'EACH AND EVERY' = 'EACH OR EVERY'

There are six unordered pairs from the word set {each, every, any, all}. They are (every, each), (every, any), (every, all), (each, any), (each, all), (any, all).

Please ignore the conventions of English pluralism, as these phrases can be correctly pluralized if you wish.

So which phrase is most logical?

1. Every and each item for sale

2. Every and any...

3. All and every...

4. Each and any...

5. Each and all...

6. Any and all...

Answer: They are all equally logical but none have been reduced to the simplest form:

1. Every item for sale

2. Each...

3. Any...

4. All items...

Every term is logically equivalent to every other term, except in arcane corners of the field of logic that are rarely visited by anyone, especially ordinary speakers and writers.

If you are a clever marketer, the phrase "each and every" might add a dash of panache. As I have no need to market anything, I have no need to mess any further with that phrase.

Except... to explain the headline.

There are two types of "or" in most forms of logic.

1. The "exclusive or" (sometimes written xor): (A & ~B) xor (~A & B).

2. The "inclusive or," which most mathemticians use. (A xor B) xor (A & B). Don't worry, when we do the truth tables, it all works out.

About the headline: Since 'each and every' is equivalent to 'each and each,' we then have 'each and each' equiv to 'each' and 'each or each' equiv to 'each.' And certainly 'each' equiv to 'each.'

So, the headline stands.

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Minor non-grammatical point:

Isn't 'xor' in the two definitions of 'or' above self-referencing? Yes, but only in a minor way.

That is, I have defined the "inclusive or" with an "exclusive or" (also written, +) that we might envision stacking up to heaven ad infinitum. Same problem for the "exclusive or." But no worries!

Consider (A & ~B) & (~A & B) where we have chosen a case that exists for the "inclusive or."

But this can be rewritten A & ~A & B & ~B, a double contradiction! But contradictions are not permitted. Thus we have no choice but to either claim that in this specal case
v --> xor
or equivalently to say that (A & ~B) & (~A & B) is universally false and so can't be used in the definition.