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Arguments in Propositions and Syllogism

• Compound Propositions and Syllogism:
  • Arguments often involve compound propositions, specifically focusing on types of syllogism.

Kinds of Hypothetical Syllogism

• Two Main Types:
  1. Pure Hypothetical Syllogism
  2. Mixed Hypothetical Syllogism

Pure Hypothetical Syllogism

• Definition: Involves two conditional (if-then) statements to derive a conclusion.
  • Example:
  1. If I study hard, I will pass the exam.
  2. If I pass the exam, I will graduate.
  3. Therefore, if I study hard, I will graduate.

Mixed Hypothetical Syllogism

• Definition: Combines categorical statements with affirmative or negative premises, leading to a conclusion.
  • Example:
  1. If it rains, the ground will be wet.
  2. The ground is wet.
  3. Therefore, it must have rained.

Valid Forms of Mixed Hypothetical Syllogism

1. Modus Ponens

   • Structure:
   1. If P, then Q.
   2. P is true.
   3. Therefore, Q is true.
   • Example:
   1. If I study hard, I will pass the exam.
   2. I studied hard.
   3. Therefore, I passed the exam.

2. Modus Tollens

   • Structure:
   1. If P, then Q.
   2. Q is not true.
   3. Therefore, P is not true.
   • Example:
   1. If I study hard, I will pass the exam.
   2. I did not pass the exam.
   3. Therefore, I did not study hard.

Invalid Forms of Mixed Hypothetical Syllogisms

1. Fallacious Form

   • Structure:
   1. If P, then Q.
   2. P is not true.
   3. Therefore, Q is not true.
   • Example:
   1. If I study hard, I will pass the exam.
   2. I did not study hard.
   3. Therefore, I did not pass the exam.

2. Consequent Fallacy

   • Structure:
   1. If P, then Q.
   2. Q is true.
   3. Therefore, P is true.
   • Example:
   1. If it is raining, the ground will be wet.
   2. The ground is wet.
   3. Therefore, it is raining.

Exercises

• Argument: If I study hard, I will pass the exam.
  • Modus Ponens example(validation): If I study hard, I will pass the exam. I studied hard. Therefore, I passed the exam.
  • Consequent Fallacy example(invalidation): If I study hard, I will pass the exam. The exam was passed. Therefore, I studied hard is invalid.
  • Modus Tollens example(validation): If I study hard, I will pass the exam. I did not pass the exam. Therefore, I did not study hard.
  • Fallacious Form example(invalidation): If I study hard, I will pass the exam. I did not study hard. Therefore, I did not pass.

Types of Hypothetical Syllogism

• Three Types:
  1. Disjunctive
  2. Conditional
  3. Conjunctive

Disjunctive Syllogism

• Definition: Involves one premise stating a disjunction and eliminates one option to conclude the other option.
  • Example:
  1. Either I will go for a walk, or I will stay indoors.
  2. I did not go for a walk.
  3. Therefore, I will stay indoors.

Conditions of Disjunctions

1. Simple Disjunction - Example: Either it will rain tomorrow, or it will be sunny tomorrow.
2. Disjunction with False Outcomes - Example: I either went to the gym or stayed at home. I did not go to the gym; hence, I stayed home.
3. Disjunction with Two True Statements - Example: Either I will have pizza or pasta for dinner. I will have pizza, thus I am not having pasta.
4. Disjunction with Both False Statements - Example: Either I will go to the park or library. I did not go to either, so both statements are false.

Conditional Syllogism

• Definition: Involves reasoning with if-then statements.
  • Valid Forms: Modus Ponens, Modus Tollens
  • Invalid Forms: Affirming the Consequent, Denying the Antecedent

Conjunctive Syllogism

• Definition: Involves reasoning based on conjunctions, one premise confirms the conjunction, the other denies a part, leading to a conclusion about the other part being true.
• Structure:
  1. P and Q.
  2. Not P.
  3. Therefore, Q.
• Example:
  1. I will go to the park and visit the museum.
  2. I did not go to the park.
  3. Therefore, I will visit the museum.