Glossary

3.1 Rules of Inference

3 min read•july 25, 2024

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 to , 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→Q)∧P∴Q(P \rightarrow Q) \land P \therefore Q (If it rains, the ground is wet. It's raining. Therefore, the ground is wet)
  • (MT) states if P implies Q and Q is false, then P must be false (P→Q)∧¬Q∴¬P(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)
  • (HS) combines two conditional statements to form a new one (P→Q)∧(Q→R)∴(P→R)(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∨Q)∧¬P∴Q(P \lor Q) \land \neg P \therefore Q (Either it's raining or snowing. It's not raining. Therefore, it's snowing)
  • (Conj) combines two true statements P∧Q∴(P∧Q)P \land Q \therefore (P \land Q) (It's cold. It's windy. Therefore, it's cold and windy)
  • (Simp) extracts one part of a conjunction (P∧Q)∴P(P \land Q) \therefore P (It's raining and windy. Therefore, it's raining)
  • (Add) introduces a disjunction from a single true statement P∴(P∨Q)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 , , ()

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→Q)∧Q∴P(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→Q)∧¬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

Key Terms to Review (17)

Addition: Addition is a fundamental mathematical operation that combines two or more numbers to obtain a total or sum. It serves as a basic building block for various mathematical concepts and operations, playing a crucial role in the development of arithmetic, algebra, and logic. Understanding addition is essential for solving equations and deriving conclusions through logical reasoning.
Alfred Tarski: Alfred Tarski was a Polish-American logician and mathematician renowned for his contributions to formal semantics, particularly in relation to truth and definability in formal languages. His work laid the groundwork for understanding the relationship between language, logic, and mathematical structures, and is pivotal in exploring the foundations of formal systems, the semantics of first-order logic, and the rules of inference that govern logical reasoning.
Conjunction: In logic, a conjunction is a compound statement formed by combining two or more propositions using the logical connective 'and'. The conjunction is true only when all of its constituent propositions are true, serving as a fundamental operation in building more complex logical expressions and analyzing their truth values.
Contradiction: A contradiction is a logical statement that asserts the simultaneous truth and falsity of a proposition, leading to an inherent inconsistency. It serves as a crucial element in understanding formal systems, where the presence of contradictions can indicate inconsistency and threaten the validity of the system. In logical reasoning, identifying contradictions is essential to apply rules of inference correctly and ensure coherent conclusions.
Direct proof: A direct proof is a method of demonstrating the truth of a mathematical statement by logically deducing it from previously established facts, axioms, or theorems without any assumptions or indirect reasoning. This approach is essential for establishing equivalence relations and can be applied across various proof strategies in mathematics. It serves as a foundation for further proofs and reasoning in first-order logic and is built on the rules of inference that dictate how conclusions are drawn from premises.
Disjunctive syllogism: Disjunctive syllogism is a valid form of argument in propositional logic that asserts if one of two statements is true, and the other is false, then we can conclude the truth of the remaining statement. This logical rule is vital for reasoning within natural deduction systems and rules of inference as it allows for deriving conclusions from disjunctive premises effectively.
Gottlob Frege: Gottlob Frege was a German philosopher, logician, and mathematician, often considered the father of modern logic and analytic philosophy. He introduced key concepts such as the distinction between sense and reference, which are crucial for understanding meaning in both natural and formal languages. His work laid the groundwork for later developments in logic, particularly in natural deduction and semantics.
Hypothetical syllogism: Hypothetical syllogism is a rule of inference that allows one to draw a conclusion from two conditional statements when the conclusion of one serves as the premise for the other. This logical rule can be expressed in the form: if 'P implies Q' and 'Q implies R', then 'P implies R'. It is a fundamental aspect of reasoning that connects different statements through their implications, often used in natural deduction and other logical frameworks.
Law of Excluded Middle: The law of excluded middle states that for any proposition, either that proposition is true or its negation is true. This principle is a foundational concept in classical logic and emphasizes that there is no middle ground between truth and falsehood. It serves as a critical rule of inference, allowing for conclusions to be drawn based on the truth value of propositions.
Modus ponens: Modus ponens is a fundamental rule of inference in propositional logic that allows one to derive a conclusion from a conditional statement and its antecedent. It states that if we have a statement of the form 'If P, then Q' (P → Q) and we know that P is true, we can conclude that Q must also be true. This reasoning forms the basis for various proof strategies, including those used in natural deduction and first-order logic.
Modus tollens: Modus tollens is a fundamental rule of inference in propositional logic that states if a conditional statement is true and the consequent is false, then the antecedent must also be false. This logical form is essential for reasoning and proofs, providing a method to derive conclusions based on the truth values of premises, particularly in mathematical and formal contexts.
Proof by Cases: Proof by cases is a logical strategy where a statement is proven by considering multiple scenarios, or 'cases,' under which the statement can hold true. This method breaks down a complex proof into simpler parts, allowing for a more manageable approach to proving the overall statement.
Proof by Contradiction: Proof by contradiction is a logical argument technique where one assumes the opposite of what they want to prove, and then shows that this assumption leads to a contradiction. This method is useful in various mathematical contexts, allowing for a clearer understanding of statements and their validity by demonstrating the impossibility of the negation.
Simplification: Simplification is a rule of inference used in logical reasoning that allows one to conclude a single component of a disjunction when at least one of its components is known to be true. It serves as a method for reducing complex propositions into simpler statements, thereby making the reasoning process clearer and more manageable. This technique is essential for streamlining arguments and making deductions easier to follow.
Sound Argument: A sound argument is a logical argument that is not only valid but also has all true premises. This means that if the premises are true, the conclusion must also be true. Sound arguments are crucial because they ensure that reasoning is both structurally correct and factually accurate, leading to reliable conclusions.
Tautology: A tautology is a statement that is always true, regardless of the truth values of its components. It plays a crucial role in logical reasoning as it helps in validating arguments and ensuring consistency within formal systems. By understanding tautologies, one can explore logical equivalence, as they are fundamental in establishing the relationships between different propositions and their validity.
Valid argument: A valid argument is a logical structure where, if the premises are true, the conclusion must also be true. This concept is foundational in assessing the strength of logical reasoning, as it ensures that the argument's form guarantees the truth of the conclusion based on the truth of its premises. Understanding valid arguments is crucial in distinguishing sound reasoning from fallacies and understanding how rules of inference function within logical systems.
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