There are some odd books in the world. I found this one:

Alternative Picture of the World, by Leonid G. Kreidik and George P. Shpenkov in a library sale and picked up for £1.50. 

Here's a link to the guy's own website: 

It's not work that's really been picked up anywhere or particularly used, and I can definitely say I don't understand very much of it. 

But what little I do understand is this: 

They suggest a new form of logic based on judgements, such as affirmation (yes in 
English or Si in some logics) and negation (no or No), which relate to measures. The measures may be quantitative, qualitative, or a weird combined quantity qualitative-qualitative and these measures a properties of any 'object of thought'. 

All quite abstract, but then, this is logic. 

Still, it occurred to me that this is the perfect tool for thinking about counting infinities.

Imagine that there is a restaurant with infinite number of places for diners, there are then an infinite number of chairs and an infinite number of plates. How many items of cutlery are there? 

Say it's a relatively posh restaurant, posh enough to give you two knives and two forks, but not posh enough to make them different sizes. 

So, there are 2 x infinity knives and 2 x infinity forks or 4 x infinity items of cutlery. 

I've also heard the example of the chair legs. If there is an infinite number of chairs, lets call this number A, there are 4 x infinity chair legs, lets call this number B. 

But, intuitively, since infinity is supposed to be the biggest number possible, this can't be true. 

Well, there are different sizes of infinity, there has to be, because the number of chair legs infinity must be 4 times that of the number of chairs. 

So, is A (the number of infinite chairs) equal to B (the number of legs on the infinite chairs), or not?

Kreidik and Shpenkov's judgements can explain this quite neatly. 

Qualitatively, the infinities (of chair legs to chairs) are equal, because they are both infinite, ie qualitatively A = B

Quantitatively, the infinities are not equal, because quantitatively B = 4A

And perhaps the best way of encapsulating the dual nature of the problem, is A equal to B, is to write is as a quantitative-qualitative judgement:


ie No, not quantitatively, yes, qualitatively. 

I think that this can be a good way to reason about some things.  

Their reasoning is interesting too. 

They suggest that the world is perceived in a concrete-abstract way. The part of the brain involved with concrete thought perceives the world and then it is converted to abstract ideas by part of the brain that does that. So they think that reasoning in these seemingly contradictory ways is more natural. 

In this example, since it is so abstract, I think qualitative-quantitative makes more sense as the first answer to the question is yes, they're both infinities and therefore equal, it is only after a moments reflection that you might realise the problem with that statement. 
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