Logic Fundamentals and Counterexamples

Overview

This lecture introduces fundamental concepts in logic, focusing on how to prove arguments invalid using the counterexample method and translating ordinary arguments into their formal structures.

Basics of Validity and Invalidity

• An invalid argument is one where true premises can lead to a false conclusion.
• Validity depends on argument form, not content.
• To assess validity, arguments are rewritten in formal logical structure.

Symbolizing Arguments

• Identify the conclusion and premises in an argument.
• Isolate the logical terms (the main nouns/subjects of each statement).
• Assign letter symbols to each unique logical term.
• Rewrite the argument using only the symbolic letters and logical connectors.

Counterexample Method (for Invalidity)

• The counterexample method proves only invalidity, not validity.
• Steps:
  1. Determine the conclusion and premises.
  2. Symbolize the form of the argument using letters.
  3. Create a substitution instance—replace letters with different terms—to get true premises and a false conclusion.
  4. If successful, the argument form is invalid, and thus, the original argument is invalid.

Examples of Counterexample Method

• Substitute terms so that the conclusion is obviously false and check if premises can remain true.
• Example: "All dogs are animals. All cats are animals. Therefore, all dogs are cats." (True premises, false conclusion.)
• Try various substitutions if your first attempt doesn’t yield the result.

Tips for Using the Counterexample Method

• Start substitution with the conclusion to make it clearly false.
• Work upward to find premises that can be true with your chosen terms.
• It may take multiple substitution attempts to find a working set.

Key Terms & Definitions

• Premises — Statements offered as evidence in an argument.
• Conclusion — The claim or takeaway the argument seeks to establish.
• Validity — An argument form where, if all premises are true, the conclusion must be true.
• Invalidity — An argument form that allows for true premises and a false conclusion.
• Form — The logical structure or “skeleton” of an argument, independent of content.
• Counterexample Method — A technique to prove invalidity by finding a substitution instance with true premises and a false conclusion.
• Substitution Instance — Assigning new terms to the variables in an argument form to test validity.

Action Items / Next Steps

• Read the relevant textbook section on the counterexample method.
• Complete exercises using the counterexample method to prove categorical syllogisms invalid.
• Prepare for material on categorical syllogisms and propositions in the next chapter.