Deductive Argumentation Review

Deductive Argumentation Overview

• Date: Week 6, W 2/18/2026

Conditional Propositions

• The statement: "If it is a Living thing, it must have a Metabolism" can be symbolized as:

  • $L \to M$

• This can also be expressed in multiple ways:

  1. All L is M.

  2. If something is L, it must M.

  3. Something M if it is L.

  4. Any L you will find must M.

  5. Whenever something L, it must M.

  6. Unless it is L, it will not M.

Relationship between Antecedent and Consequent

• Sufficient Condition: When the truth of the antecedent guarantees the truth of the consequent.

  • Example: "If it is a Living thing, it must have a Metabolism" ensures that being a living thing guarantees metabolism.

• Necessary Condition: When the falsity of the consequent ensures the falsity of the antecedent.

  • Example: If something doesn’t metabolize, then it isn’t a living thing. Hence, metabolism is a necessary condition for living.

Negation

• Symbolized with the ‘~’ sign, meaning ‘no’ or ‘not.’

• More accurately, it represents:

  • A change in truth value

  • Example: $\sim X$ means ‘not X’ or ‘the opposite truth value of X.’

Logical Forms

1. Modus Ponens (MP)

   • Structure:

     • P1: $\sim X \to Y$

     • P2: $\sim X$

     • Conclusion: $Y$

   • This fits a logical form.

2. Modus Tollens (MT)

   • Structure:

     • P1: $X \to \sim Y$

     • P2: $Y$ (this is really $~~Y$)

     • Conclusion: $\sim X$

Practical Examples

• Example Arguments:

  1. The food in that restaurant stinks, and the portions are too small!

  • $F \land P$

  1. Your ice is not cold.

  • $\sim C$

  1. If my stock portfolio is weak, then I am losing money.

  • $S \to \sim M$

  1. My car doesn’t look great, but it gets great gas mileage.

  • $\sim L \land G$

  1. If you feel great, then you look great.

  • $F \to L$

  1. My test score was high or I am mistaken.

  • $T \lor M$

  1. You passed the exam only if you got a C.

  • $C \to P$ (note that ‘only if’ flips antecedent and consequent)

Categorical Statements

• These statements fall under deductive categorization and include propositions with 'All,' 'Some,' 'Some are not,' and 'None.'

  • Example of Categorical Syllogism:

  1. P1: All humans are mortal.

  2. P2: Socrates is a human.

  3. Conclusion: Socrates is mortal.

Categorical Syllogism Validity

• Sound Deductive Argument: Valid if it has the correct logical form and if the premises are true.

  • Example: "All humans are mortal" identifies 'humans' as subject and 'mortal' as predicate.

Categorical Constructs and Patterns

1. Combine Subject and Predicate

   • Valid combinations:

   1. Some things named Socrates are human.

   2. No thing named Socrates is human.

   3. Some things named Socrates are not human.

2. Categorical Syllogism Patterns (Moods):

   • A. All S is P (A claim)

   • B. No S is P (E claim)

   • C. Some S is P (I claim)

   • D. Some S is not P (O claim)

Categorical Syllogism Construction

• Major Term: Predicates of conclusions

• Minor Term: Subjects of conclusions

• Middle term: Terms repeated in premises but not in the conclusion identifies the figure of the syllogism.

Deductive Argument Exercises

• Feel comfortable analyzing or creating original deductive arguments.

1. Deductive Rules of Inference:

   • a. Modus Ponens (MP): $P1: X \to Y$

   • b. Modus Tollens (MT): $P1: X \to Y$

   • c. Hypothetical Syllogism (HS):

     • P1: $X\to Y$

     • P2: $X$

     • Conclusion: $Y$

   • d. Disjunctive Syllogism (DS):

     • $P1: X \lor Y$

     • P2: $\sim X \lor \sim Y$

     • Conclusion: $Y \lor X$

Analyzing Real-World Assertions

• Analyze arguments from credible sources, such as:

  • Letter to the Editor by Randi Weingarten discussing education policies and their effectiveness.

  • Contains a core conditional statement:

  • "If policymakers want increased educational achievement, the path is clear: Invest in educators, provide them with evidence-backed curriculums…"

• Analysis Process:

  1. Identify standard conditional format for logical clarity.

  2. Explore Modus Ponens (MP) vs Modus Tollens (MT) for stronger argument construction.

  • Example for MP:

    • P1: If policymakers want increased educational achievement, then we need to invest…

    • P2: Policymakers do want to increase achievement.

    • Conclusion: We need to invest…

  • Example for MT:

    • P1: If policymakers want increased achievement then we need…

    • P2: We don't invest…

    • Conclusion: Policymakers don’t want increased achievement.

Categorical Statements in Arguments

• Identify claims:

  1. All Americans should support public education needs.

  • Creating a Categorical Syllogism:

  • P1: All M should give public education what it needs.

  • P2: All Americans should M.

  • Conclusion: All Americans should give public education what it needs.

  • M: Investors/taxpayers/civilized societies.

Justification and Evaluation

• Justify opinions on letter:

  • Use a sound Categorical Syllogism.

  • Example:

  • P1: All M is good.

  • P2: Some of her letter was M.

  • Conclusion: Some of her letter was good.

  • M = truthful/progressive/practical

Evaluating Subscriptions' Claims

1. Claim: Some subscriptions are good (I claim).

   • P1: All M is good.

   • P2: Some subscriptions are M.

   • Conclusion: Some subscriptions are good.

2. Opposing Claim: Some subscriptions are not good.

   • P1: No good things are M.

   • P2: Some M are subscriptions.

   • Conclusion: Some subscriptions are not good.

Rules of Inference Applications

• Create sound arguments utilizing deductions effectively.

Note: All created syllogisms and inferences must be tested against set rules to verify their validity.