6.2 Rules of Inference for Quantifiers

Quantifier rules are crucial tools in logical reasoning, allowing us to make inferences about universal and existential statements. These rules help us draw conclusions from general statements about all members of a set or specific instances.

Understanding quantifier rules is essential for constructing valid arguments and avoiding common fallacies. By mastering universal instantiation, existential generalization, universal generalization, and existential instantiation, we can navigate complex logical proofs with confidence.

Rules of Inference for Quantifiers

Rules of inference for quantifiers

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• Universal instantiation (UI) allows inferring a specific instance from a universally quantified statement
  • If $\forall x P(x)$ is true, then $P(c)$ is true for any constant $c$ in the domain (humans, animals)
• Existential generalization (EG) allows inferring an existentially quantified statement from a specific instance
  • If $P(c)$ is true for some constant $c$ in the domain, then $\exists x P(x)$ is true (Socrates, Plato)
• Universal generalization (UG) allows inferring a universally quantified statement from a specific instance
  • If $P(c)$ is true for an arbitrarily chosen constant $c$ in the domain, then $\forall x P(x)$ is true (triangles, squares)
• Existential instantiation (EI) allows introducing a new constant to represent an existentially quantified variable
  • If $\exists x P(x)$ is true, then $P(c)$ is true for some new constant $c$ not previously used in the proof (prime numbers, even numbers)

Application of quantifier inference rules

• Identify the given statements and the conclusion to be proved
• Determine the appropriate rule of inference to apply based on the structure of the statements
  • UI for universally quantified statements (all, every)
  • EG for specific instances to infer existential statements (some, at least one)
  • UG for specific instances to infer universal statements (arbitrary constants)
  • EI for existentially quantified statements to introduce new constants
• Instantiate variables using constants or introduce new constants as needed
• Apply the rule of inference to derive new statements
• Repeat the process until the conclusion is reached (proof complete)

Fallacies in quantifier inference

• Existential fallacy incorrectly assumes a property holds for all members of a set based on its existence for some members
  • If some students are intelligent, it does not follow that all students are intelligent
  • If some politicians are corrupt, it does not mean all politicians are corrupt
• Universal fallacy incorrectly assumes a property holds for some members of a set based on its holding for all members
  • If all dogs are mammals, it does not follow that some dogs are mammals (as all dogs are mammals)
  • If all squares are rectangles, it does not mean some squares are rectangles (they all are)

Validity proofs using quantifier rules

1. Identify the premises and the conclusion of the argument
2. Apply the appropriate rules of inference for quantifiers to the premises
   • Use UI to derive specific instances from universally quantified statements (all humans are mortal)
   • Use EG to infer existentially quantified statements from specific instances (Socrates is human)
   • Use UG to infer universally quantified statements from arbitrarily chosen instances (arbitrary triangle ABC)
   • Use EI to introduce new constants representing existentially quantified variables (some prime number p)
3. Derive new statements using the rules of inference until the conclusion is reached
4. Ensure each step in the proof is valid and follows logically from the previous steps
   • Justify each step by citing the rule of inference used (UI, EG, UG, EI)
   • Maintain consistency in variable and constant names throughout the proof