Introduction to Logic

Instructor: Mark Thorsby

• Course overview: Basics of categorical, propositional, and predicate logic
• Focus today: Propositional logic, specifically natural deduction and rules of implication

Propositional Logic

Natural Deduction

• Method for building proofs in propositional logic
• Goal: Prove valid arguments using specific argument forms

Review: Truth Table Method

• Determines the validity of arguments by identifying invalid instances (all true premises and a false conclusion)
• Example: If P then Q; P; therefore Q
• If no invalid instance in truth table, argument is valid

Key Argument Forms in Natural Deduction

1. Modus Ponens

   • Form: If P then Q; P; therefore Q
   • Example: If Joey joins the circus, then he'll be a clown; Joey joins; therefore, Joey is a clown

2. Modus Tollens

   • Form: If P then Q; not Q; therefore not P
   • Example: If it rains, you'll get wet; you're not wet; therefore, it must not be raining

3. Hypothetical Syllogism

   • Form: If P then Q; If Q then R; therefore, if P then R
   • Example: If Noah drops milk, his mother gets mad; if mad, Noah gets in trouble; therefore, if Noah drops milk, he gets in trouble

4. Disjunctive Syllogism

   • Form: P or Q; not P; therefore Q
   • Example: Either you get an A or you don't; you don't get an A; therefore, you get a B

Building Propositional Proofs

• Learning the argument forms is crucial—must memorize them
• Proof construction involves using these forms to deduce conclusions
• Order of premises is irrelevant
• Conclusion written on the right side of a forward slash
• Note every derived deduction including the form and lines used
• Look for substitution instances

Strategies for Constructing Proofs

• Start with the conclusion to understand what needs proving
• Use substitution instances where parts of formulas stand for larger compound statements
• Validity in proofs only; cannot prove invalidity, unlike truth tables

Example Problems

• Example 1: Given premises, deduce a conclusion using rules of implication (e.g., Modus Ponens, Modus Tollens)
• Example 2: Identify missing premises in a given deduction

Homework and Practice

• Demonstrate understanding of argument forms by recognizing substitution instances and building proofs
• Practice with multiple examples to become proficient in recognizing patterns

Conclusion

• The importance of memorizing argument forms for effective proof construction
• Future lessons will introduce more rules and complex problems

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Note: The skills learned in this section include analyzing and constructing logical arguments using a structured method, which is valuable for developing critical thinking and problem-solving abilities in logic and beyond.