"How to write proofs: a quick guide"

In introductory logic, students are asked to answer problems like,

• Show that the formula $P \rightarrow (P \rightarrow Q)$ is equivalent to $P \rightarrow Q$.

So, the student writes down a truth table with sentence letters $P$ and $Q$, and a column $P \rightarrow (P \rightarrow Q)$ and a column for $P \rightarrow Q$ and checks that the truth values of these two columns all match. Alternatively, a student might be asked to give a formal derivation of $P \rightarrow Q$ from $P \rightarrow (P \rightarrow Q)$ and vice versa.

In intermediate logic, students are asked to answer problems like

• Suppose $S_0$ is $P \rightarrow Q$ and $S_{n+1}$ is $P \rightarrow S_n$. Show that, for all $n$, $S_n$ is equivalent to $P \rightarrow Q$

This involves something like a genuine mathematical proof, using induction. When philosophy students step up from introductory logic to intermediate logic, they often find it challenging to come up with informal mathematical proofs of such claims. For philosophy students who do not intend to focus on theoretical philosophy, this needn't matter (though I believe that, increasingly, it will). But for advanced philosophy students who want to focus on topics in logic and parts of metaphysics, philosophy of language, mathematics and science, at some point it becomes necessary to be able to understand, and write out, informal proofs of a mathematical nature.

Here is a link to a short guide on writing proofs, for mathematics students, by Eugenia Cheng, a category theorist at The University of Sheffield.