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If I have two propositions $p$ and $q$, what is the simplest standard notation for expressing that they are logically consistent with each other—in other words, that no contradiction can be derived from assuming them both.

Something like $\,p,q \,\nvdash\! \bot$ seems clunky, especially if we use the notation many times, and $p \parallel q$ has unwanted connotations from geometry. Are there better options?

META: This is a very basic question about mathematical notation. From perusing the various Stack Exchange sites, it appears math.se is the best place to ask it, but feel free to correct me if I'm wrong.

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    $\begingroup$ This is a good question for the site. I don't have an answer, but it probably won't be closed. $\endgroup$
    – Rushabh Mehta
    Commented Sep 13, 2019 at 14:32
  • $\begingroup$ See whether this helps you. $\endgroup$
    – user371838
    Commented Sep 13, 2019 at 14:42

1 Answer 1

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I'm sorry to disappoint your expectations, but the widely used notation to say that two propositions $p$ and $q$ are consistent with each other is $p,q \not\vdash \bot$ or $\text{Con}(\{p,q\})$. This is the same notation used more in general to say that a set $\Gamma$ of propositions is consistent ($\Gamma \not\vdash \bot$ or $\text{Con}(\Gamma)$).

As far as I know, there is no special (and widely used) notation for consistency of just two propositions.

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    $\begingroup$ +1. To the OP, it's worth noting that joint consistency isn't actually all that great a notion. For example, it's not transitive: we can have $a,b\not\vdash \perp$ and $b,c\not\vdash \perp$ but $a,c\vdash \perp$. The point is that while its specific instances may be (indeed, are often) important and interesting, its general behavior is too weak to be of much use in the abstract. And this is probably why we don't have a snappier dedicated notation for it. $\endgroup$
    – Noah Schweber
    Commented Sep 13, 2019 at 16:39
  • $\begingroup$ @NoahSchweber - Good point! I totally agree. $\endgroup$
    – Taroccoesbrocco
    Commented Sep 13, 2019 at 16:41

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