Should Modus Ponens and Modus Tollens Be Viewed As Implications?

Question

Should Modus Ponens and Modus Tollens be viewed as implications? I ask because of the following. It seems that Modus Ponens can be written as $\left(P\rightarrow Q\right)\land P\rightarrow Q$ And Modus Tollens can be written as $\left(P\rightarrow Q\right)\land\lnot Q\rightarrow\lnot P$

Answer 1

I should really put my comment as an answer:

Inference rules, like Modus Ponens and Modul Tollens, etc, are not implications in the traditional sense, but sequents, also called assertions.

A sequent is of the form $$\{A_1,\dots, A_m\}\vdash \{B_1,\dots,B_n\}$$ where $\{A_1,\dots, A_m\}$ is the sequence of antecedents and $\{B_1,\dots,B_n\}$ is the sequence of consequents. Whether these are considered sets vs. sequences are a matter of debate amongst logicians (that is to say, it depends on if the order of antecedents and consequents matters to you). Usually the curly brackets are dropped for brevity.

But in any case, a sequent is understood as a transformation rule which reads as

If all the antecedents are true, then at least one of the consequents are true.

In the case of Modus Ponens and Modus Tollens, there is only one consequent, thus they are specifically simple conditional assertions.

Thus, Modus Ponens can be denoted as $$P,P\to Q\vdash Q,$$ and Modus Tollens can be denoted as $$P\to Q,\neg Q \vdash \neg P.$$

However, by the deduction theorem, we can realize $$(P,P\to Q\vdash Q)\iff \vdash (P\wedge (P\to Q))\to Q.$$

I.e; $\vdash (P\wedge (P\to Q))\to Q$ is an equivalent formulation, in which the initial $\vdash$ is often dropped for brevity; hence you can intepret the sequent only in terms of logical operators.