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Let's say we have a superset called "People" (P). In that superset we have two subsets, "Men" (M) and "Women" (W). We can say that:

$P \supset M$

$P \supset W$

In logical language, this is equivalent to:

  1. $P \Rightarrow M$
  2. $P \Rightarrow W$

But translated to human language, (1) means that being a Person implies being a Man, and (2) means that being a Person implies being a Woman, which does not make any logical sense.

Shouldn't the equivalence between sets and logic be the other way round?

  1. $P \Leftarrow M$
  2. $P \Leftarrow W$

Now, being a man, obviously implies being a Person, and being a woman obviously implies being a Person.

Mauro ALLEGRANZA
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mateleco
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    Yes, the logical symbol $\supset$ for $\implies$ is indeed confusing, and I guess, even its origin is probably independent from the set theoretic symbol $\subseteq$.. – Berci Oct 18 '19 at 10:46

1 Answers1

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We have that $M \subset H$, that reads : the set of Males is a subset of the set of Humans", is defined by the formula :

$\forall x (\text Mx \to \text H (x))$,

that reads : "every Male is Human".

The relationship between the symbol $\subset$ for set inclusion and $\supset$ for the conditional connective ("if…, then…") is historical; see the post Why is there this strange contradiction between the language of logic and that of set theory? as well as the post Is there any connection between the symbol $\supset$ when it means implication and its meaning as superset?

In a nutshell :

Ernst Schröder, into Vorlesungen uber die Algebra der Logik, Vol. I (Leipzig, 1890), used $\subset$ for "is included in" (untergeordnet) and $\supset$ for "includes" (ubergeordnet).

Giuseppe Peano expressed the relations and operations of logic in Volume I of his Formulaire de mathematiques (1895) by the signs $\in$, c, ɔ, [...] the meanings of which are, respectively, "is" (i.e., is a member of), "contains," "is contained in," [...].

Peano "inverted c" becomes $\supset$ :

On pourrait indiquer la relation $p \supset q$ par le signe $q \text C p$ qu'on lira "$q$ est conséquence de $p$".

The diffusion of the use of $\supset$ to deonte the conditional is due to Whitehead & Russell's Principia Mathematica (1910-1913).

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Mauro ALLEGRANZA
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