The two characters '⊃' and '→' are in use in mathematical logic, as far as I can tell without any difference.

Maybe the character '⊃' is progressively abandoned in favour of the character '→', but it seems to be a question of typographical preference.

In mathematics outside mathematical logic, it is the character '⇒' which is often used.

Is there any subtle difference which would explain or justify the preference of many mathematicians for using the character '⇒'?

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    $\begingroup$ I don't really know, but I've got a guess: the symbol $\to$ is quite heavily overloaded outside of mathematical logic. You use it in contexts such as $f:X\to Y$ (function that maps $X$ to $Y$) and also in contexts such as $f(x)\to\infty$ when $x\to 0$ (i.e. $\lim_{x\to 0}f(x)=\infty$). $\endgroup$ Jan 15 '20 at 9:45
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    $\begingroup$ Unfortunately, no standard practice in ML; see e.g. the following post $\endgroup$ Jan 15 '20 at 9:46
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    $\begingroup$ I haven't seen $\supset$ in that context since Principia Mathematica. Nowdays $\supset$ is used between sets and $\rightarrow$ or $\Rightarrow$ is used between statements. $\endgroup$
    – skyking
    Jan 15 '20 at 9:48
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    $\begingroup$ See also the following post as well as this one $\endgroup$ Jan 15 '20 at 9:53

Unfortunately, we have no standard practice in mathematical logic.

Basically, the first two are used with the same meaning: the conditional connective.

The "horseshoe" was used by W&R in their groundbreaking treatise Principia Mathematica (1910-13) but has been superseded for obvious typographical reasons.

It is still used by some author in the context of Sequent Calculus where the arrow is used for the sequent notation: $\Gamma \to \Delta$ (alternative notation: $\Gamma \vdash \Delta$).

Also $\Rightarrow$ is used as a synonym of $\to$, but in some cases we have $→$ and $↔$ used as conenctives in the object language while $⇒$ and $⇔$ are used in the meta-language (see e.g. van Dalen's textbook).

  • $\begingroup$ What were the "obvious typographical reasons" for eschewing the horseshoe? $\endgroup$
    – Rob Arthan
    Jan 17 '20 at 23:59

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