In the dunce’s corner …

I’m spending a bit of quality time in the dunce’s corner, having dashed off a ‘solution’ to a problem in propositional calculus (yes propositional calculus, for heaven’s sake) over at math.stackexchange and got it quite wrong. And as fate would have it, I was off-line for a while, staying over with my aged mother, so couldn’t quickly delete it. Ho hum. So much for totting up “reputation points” eh?

OK: here’s the problem for you (from Enderton’s Math Logic book): let # be the three-place connective such that #(A, B, C) is true so long as the majority of A, B, C are true. Show that {#, ¬} is not an expressively adequate set of connectives.

Knowing I’m quite capable of that sort of thing makes the business of doing a final read of  Gödel 2 for thinkos a bit fraught. Even at this last minute stage I’ve found a silly omission. Ah well ….

Still, there are mistakes and mistakes. I had occasion a couple of days ago to look at James Robert Brown’s Philosophy of Mathematics where he makes a complete hash of understanding why the Intermediate Value Theorem is non-trivial (no, a diagram does not cut it). I was wondering about whether to bang on about that here, when — by one of those coincidences — I came across the ever-excellent T.W. Körner‘s Companion to Analysis where he takes a sideswipe at Brown: “The author cannot understand the problems involved in proving results like the intermediate value theorem and has written his book to share his lack of understanding with a wider audience.” Körner than proceeds to give a presentation which should be enough for most readers to work out (at least some of) what’s gone so wrong in Brown’s discussion. Job done.

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