While I was reading definition of a transitive relation on the set $X$ on Wikipedia: $$\forall a,b,c:(aRb\land bRc)\implies aRc$$ I noticed that it uses logical implication instead of material implication, but isn't the logical implication a sort of "meta-concept"? If it is, doesn't it mean that we can't use it inside of definition in our theory (which can be taken to be set theory)? Or is it just that sometimes $\implies$ is used as material implication?
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2$\begingroup$ Is this just an issue with notation? I wonder if you assume that $\to$ and $\Rightarrow$ have always the same meaning? $\endgroup$– John StellMar 31, 2021 at 19:17
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$\begingroup$ @JohnStell I assumed $\to$ is used as material implication and $\implies$ is used as logical implication, and I didn't assume that they have the same meaning. $\endgroup$– Юрій ЯрошMar 31, 2021 at 19:21
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1$\begingroup$ Sorry, what I wrote was ambiguous and unclear. I didn't mean to ask if they meant the same as each other. What I meant was to point out that they have no fixed meaning in mathematical culture. Sometimes they are used in the same way and sometimes not $\endgroup$– John StellMar 31, 2021 at 21:04
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$\begingroup$ @JohnStell In that case, it kind of is just an issue with notation. I am trying to figure out whether in the Wikipedia definition $\implies$ means material implication, and if not, why it is the case. $\endgroup$– Юрій ЯрошMar 31, 2021 at 21:20
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2 Answers
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In the Wikipedia definition, you should take both the $\wedge$ and the $\Rightarrow$ as being formal logical connectives in the syntax of first order logic. They are not things at the meta level.
You are correct that sometimes $\to$ and $\Rightarrow$ are used denote quite different things, and that sometimes the distinction is between object level and meta level. There is no consistent convention on this and it is all very confusing for people learning about logic.
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Short answer: Sometimes ⟹ is used as material implication.
Long answer: See e.g. my answer here: Implies ($\Rightarrow$) vs. Entails ($\models$) vs. Provable ($\vdash$) As I remark there, "⟹" is used in at a number of different ways, and is probably best avoided!
answered Mar 31, 2021 at 19:19
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$\begingroup$ So, in the definition I mentioned, does $\implies$ mean material implication ? $\endgroup$ Mar 31, 2021 at 21:21