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From time to time, although not so commonly, I see $A \supset B$ written with seemingly the same meaning as $A \Rightarrow B$.

Ex. in Constructive Type Theory and Interactive Theorem Proving, on page 8.

A proof of $A \supset B$ is a method which transforms a proof of $A$ to a proof of $B$.

This suggest $A \supset B$ means the same as $A \Rightarrow B$, isn't it?

$A \Rightarrow B$ means: whenever $A$ holds, then $B$ holds. However, this also means $B$ may holds when $A$ do not. Which in turn means the set of the cases where $A$ holds is a subset of the set of the cases where $B$ holds. Is this OK? If that's OK, $A \Rightarrow B$ should be written as $A \subset B$ instead of $A \supset B$.

Why is this not so? Or am I wrong in some way? If I'm wrong, where am I so?

Edit: oops, duplicate of this one: Using $p\supset q$ instead of $p\implies q$ . Sorry.

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Hibou57
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1 Answers1

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The symbol $\supset$ (called "horseshoe") was used a century ago for the conditional connective "if-then"(see The Notation in Principia Mathematica , and after replaced (mainly) by $\rightarrow$ or $\Rightarrow$.

It does not mean that the sentence $A$ is "included" into the sentence $B$.

Of course, there is a "relation", through The Algebra of Logic Tradition and Giuseppe Peano.

Note

In the specific example you are referencing, the Author wants to differentiate the "if-then" connective from the (more heavy used in his presentation) symbol for "mapping" : $\rightarrow$.

Thus, he pulled out from the attic the symbol $\supset$.

Mauro ALLEGRANZA
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  • By the way, I'm also confused about why at least some people say “$\rightarrow$” is the right notation, while “$\Rightarrow$” is not. I may understand it in the context of intuitionist/constructive logic, however not in the large. – Hibou57 Jul 29 '14 at 16:17
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    @Hibou57 - In some book, $\Rightarrow$ is used for the "metalogical" relation of entailment (i.e. "implies") and $\rightarrow$ is used for the "if-then" connective (see van Dalen's book), but the practice is not uniform (due also to the fact that the two are "related"), so you can find both: Mendelson uses $\Rightarrow$ for "if-then", while Enderton uses $\rightarrow$. – Mauro ALLEGRANZA Jul 29 '14 at 16:24
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    It can also be convenient to use $\supset$ for implication/conditional (i.e., exponential) objects in working with category theory, because $\to$ is so often used to denote arrows/morphisms. E.g., $\displaystyle{\mathrm{eval}_{A,B}\colon (A \supset B) \wedge A \to B}$ and $\dfrac{f\colon A \wedge B \to C}{\lambda f\colon A \to B \supset C}$. – Joshua Taylor Jul 31 '14 at 01:13
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    @JoshuaTaylor - well done ! Also in sequent calculus we have the same problem; the "arrow" $\rightarrow$ is used for the main conncetive; thus we need something else for conditional. – Mauro ALLEGRANZA Jul 31 '14 at 13:00