Modus ponens and modus tollens are ways to solve logical implications. A logical implication is a clause in the following Form:

p → q

And it means:

If p is true, then q is also true.

Modus ponens

The modus ponens occurs when we have this:

p → q
p

Which means:

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

And so the logical consequence is:

q is true.

And so, the modus ponens is defined like this:

p → q
p
-----
∴ q

Modus tollens

Already the modus tollens occurs when we have this:

p → q
q

Which 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 was true, so q would also have to be true. But since q is false, it is therefore not possible that p is true, and therefore p can only be false.

And so, the modus tollens is defined like this:

p → q
q
-----
P p

Differentiation

Thus, being the main way to differentiate is that:

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

• In modus tollens , you use the implication to prove that the premise is false by demonstrating that the consequence is false.

That is, they are almost opposite, seeing about that angle.

In addition, it can be said that the mathematical process of demonstration by inference is an application of modus ponens, while the process behind demonstration by reduction to absurdity is an application of modus tollens.

_{note: the symbol " ∴ "is read as" therefore", "logo", "it is concluded that " or " and then ".}