1.6 – Modus Ponens & Modus Tollens

Modus Ponens and Modus Tollens are rules of inference in propositional logic that allow you to draw conclusions based on conditional statements and their relationships with other propositions. Both rules are fundamental to logical reasoning and are widely used in constructing proofs and arguments in various fields, including mathematics, philosophy, and computer science.

 

Modus Ponens (also known as Affirming the Antecedent or Law of Detachment): 

Modus Ponens is a rule of inference that states that if you have a conditional statement “If P, then Q” (P → Q) and you know that P is true, then you can conclude that Q must be true. The argument takes the following form:

 

      1. P → Q           (Premise 1: Conditional statement)
      2. P                    (Premise 2: Affirmation of the antecedent)
      3. Q                   (Conclusion)

 

Example of Modus Ponens:

 

      1. If it is raining, then the ground is wet. (R → W)
      2. It is raining. (R)
      3. The ground is wet. (W)

 

Modus Tollens (also known as Denying the Consequent):

Modus Tollens is a rule of inference that states that if you have a conditional statement “If P, then Q” (P → Q) and you know that Q is not true (¬Q), then you can conclude that P must not be true (¬P). The argument takes the following form:

 

      1. P → Q            (Premise 1: Conditional statement)
      2. ~Q                  (Premise 2: Negation of the consequent)
      3.  ~P                 (Conclusion)
  1.  

Example of Modus Tollens:

      1. If it is raining, then the ground is wet. (R → W)
      2. The ground is not wet. (~W)
      3. It is not raining. (~R)

Understanding Modus Ponens and Modus Tollens is crucial for working with logical expressions, analyzing propositions, and constructing proofs in various domains.