false). "What time is it?" is not a proposition (it's a question, not a true/false statement).
Notation
We use letters (p, q, r, ...) to represent propositions. We assign truth values: T (true) or F (false). The truth value of a compound proposition depends on the truth values of its components and how they're connected.
2. Logical Connectives
Conjunction (AND)
p ∧ q is true only when both p and q are true. Truth table: T∧T=T, all others are F. In English: "p and q" both hold.
Disjunction (OR)
p ∨ q is true when at least one of p, q is true. It's false only when both are false. Truth table: F∨F=F, all others are T. In English: "p or q" (or both).
Negation (NOT)
¬p is true when p is false, and false when p is true. It flips truth value. Truth table: ¬T=F, ¬F=T.
Implication (IF-THEN)
p → q reads "if p then q" or "p implies q." It's false only when p is true and q is false. Truth table: T→F=F, all others are T. Note: when p is false, p→q is true (vacuously true).
Biconditional (IF AND ONLY IF)
p ↔ q reads "p if and only if q." It's true when p and q have the same truth value. Truth table: p↔q is T when (p=T and q=T) or (p=F and q=F); otherwise F.
3. Truth Tables
Building Truth Tables
For a compound proposition with n simple propositions, create a table with 2^n rows (one for each truth assignment). Compute truth values column by column, starting with simplest components and building to more complex ones.
Example
For (p ∧ q) → ¬r with three propositions, create 8 rows. Compute p∧q, then ¬r, then the implication. This systematic approach prevents errors.
4. Logical Equivalence
Definition
Two propositions are logically equivalent (written ≡ or ⇔) if they have the same truth value for all possible truth assignments. For example, p→q ≡ ¬p ∨ q (implication equals "not p or q").
De Morgan's Laws
¬(p ∧ q) ≡ ¬p ∨ ¬q and ¬(p ∨ q) ≡ ¬p ∧ ¬q. Negation distributes over AND/OR but flips the connective.
Other Important Equivalences
p ∨ p ≡ p (idempotent), p ∧ T ≡ p (identity), p ∨ ¬p ≡ T (law of excluded middle), p ∧ ¬p ≡ F (contradiction).
5. Tautologies and Contradictions
Tautology
A tautology is a proposition that's true for all truth assignments. Example: p ∨ ¬p (either p or not-p; always true). Tautologies represent logical truths independent of content.
Contradiction
A contradiction is false for all truth assignments. Example: p ∧ ¬p (p and not-p; always false). Contradictions represent logical impossibilities.
Contingency
A contingent proposition is sometimes true, sometimes false. Most real propositions are contingent; their truth depends on specific circumstances.
6. Arguments and Validity
What is an Argument?
An argument is a sequence of propositions (premises) followed by a conclusion. The argument is valid if whenever all premises are true, the conclusion must be true.
Testing Validity
Create a truth table with all propositions. Look for rows where all premises are true. In such rows, is the conclusion also true? If yes, the argument is valid. If there's even one row where premises are true but conclusion is false, it's invalid.
Conclusion
Symbolic logic provides a rigorous framework for reasoning. Mastering logical connectives, truth tables, equivalences, and argument validity prepares you for mathematical proof and rigorous thinking in any field.