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I need to create a few truth tables and I got confused by the logic symbols as some of the questions use either one or the other which is really confusing especially if they all mean the same thing.

I have found this link: http://signs-symbols.blogspot.com/2012/12/logic-symbols.html

From which I understand that these symbols mean the same thing and can be used vice versa?

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  • $\begingroup$ The short answer is they have analogous meanings, but they are different. $\endgroup$
    – Git Gud
    Mar 21, 2013 at 20:29
  • $\begingroup$ So if I have to write a truth table for A=>B would a truth table for A→B be the same thing? $\endgroup$
    – Hazzle
    Mar 21, 2013 at 20:35
  • $\begingroup$ No, as I said, they are analogous but definitely not the same thing. If you know about natural deduction systems you'll know that $A\rightarrow B$ means what whatever is on the right of $\rightarrow$ can be deducted from the hypothesis on the left using rules of inference, where as $\Longrightarrow$ means that whenever what is on the left is true (as in truth table true) then what's on the right is also true. $\endgroup$
    – Git Gud
    Mar 21, 2013 at 20:40
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    $\begingroup$ Apparently you're using $\rightarrow$ in a different way than what I meant on my comment, which is something Peter Smith mentioned it can happen. $\endgroup$
    – Git Gud
    Mar 21, 2013 at 20:53
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    $\begingroup$ Alright, whoever set the questions has used both symbols which confused me. Thanks for your time. $\endgroup$
    – Hazzle
    Mar 21, 2013 at 21:01

1 Answer 1

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The short answer is: Symbolic practice varies a bit, and you won't be going wrong if you use any of those symbols for the truth-functional conditional. Just make your choice of symbol explicit, and then no-one can be misled.

The longer answer is: There is some historical precedent for using $\supset$ to mean an [object-language] connective defined from the outset as having the truth-table of the material conditional.

And there is some precedent for using $\to$ for an [object-language] conditional connective more generally (perhaps introduced as governed by certain rules of inference). If you choose the classical rules, it will then be a result (not a mere matter of definition) that this connective is none other that the material conditional again.

As for $\Rightarrow$, this has been used as a sequent former in formal sequent calculi; but also seems often to be used (in some places, at any rate) as a metalinguistic symbol (i.e. not part of a formal object language, but as shorthand in mathematical English) to mean "logically entails" (so something stronger than the material conditional).

But, as I say, practice varies. So just make your usage clear, up front.

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  • $\begingroup$ That's the issue. I am not sure as I have a few truth tables to write down and both symbols have been used in the questions, the → in most of them and ⇒ in one of them which makes no sense to me, especially that I'm not very good at logic. $\endgroup$
    – Hazzle
    Mar 21, 2013 at 20:52
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    $\begingroup$ From what I understand, $\supset$ was used by Russell and Whitehead, $\to$ by Hilbert, and $\Rightarrow$ by Bourbaki. $\endgroup$
    – Zhen Lin
    Mar 28, 2013 at 0:14

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