On equality of propositions

On equality of propositions

After the observation that equality of propositions seems to be underused in mathematical reasoning, I discovered that I myself was --perhaps not surprisingly-- only vaguely familiar with all sorts of equalities. This note --which is not deep at all-- is written to remedy that situation. I shall not repeat that ∧ and ∨ are symmetric, associative and mutually distributive; neither shall I repeat de Morgan's laws.

The following expressions (E0.i) are all equal.

With the possible exception of the last two, (the so-called "Laws of Absorption") this is very familiar ground. Also expressions (E1.i) are all equal.

In the following, = has the lowest binding power

Also expressions (E2.i) are all equal; since (E2.0) is symmetric, they occur in pairs.

The last two were absolutely new for me; their discovery --in a completely different context-- provided the incentive to write this note.