Overview
This topic introduces logical reasoning skills, including evaluating statements, constructing arguments, and identifying logical fallacies. You'll learn the structure of conditional statements and how to use logic to determine validity, equivalence, and truth value in reasoning.
Key Concepts and Structures
- Statements: Sentences that are either true or false. Not questions or commands.
- Logical Connectives:
- Conjunction (and):
p ∧ q
- Disjunction (or):
p ∨ q
- Negation (not):
¬p
- Conditional:
p → q (if p, then q)
- Biconditional:
p ↔ q (p if and only if q)
- Truth Tables: Used to determine the truth values of compound statements.
- Logical Equivalence: Two statements that have the same truth value in all cases.
- Converse:
q → p
- Inverse:
¬p → ¬q
- Contrapositive:
¬q → ¬p
- Valid Argument: Conclusion follows logically from premises.
- Counterexample: An example that disproves a universal statement.
Step-by-Step Example
Problem: Write the converse, inverse, and contrapositive of the statement: "If it is raining, then the ground is wet."
Step 1: Identify p = it is raining, q = the ground is wet
Step 2:
- Converse: If the ground is wet, then it is raining.
q → p
- Inverse: If it is not raining, then the ground is not wet.
¬p → ¬q
- Contrapositive: If the ground is not wet, then it is not raining.
¬q → ¬p
Final Summary: Logical statements can be rewritten to examine their structure and validity. Only the contrapositive is always logically equivalent to the original.
Quick Tip
To test logical equivalence or validity, use a truth table or counterexample. The contrapositive of a statement always shares the same truth value as the original.