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There are three propositions A, B, C and another condition "some two of these propositions are true and the third one is false", or, in other words, "exactly 2 of 3 propositions are true". Using truth tables and a Karnaugh map (as discussed at How to find the logical formula for a given truth table?) i deducted the Boolean expression for this: ABC' + AB'C + A'BC. Is there any more succinct notation for this expression in any branch of logic?

Edit: Obviously using proposition calculus notation the above statement may be represented as: $(A \wedge B \wedge \neg C) \vee (A \wedge \neg B \wedge C) \vee (\neg A \wedge B \wedge C)$. I am sorry if that misguided you. I'm still interested, if any more succinct notation is possible.

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    If you are using the symbol for your own purposes (taking notes, studying, etc.), then invent any symbol you like. If you are planning to use this symbol on a manuscript you are expecting _anyone_ else to read, then I strongly suggest you do not use such a symbol. I personally believe that one should do everything possible to make their own papers as easy to read as possible. In my opinion, if you are not willing to put in the work to make your paper as easy to digest as possible, then why should others put in the work to read the paper? I would highly recommend simply using words.2012-08-08

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Using the Iverson bracket, $ [A]+[B]+[C]=2 $

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The most symmetric definition of 'exactly one of three' I know is $ \text{exactly one of } P, Q, R \text{ is true} \;\equiv\; (P \equiv Q \equiv R) \land \lnot (P \land Q \land R) $ This uses the fact that equivalence ($\;\equiv\;$) is associative.

Using this, we can write \begin{align} & \text{exactly two of } A, B, C \text{ are true} \\ = & \;\;\;\;\;\text{"invert the count"} \\ & \text{exactly one of } A, B, C \text{ is false} \\ = & \;\;\;\;\;\text{"$\;P \equiv \text{false}\;$ is the same as $\;\lnot P \equiv \text{true}\;$ (three times)"} \\ & \text{exactly one of } \lnot A, \lnot B, \lnot C \text{ is true} \\ = & \;\;\;\;\;\text{"the above definition"} \\ & (\lnot A \equiv \lnot B \equiv \lnot C) \land \lnot(\lnot A \land \lnot B \land \lnot C) \\ = & \;\;\;\;\;\text{"simplify"} \\ & \lnot(A \equiv B \equiv C) \land (A \lor B \lor C) \\ \end{align}

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    +1 for firmness, although obviously I disagree: if one uses logic as a tool (as opposed to an object of study) then this is one of the things that is actually helpful. Therefore I prefer the other way to avoid confusion, which is to never write something like $\;\ldots = \ldots = \ldots\;$, and instead always use the calculational multi-line format as I used in this answer.2013-11-18