# Preferred character to use for the implication: '⊃', '→' and '⇒'

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 '⇒'?

• 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. $\lim_{x\to 0}f(x)=\infty$). Jan 15 '20 at 9:45
• Unfortunately, no standard practice in ML; see e.g. the following post Jan 15 '20 at 9:46
• 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. Jan 15 '20 at 9:48
• See also the following post as well as this one Jan 15 '20 at 9:53

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).