Logic Notes: Deductive & Inductive Arguments

Key Concepts and Definitions

  • Logic: study of reasoning and the evaluation of arguments.

  • Argument: a series of statements (sentences that are either true or false) where at least some are premises and one statement is the conclusion such that the premises are intended to give reasons for the conclusion.

    • Textbook definition: a group of statements in which the conclusion is claimed to follow from the premise.

  • Standard Form: Argument Form

    • Premise 1

    • Premise 2

    • Premise 3

    • …

    • Conclusion

  • Point of an Argument

    1. Persuasion

    • Change beliefs/actions (e.g., professor arguing why you should take this course)

    • Intend to persuade and change mind

    • Could include bad reasons, false premises and fallacies

    1. Justification

    • Intended to give reasons

    • Intend to be true, valid and sound

  • Explanations: Give reasons for why or how an event occurred; explain why something is true or help someone understand; aim to fit a phenomenon into a general pattern to remove bewilderment.

    • Example: a professor explaining why this course is offered at Lake Forest College.

  • Types of Arguments

    • Deductive: the conclusion follows necessarily from the premises; if the premises are true, the conclusion is impossible to be false.

    • Premise-Conclusion structure.

    • Type labels: Valid, Invalid, Sound, Unsound

    • Inductive: the premises make the conclusion probable; if the premises are true, the conclusion is unlikely to be false.

    • The conclusion is not guaranteed to be true; it is considered likely or probable given the premises.

Deductive vs Inductive Arguments

  • Deductive Arguments

    • Valid: if premises are true → conclusion must be true (impossible for it to be false).

    • Invalid: even if premises are true → conclusion can still be false.

    • Sound: valid reasoning with all true premises.

    • Unsound: invalid reasoning, or at least one false premise (or both).

    • Note: a conclusion does not have to follow from the premises if the argument is invalid.

    • Important: "Soundness" is a property of the argument; "Validity" is a property of the argument.

    • Statement is not a property; statements can be true or false.

    • Formalization: a deductive argument is valid iff the conjunction of premises implies the conclusion is a tautology:
      (P<em>1∧P</em>2∧⋯∧Pn)→C(P<em>1 \land P</em>2 \land \dots \land P_n) \rightarrow C
      is a tautology.

  • Inductive Arguments

    • Strong: if the premises are true, then the conclusion is probable; or if the premises are true, then it is improbable that the conclusion is false.

    • Weak: if the premises are true, then the conclusion is not probable.

    • Cogent: a strong inductive argument in which the premises are true.

    • Uncogent: either a weak argument or at least one false premise.

    • Note: There’s no such thing as a valid inductive argument in the strict sense; validity applies to deductive arguments.

    • Statement about truth: the property terms for inductive arguments are strength of support (cogency) rather than validity.

Important Clarifications

  • There’s no such thing as a valid statement. Validity is a property of arguments, not of individual statements.

  • Statement property: truth or falsehood; Arguments have properties like validity, soundness, cogency, and strength.

Patterns and Structures in Arguments

  • The abstract pattern or structure of reasoning:

    • Example pattern:

    • All A are B.

    • All B are C.

    • Therefore, all A are C.

  • Statement Form: the general template of a statement with variables (A, B, C) instead of concrete terms.

  • Substitution Instance: a specific case of a statement form when variables are replaced with actual terms.

    • Example:

    • All dogs are mammals.

    • All mammals are animals.

    • Therefore, all dogs are animals.

  • Counterexample: an example that shows an argument form is invalid by making premises true and the conclusion false.

Enthymemes

  • Enthymemes: arguments with an unstated (implicit) premise or conclusion.

    • Example: “Socrates is human, so he is mortal.”

    • Unstated premise: All humans are mortal.

Theories of Interpretation and Evaluation

  • Principle of Charity: interpret an argument in its strongest, most reasonable form before evaluating it.

    • Purpose: prevent misrepresentation (straw manning) someone’s reasoning.

    • Note: a valid argument can have all false premises; validity concerns logical structure, not truth.

Quick Reference: Key Terms and Relationships

  • Statement: a sentence that can be true or false.

  • Argument: a set of statements with a conclusion supported by premises.

  • Premises: statements assumed to be true for the purpose of argument.

  • Conclusion: the statement that follows from the premises.

  • Validity: structural property of an argument; if the premises are true, the conclusion must be true.

  • Soundness: validity plus all premises true.

  • Cogency: inductive analogue of soundness; strong + true premises.

  • Strength/Weakness (Inductive): degree to which premises support the conclusion.

  • EnTHOOK/Enthymeme: an argument with an implicit premise or conclusion.

  • Substitution Instance: applying a general form to a specific case.

  • Counterexample: a case that refutes a form by showing premises true but conclusion false.

  • Pattern terminology: Specific-General structure (for inductive arguments).

  • Commitment to truth: Validity does not guarantee truth of premises; it guarantees the logical connection.

Quick Examples for Review

  • Deductive Pattern (All A are B; All B are C; Therefore All A are C)

  • Substitution Instance: All dogs are mammals; All mammals are animals; Therefore all dogs are animals.

  • Enthymeme: Socrates is human, so he is mortal (unstated premise: All humans are mortal).

  • Principle of Charity: Always reconstruct the strongest form before judging.

Notes on Logical Evaluation

  • A valid deductive argument can have all false premises and still be valid (the conclusion would also be false in that case).

  • Validity concerns only the logical structure; truth of premises is a separate issue.

  • Inductive arguments are assessed by strength (strong/weak) and by cogency (true premises + strong support).

  • Enthymemes require identifying the hidden premise or conclusion to assess validity and soundness.