Deductive Reasoning

Deductive Reasoning Formula Sheet

  • Categorical Syllogisms

    • Definition: A categorical syllogism is a form of logical reasoning that consists of three categorical terms: a major term, a minor term, and a middle term. Its primary function is to establish a conclusion derived from the premises provided by the terms used.

    • Major Term:

      • Position: The major term is the predicate of the conclusion, representing the attribute or property that the subject possesses or lacks.

      • Occurrence: It appears only once in the major premise, which is the first premise in a syllogism.

    • Minor Term:

      • Position: The minor term serves as the subject in the conclusion and denotes the individual or group being evaluated.

      • Occurrence: It appears once in the minor premise, which is the second premise in the syllogism.

    • Middle Term:

      • Occurrences: This term is crucial in connecting the major and minor premises; it appears once in both the major and minor premises.

      • Note: The middle term is not included in the conclusion, as it serves solely to bridge the argument between the major and minor premises.

Mood of Categorical Syllogisms

  • Notation: Each mood of syllogism can be expressed through standardized symbols for clarity.

    • A - Represents the categorical proposition "All S are P," indicating a universal affirmative.

    • E - Represents the proposition "No S are P," indicating a universal negative.

    • I - Represents the proposition "Some S are P," a particular affirmative which may be illustrated with the symbol .

    • O - Represents "Some S are not P," indicating a particular negative.

Valid Syllogisms

Modus Ponens

  • Proposition: This rule of inference allows one to draw a certain conclusion when the antecedent is affirmed:

    • If a universal affirmative claim is made (All S are P), then:

      • The conclusion drawn can either be that All S are P, No S are P, Some S are P, or Some S are not P based on the logical structure.

Modus Tollens

  • Structure: A rule that enables the rejection of a hypothesis:

    • If A then B (A → B)

    • A is affirmed

    • Therefore, B must also be affirmed due to the logical consequence of the original proposition.

Fallacies

  • Affirming the Consequent: This common fallacy occurs when the consequent is affirmed without validating the antecedent:

    • Structure:

      • If A then B (A → B)

      • B is affirmed

      • Therefore, A is concluded, which is logically unsound.

Distribution of Terms

Terms Distributed:

  • S: In a syllogism, S is distributed in the major premise and serves as the subject in the conclusion, ensuring clarity of logical relationships.

  • S and P: The distribution status of S and P is context-dependent, requiring careful analysis during reasoning.

  • None: When no specific distribution is prescribed, logical coherence must still be maintained.

  • P: This term appears in the premise and signifies the predicate in categorical statements.

Types of Reasoning

Hypothetical Reasoning

  • Structure: A form of reasoning that involves implications:

    • If A then B (A → B)

    • If A then C

    • Highlights the logical connections and necessary conclusions derived from established premises.

  • Disjunctive: This reasoning style involves alternatives where one option negates the other:

    • Either A or B

    • If Not B, then it follows that Not A must be concluded, providing an avenue for logical deduction.

Denying the Antecedent

  • Structure: This method of reasoning mistakenly infers the negation of the consequent:

    • If A then B (A → B)

    • Not A is established

    • Therefore, Not B is concluded, which is often fallacious as it negates possible conditions that can still affirm the conclusion.

Six Rules for Syllogisms

  1. Three Categorical Terms: A valid syllogism must comprise three distinct categorical terms, each remaining constant in meaning throughout the syllogistic argumentation.

  2. Middle Term Distribution: The middle term must be distributed at least once within one of the premises to uphold logical validity.

  3. Distribution Requirement: Established distribution in conclusions must reflect in their corresponding premises to maintain logical integrity.

  4. Negative Premises: A proper syllogism cannot possess two negative premises as it leads to ambiguity in the conclusions drawn.

  5. Negative Conclusions: When concluding with a negative statement, at least one of the premises must also state a negative to justify the conclusion logically.

  6. Universal Premises: A universal conclusion cannot be drawn from two universal premises alone, as it requires particular premises for valid conclusions.