Modus ponens and modus tollens are ways of solving logical implications. A logical implication is a clause in the following form:

And it means:

  

If p is true, then q is also true.

Modus ponens

Modus ponens occurs when we have this:

What it means:

  If p is true, then q is also true. p is true.

And so, the logical consequence is:

  

q is true.

And so, modus ponens is defined this way:

  

p
q   -----
  ∴ q

Modus tollens

modus tollens occurs when we have this:

  

p → q
  ¬ q

What it means:

  If p is true, then q is also true. q is false.

And so, the logical consequence is:

  

p is false.

The reason for this is because if p is true, then q would also have to be true. But since q is false, then it is not possible that p is true, so p can only be false.

And so, modus tollens is defined as follows:

  

p → q
  ¬ q
  -----
  ∴ ¬ p

Differentiation

So, the main way to differentiate is that:

In modus ponens , you use the implication to prove that the consequence is true in demonstrating that the premise is true .
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