Is there any connection between the symbol $\supset$ when it means implication and its meaning as superset?

Question

A rather old-fashioned symbol for logical implication is $\supset$ (see list of logic symbols). For example $p \supset q$ means $p \implies q$ or $p \rightarrow q$ in more recent notations.

Is there any semantic connection between using the symbol to show implication and using it to represent subset/superset relationships?

I can see a connection myself but the problem is I see it in the opposite direction ($\subset$ instead of $\supset$). For example, I see $x \in A \implies x \in B$ is (somewhat) equivalent to $A \subset B$. Therefore, if I had to reuse the subset/super set symbol for logical implication, I would have used $p \subset q$ to denote $p \implies q$. Is there any historical or semantic reason for using $\supset$?

Answer 1

For the "devious" evolution of the symbolism, we can see :

• Florian Cajori, A history of mathematical notations (1928) : SYMBOLS IN MATHEMATICAL LOGIC, §667-on :

  [§674] Signs of Gergonne : A theory of the meccanisme du raisonnement was offered by J.D. Gergonne in an Essai de dialectique rationnelle (1816-1817); there the symbol $H$ stands for complete logical disjunction, $X$ for logical product, $I$ for "identity," C for "contains," and Ɔ ("inverted C") for "is contained in."

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

  [§689] Peano expresses the relations and operations of logic in Volume I of his Formulaire de mathematiques [of 1895] (Introduction, p. 7) by the signs $\in$, c, ɔ, [...] the meanings of which are, respectively, "is" (i.e., is a member of), "contains," "is contained in," [...].

  [§690] Some additional symbols are introduced [into Number 2 of Volume II of Peano's Formulaire]. Thus "inverted c" becomes $\supset$. By the symbolism $p.\supset x \ldots z. q$ is expressed "from $p$ one deduces, whatever $x \ldots z$ may be, and $q$."

We have in Peano, Formulaire, vol.2, page 26 :

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

See also in :

• Jean van Heijenoort, From Frege to Gödel : A Source Book in Mathematical Logic (1967), Giuseppe Peano, The principles of arithmetic (Arithmetices principia: nova methodo exposita, 1899), page 87.

The original edition has a Signorum Tabula [page vi] with Ɔ : deducitur aut continetur.

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You can see also Earliest Uses of Symbols of Set Theory and Logic.

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In spite of the seemingly inconsistency, we can conjecture an historical explanation for this sort of "inversion" that used the "horseshoe" ($\supset$) where we (today) expect to find the "inclusion" ($\subset$).

According to Peano, we have $q\mathrel{C}p$ for "$q$ is a consequence of $p$", and then, through W&R's Principia Mathematica to $p \supset q$ and the modern $p \to q$.

For the modern formal logic, propositional logic is "prior to" first-order logic; but from ancient Greece until Early Modern times, formal logic was mainly the Syllogistics which, from a modern point of view, id monadic first-order logic.

The basic "building block" of syllogism is the assertion :

"all $S$'s are $P$'s"

that we translate as $\forall x(Sx \to Px)$, and thus the "set equivalent" : $S \subset P$, where we have silently moved from the predicates (or attributes) $S,P$ to the corresponding "extension" : the sets $S,P$.

Thus, we have a "symbolic" inconsistency : $Sx \supset_x Px$ corresponds to $S \subset P$.

But in "traditional" logic there is also another possible reading of the categoricals sentences : "all $S$'s are $P$'s". In Prior Analytics, Aristotle says :

"$P$ belongs to $S$".

The usual example : "all $\mathit{Men}$ are $\mathit{Mortal}$", that we read "estensionally" as "the set-of $\mathit{Men} \subset$ the set-of $\mathit{Mortal}$", can also be read "intensionally" as "$\mathit{Mortality}$ belongs to $\mathit{Humanity}$".

We can see in :

• Antoine Arnauld & Pierre Nicole, Logic or the Art of Thinking (La Logique ou l'art de penser, 1st ed 1662, ed.Jill Vance Buroker, 1996), page 39:

  there are two things which it is most important to distinguish clearly, the comprehension and the extension.

  I call the comprehension [i.e. intension] of an idea the attributes that it contains in itself, and that cannot be removed without destroying the idea. For example, the comprehension of the idea of a triangle contains shape, three lines, three angles, and the equality of these three angles to two right angles, etc.

According to this theory, $\mathit{Mortality}$ belongs to $\mathit{Humanity}$ exactly because the "attribute" of $\mathit{Mortality}$ is contained into the idea of $\mathit{Humanity}$.

So, from the idea of $Humanity$ we can "deduce" the attribute of $\mathit{Mortality}$.

In conclusion, we can trace a path from :

"all $\mathit{Men}$ are $\mathit{Mortal}$"

to, from one side : $\mathit{Men}(x) \supset_x \mathit{Mortal}(x)$, that we read "estensionally" as : $\mathit{Men} \subset \mathit{Mortal}$, and, from the other side to : $\mathit{Mortality}$ belongs to $\mathit{Humanity}$, that we read as : from $\mathit{Humanity}$ we can deduce $\mathit{Mortality}$, i.e. using Peano's symbols : $\mathit{Mortality} \mathrel{C} \mathit{Humanit}y$, then reverted into :

$\mathit{Humanity} \supset \mathit{Mortality}$.

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Addendum

See :

• Leila Haaparanta (editor), The Development of Modern Logic (2009), Ch.3. Logic and Philosophy of Logic from Humanism to Kant by Mirella Capozzi and Gino Roncaglia, page 78-on: §10. Logical Calculi in the Eighteenth Century, page 129 :

  [the] work Specimen Logicae universaliter demonstratae [1740] written by the mathematician and scientist Johann Andreas Segner (1704–1777) with the explicit aim of treating syllogistic by way of a calculus (per calculum) based on the example of algebra.

  To this end, Segner builds an axiomatic system consisting of 16 definitions, 3 postulates, and 2 axioms. The definitions introduce ideas, their relations, their arrangement in a hierarchy of genera and species, and the operations for forming ideas. Segner defines idea as a mental representation of something. If the idea is simple, its contents are obscure ideas and the simple idea is confuse for us; if the idea is composite, its contents are clear ideas and the composite idea is distinct for us. Consequently, by definition, every idea contains some idea within itself. In this way, Segner can presuppose the relation of containment (viewed from an intensional perspective) as the basic relation between two ideas. But it must be clear that Segner does not identify the content of an idea with its comprehension in the sense of the Port-Royal Logic. He simply says that given two ideas $A$ and $B$, $A$ is contained (or involved) in $B$ if, whenever $B$ is posited, $A$ is also posited [and Segner writes : $A < B$].

  [...]

  Axiom II : If $A$ contains $B$, then $AB = A$.

  Note : clearly, $AB = A$, in our modern "extensional" reading (as sets) means "$A$ is subset of $B$", and thus we have $A \subset B$. Segner, instead, says : "$A$ contains $B$".

  [...]

  “All $A$ are $B$” means either $A = B$ or $A < B$.

  Note : again, for us "all $A$ are $B$" means that the sets of $A$'s is a subset of the set of $B$'s, while for Segner means "the idea $A$ contains the idea $B$".

Thus, $\mathit{idea}_A$ is contained in $\mathit{idea}_B$ when "if $\mathit{idea}_B$, then $\mathit{idea}_A$"; according to my "interpretation" :

$\mathit{idea}_A \mathrel{C} \mathit{idea}_B$ when $\mathit{idea}_B \supset \mathit{idea}_A$ (here $\supset$ is the "horseshoe").

See page 133 :

In his Essai de dialectique rationelle Joseph Diez Gergonne (1771–1859) considers five idea-relations using the notion of containment as basic but giving it an extensional interpretation: “the more general notions are said to contain the less general ones, which inversely are said to be contained in the former; from this the notion of relative extension of two ideas originates” (Gergonne 1816–17, 192). This extensional interpretation of the notion of containment is used by Gergonne to classify the relations between two ideas on a par with the circles of Leonhard Euler and to designate each of them by a symbol. Two ideas $A$ and $B$, where $A$ is the less general idea and $B$ is the more general one, can [...] (4) be such that $A$ is contained in $B$, so that they stand in the relation $C$; (5) be such that that $B$ is contained in $A$, so that they stand in the relation "inverted C".

Thus, $\mathit{idea}_A$ is contained in $\mathit{idea}_B$ when $A$ is less general than $B$, i.e. :

$\mathit{idea}_A \mathrel{C} \mathit{idea}_B$ when $\mathit{extension}_A \subset \mathit{extension}_B$ (here we have the "modern" reading).