Query: Can the discrepancy be used as a quantifier?

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Eric Macaulay

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Dec 20, 2016, 7:39:56 AM12/20/16
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Shalom!

I was wondering, the equivalence is symmetric, associative with unit true and serves as a quantifier. Now the discrepancy (or inequivalence, if you prefer) is also symmetric, associative with unit false. Can it not too serve as a quantifier?

Agape,
Eric

Bastiaan Braams

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Jul 15, 2017, 2:56:24 PM7/15/17
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In a model that has false represented by 0 and true represented by 1 the discrepancy corresponds to addition modulo 2 and conjunction corresponds to multiplication; these are the operations of a boolean ring. We are happy to use quantifier notation for addition and it should be uncontroversial, I think, to use it for addition modulo 2.
In fact, the original poster provides broader reasons for admitting quantifier notation for discrepancy: the operation is symmetric, associative, and has a unit (or neutral) element. It is defined for any finite bag of arguments. I have understood Dijkstra's notation for quantified expressions to apply in precisely that situation.
-Bas Braams (new here and therefore replying a bit late)

Juan Michelini

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Jul 23, 2017, 10:02:35 PM7/23/17
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David Gries' discusses this in A Logical Approach to Discrete Math, Chapter 8: Quantification.
He generalizes quantification and proves various properties about it. Have fun!
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