3.1 Rules of Inference

Inference rules are the building blocks of logical reasoning. They allow us to draw valid conclusions from given premises, forming the foundation for constructing sound arguments. Understanding these rules is crucial for navigating complex logical problems.

From Modus Ponens to Disjunctive Syllogism, each rule serves a specific purpose in logical deduction. By mastering these tools, we can construct valid arguments, determine the validity of existing ones, and avoid common fallacies that lead to faulty reasoning.

Basic Rules of Inference

Basic rules of inference

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• Modus Ponens (MP) asserts if P implies Q and P is true, then Q must be true $(P \rightarrow Q) \land P \therefore Q$ (If it rains, the ground is wet. It's raining. Therefore, the ground is wet)
• Modus Tollens (MT) states if P implies Q and Q is false, then P must be false $(P \rightarrow Q) \land \neg Q \therefore \neg P$ (If it's sunny, it's not raining. It's raining. Therefore, it's not sunny)
• Hypothetical Syllogism (HS) combines two conditional statements to form a new one $(P \rightarrow Q) \land (Q \rightarrow R) \therefore (P \rightarrow R)$ (If I study, I'll pass the exam. If I pass the exam, I'll graduate. Therefore, if I study, I'll graduate)
• Disjunctive Syllogism (DS) eliminates one option from a disjunction $(P \lor Q) \land \neg P \therefore Q$ (Either it's raining or snowing. It's not raining. Therefore, it's snowing)
• Conjunction (Conj) combines two true statements $P \land Q \therefore (P \land Q)$ (It's cold. It's windy. Therefore, it's cold and windy)
• Simplification (Simp) extracts one part of a conjunction $(P \land Q) \therefore P$ (It's raining and windy. Therefore, it's raining)
• Addition (Add) introduces a disjunction from a single true statement $P \therefore (P \lor Q)$ (It's sunny. Therefore, it's sunny or cloudy)

Construction of valid arguments

• Steps for constructing valid arguments:
  1. Identify given premises
  2. Determine desired conclusion
  3. Apply appropriate inference rules to connect premises to conclusion
  4. Ensure each step follows logically from previous steps
• Strategies for argument construction include working backwards from conclusion, identifying key intermediate steps, using multiple rules in combination when necessary
• Common argument structures encompass direct proof, proof by contradiction, proof by cases (law of excluded middle)

Validity determination in arguments

• Criteria for argument validity require all steps follow logically from premises or previous steps, each step uses a valid inference rule, conclusion properly derived from given premises
• Methods for checking validity include truth tables, formal proofs, semantic tableaux (tree method)
• Invalid argument characteristics involve premises not supporting conclusion, incorrect application of inference rules, hidden or unstated assumptions

Common fallacies vs inference rules

• Formal fallacies violate logical structure:
  • Affirming the consequent incorrectly concludes P from Q and P→Q $(P \rightarrow Q) \land Q \therefore P$ (If it rains, the ground is wet. The ground is wet. Therefore, it rained)
  • Denying the antecedent falsely concludes not Q from not P and P→Q $(P \rightarrow Q) \land \neg P \therefore \neg Q$ (If it's sunny, it's warm. It's not sunny. Therefore, it's not warm)
• Informal fallacies include ad hominem (attacking the person), appeal to authority (accepting claim based solely on authority), slippery slope (assuming extreme consequences), false dichotomy (presenting only two options when more exist)
• Strategies to avoid fallacies involve examining logical structure of arguments, identifying unstated assumptions, considering alternative explanations or causes