Logic - Good and Bad

To help you understand good and bad examples of logical constructions, here are some examples. The basic ideas are:
  • There are two consistent logical argument constructions: modus ponens ("the way that affirms by affirming") and modus tollens ("the way that denies by denying"). Here are how they are constructed:
    • Modus Ponens: "If A is true, then B is true. A is true. Therefore, B is true."
    • Modus Tollens: "If A is true, then B is true. B is not true. Therefore, A is not true."
  • There are two related incorrect and inconsist constructions: affirming the consequent and denying the antecedent.
    • Affirming the Consequent: "If A is true, then B is true. B is true. Therefore, A is true."
    • Denying the Antecedent: "If A is true, then B is true. A is not true. Therefore, B is not true."


"A" and "B" can be anything - they can even be totally made up words. Don't let the language fool you. Focus on the CONSTRUCTION of the argument.

Here is a sensible example, illustrating each of the above:

  • "If it is a car, then it has wheels. It is a car. Therefore, it has wheels." (Modus Ponens - CORRECT)
  • "If it is a car, then it has wheels. It does not have wheels. Therefore, it is not a car." (Modus Tollens - CORRECT)
  • "If it is a car, then it has wheels. It has wheels. Therefore, it is a car." (Affirming the Consequent - INCORRECT.)
    • Comment: why is this incorrect? Well, the thing might have wheels but that doesn't mean it has to be a car. It might be a cart, or rollerblades, or a moped. It doesn't have to be a car.
  • "If it is a car, then it has wheels. It is not a car. Therefore, it does not have wheels." (Denying the Antecedent - INCORRECT)
    • Comment: why is this incorrect? Consider the argument for the "affirming the consequent" example. Rollerblades are not cars, but they DO have wheels.

    Here are less sensible examples. Can you determine whether these are examples of Modus Ponens, Modus Tollens, or one of the incorrect constructions? To get the answers, highlight the text in a line with your mouse. The answers are written with the same color as the background, but can be revealed by highlighting them.

    • "If Xyrplex is 9, Guffaw is 1. Guffaw is 2. Therefore, Xyrplex is not 9." (ANSWER: Modus Tollens)
    • "If Nagini is a Snake, Snape is a goner. Nagini is a snake. Therefore, Snape is a goner." (ANSWER: Modus Ponens)
    • "If Blurts are Flurts, Green is Grue. Green is Grue. Therefore, Blurts are Flurts." (ANSWER: INCORRECT - Affirming the Consequent)
    • "If Sagan has hair, Tyson is awesome. Sagan has hair. Therefore, Tyson is awesome." (ANSWER: Modus Ponens)
    • "If Fordham brings a ram, Peruna will kick. Fordham did not bring a ram. Therefore, Peruna did not kick." (ANSWER: INCORRECT - Denying the Antecedent)