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It seems to me that the sufficiency-necessity relation is effectively the same as the set-member relation.

Using a concrete examples to make the point:

  • A rain drop ⊃ a bit of water
  • A rain drop → a bit of water

So if all logical connectives can be built from the Sheffer stroke, and if the material conditional is a logical connective, and if it can perform the work of ⊃, then it would make sense that the Sheffer stroke could perform the work of the ⊃.

But I may have overlooked something. So, can the sheffer stroke do the work normally done with sets?

Thank you

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Given a set $U$, the power set of $U$ is a Boolean algebra when we interpret $\land$ as $\cap$ and $\lor$ as $\cup$, negation is complement and implication is indeed $\subseteq$.

So yes, we can work with the Sheffer stroke, it would be exactly the same thing as it is in Boolean algebra, $$A\mid B=\{x\in U:x\notin A\cap B\}$$

But why should we? Its usefulness in Boolean algebra is that it allows a single logical gate to be implemented instead of many different gates. In set theory it would be somewhat less convenient and obscure the meaning of an expression.

An even better reason why not to use this operation is that much like "negation" it depends on the universal set, i.e. what you are negating against. But in modern set theory the collection of sets is not a set itself, it's a proper class. So just taking a complement of a set does not result in a set. On the other hand, intersection and union do give back sets and inclusion is just a relation defined on sets regardless to the existence or nonexistence of a universal set.

So using the Sheffer stroke is possible, but it will inherently introduce a lot of clutter and problems because complement is not a well-defined notion in modern set theory. (whereas complement relative to a particular set $U$ is, but which $U$ are you going to use?)

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