What’s the difference between Modus Ponens and Modus Tollens?

Question

Again in the study of mathematical logic, how to differentiate a proposition modus Ponens of a modus Tollens?

Answer 1

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

p → q

And she means:

If p is true, then q is also true.

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Modus Ponens

The modus Ponens occurs when we have it:

p → q
p

What does it mean:

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 in this way:

p → q
p
-----
∴ q

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Modus Tollens

Already the modus Tollens occurs when we have it:

p → q
¬q

What does it mean:

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 were true, then q would also have to be true. But how q is false, so it is not possible that p be true, and therefore p can only be false.

And so, the modus Tollens is defined in this way:

p → q
¬q
-----
p

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Differentiating

So the main way to differentiate is that:

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

• In the 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 themselves on this angle.

Furthermore, it can be said that the mathematical process of proof by inference is an application of the modus Ponens, while the process behind the demonstration by reduction to the absurd is an application of the modus Tollens.

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_{NOTE: The symbol " " is read as "therefore", "soon", "it is concluded that" or "and then".}