5.3 Formal Fallacies in Propositional Logic

Formal fallacies in propositional logic are common mistakes in reasoning that can trip up even seasoned thinkers. These errors stem from misunderstanding or misapplying logical rules, leading to invalid conclusions despite seemingly sound arguments.

Understanding these fallacies is crucial for spotting flaws in arguments and avoiding logical pitfalls. By recognizing conditional and syllogistic fallacies, you'll sharpen your critical thinking skills and become a more effective reasoner.

Conditional Fallacies

Invalid Inferences from Conditional Statements

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• Affirming the consequent incorrectly concludes the antecedent is true because the consequent is true
  • Assumes the converse of a conditional statement is logically equivalent to the original statement
  • Example: "If it is raining, then the ground is wet. The ground is wet, therefore it is raining." (fails to consider other possible causes of wet ground, such as a sprinkler system)
• Denying the antecedent incorrectly concludes the consequent is false because the antecedent is false
  • Assumes the inverse of a conditional statement is logically equivalent to the original statement
  • Example: "If you study hard, you will pass the exam. You did not study hard, therefore you will not pass the exam." (fails to consider other factors that could lead to passing, such as natural aptitude or an easy exam)

Misinterpreting Conditional Statements

• Fallacy of the converse occurs when a conditional statement is confused with its converse
  • The converse of a conditional statement switches the antecedent and consequent
  • Example: "If a shape is a square, then it has four sides. Therefore, if a shape has four sides, it is a square." (fails to consider other four-sided shapes, such as rectangles or parallelograms)
• Fallacy of the inverse occurs when a conditional statement is confused with its inverse
  • The inverse of a conditional statement negates both the antecedent and consequent
  • Example: "If a number is even, then it is divisible by 2. Therefore, if a number is not even, it is not divisible by 2." (fails to consider that some odd numbers, such as 6 or 10, are still divisible by 2)

Syllogistic Fallacies

Fallacies Involving Categorical Syllogisms

• Fallacy of exclusive premises occurs when a syllogism has two negative premises
  • A valid categorical syllogism must have at least one affirmative premise
  • Example: "No cats are dogs. No dogs are birds. Therefore, no cats are birds." (the conclusion does not follow logically from the premises, as there is no connection established between cats and birds)
• Existential fallacy occurs when a particular conclusion is drawn from two universal premises
  • A valid categorical syllogism with two universal premises must have a universal conclusion
  • Example: "All mammals are animals. All dogs are mammals. Therefore, some dogs are animals." (the conclusion should be universal, stating that all dogs are animals, not just some)

Fallacies Involving Quantifiers and Existence

• Existential fallacy can also occur when a syllogism assumes the existence of something that may not exist
  • This fallacy arises from incorrectly using the existential quantifier (∃) or universal quantifier (∀) in the premises or conclusion
  • Example: "All unicorns have horns. All unicorns are magical creatures. Therefore, some magical creatures have horns." (the conclusion is invalid because it assumes the existence of unicorns, which are mythical creatures)
• Existential fallacy can lead to invalid inferences when the premises do not guarantee the existence of the subject being discussed
  • This fallacy often results from misinterpreting the scope or meaning of quantifiers in the context of the argument
  • Example: "All square circles are impossible figures. All square circles have four sides. Therefore, some impossible figures have four sides." (the conclusion is invalid because square circles do not exist, as they are self-contradictory)