Syllogism notes 2025 revised

Formal Reasoning and Categorical Syllogisms

Common Abbreviations

  • I.e.: "that is" (Latin: id est)

  • E.g.: "for example" (Latin: ergo gratia)

  • Re:: "with respect to" (Latin: ablative case of 'res')

Overview of Formal Reasoning

  • Types of Reasoning:

    • Deduction: systematic form of reasoning where conclusions necessarily follow from premises.

    • Induction: reasoning based on observations to derive conclusions.

Deductive vs. Inductive Reasoning

  • Cost/Benefit Analysis:

    • Deduction:

      • Benefit: 100% certainty if valid.

      • Cost: No new substantial knowledge.

    • Induction:

      • Benefit: Generates new knowledge.

      • Cost: Less than 100% certainty.

Deductive Arguments and Validity

Characteristics of Deductive Arguments

  • Validity: The conclusion must follow from the premises.

  • Truth: Refers to the actual existence of the elements in the argument.

  • Soundness: Means both that the argument is valid and that all premises are true.

Categorical Syllogism Example

  • Format:

    • Major Premise: All humans are mortal.

    • Minor Premise: Socrates is a human.

    • Conclusion: Socrates is mortal.

  • Structure: Two premises connect major and minor terms via a middle term.

    • Example Terms:

      • Major Term: Mortal.

      • Minor Term: Socrates.

      • Middle Term: Human.

Structure of Categorical Statements

Understanding Categories

  • Each categorical statement has:

    • Subject (first term).

    • Predicate (second term).

    • Copula: usually a form of the verb ‘to be’.

Key Properties of Categorical Statements

  • Quantity: Universal vs. Particular.

  • Quality: Affirmative vs. Negative.

    • A statement can be

      • Universal Affirmative (A): All S is P.

      • Universal Negative (E): No S is P.

      • Particular Affirmative (I): Some S is P.

      • Particular Negative (O): Some S is not P.

Distribution of Terms

Term Distribution Principles

  • Distribution indicates whether we are talking about the whole class or just part of it.

  • Senior Term: Governs the statement; the subject.

  • Implications of Distribution: Determines if statements are universal or particular based on the subject's distribution.

Categorical Configuration Summary

  • Configurations (AEIO):

    • A: All Sd is Pu. (Universal Affirmative)

    • E: No Sd is Pd. (Universal Negative)

    • I: Some Su is Pu. (Particular Affirmative)

    • O: Some Su is not Pd. (Particular Negative)

Rules for Valid Syllogisms

  1. Use only three terms.

  2. Middle term must be distributed (D=1).

  3. Major term distribution can only be 2 or 0.

  4. Minor term distribution can only be 2 or 0.

  5. A negative premise necessitates a negative conclusion and vice versa.

  6. A particular conclusion necessitates a particular premise.

  7. Valid syllogisms cannot have two negative premises.

  8. At least one premise must be universal.

Syllogism Analysis and Interpretation

  • When reviewed, syllogisms will include all three categorical statements.

  • Focus will primarily be on identifying invalid syllogisms.

Examples of Syllogisms

Example 1

  • Premises:

    • All bovines have hooves. (All S is P) {A}

    • Some bovines are domesticated animals. (Some S is P) {I}

  • Conclusion:

    • All domesticated animals have hooves. (All S is P) {A}

Example 2

  • Premises:

    • All triangles have angles. (All S is P) {A}

    • No circles have angles. (No S is P) {E}

  • Conclusion:

    • Some triangles are circles. (Some S is P) {I}

Example 3

  • Premises:

    • All backpacks are brown. (All S is P) {A}

    • Some backpacks are heavy. (Some S is P) {I}

  • Conclusion:

    • All heavy things are brown. (All S is P) {A}

Example 4

  • Premises:

    • All characters in novel X are calligraphy writers. (All S is P) {A}

    • Jane is a calligraphy writer. (All S is P) {A}

  • Conclusion:

    • Jane is a character in novel X. (All S is P) {A}

Example 5

  • Premises:

    • Some stores are small. (Some S is P) {I}

    • Some stores are dusty. (Some S is P) {I}

  • Conclusion:

    • Some dusty things are small. (Some S is P) {I}

Combination of Reasoning Types

The Interrelationship of Deduction and Induction

  • Despite traditional views, valid deductive arguments often rely on inductive observations.

  • Deductive arguments final outcome does not exceed knowledge from premises.

  • In daily reasoning (induction), error can occur but drives the new knowledge gained, particularly in the sciences.

  • Induction is foundational for the scientific method, highlighting the importance of checking and validating observations.