Paradox of cheese with holes

This Paradoxe is generally known under name paradox of the Gruyere , but as it is desirable to be precise when the paradox is driven out and that actually the Gruyère does not have holes (with the difference, inter alia, of the Emmental), we will speak about cheese (implied: with holes). It is stated then:

Plus there is cheese, more there are holes;
gold more there are holes, less there is cheese;
thus more there is cheese, less there is cheese.

It is most famous of the paradoxes builds according to a mode of syllogism called B' a' rb' a' r' a' (or Syllogisme in A-A-A). It highlights that the application without understanding of the syllogism is source of aberrations.

Actually, it takes place here a subtle slip Sémantique between the average terms of both premises; what prohibits the conclusion. This slip is due to a contextual difference. Indeed, for each premise, there exists a Contexte implied which makes possible the acceptance of the premise as a truth. However these implicit contexts are opposed and invalidated the syllogism.

In detail, one can schematize the thing by considering the volume cheese (packed, i.e. holes included/understood) and the density of the cheese (the matter ratio/volume).

Is not there average to conclude? Yes; if one adopts the point of view of the Logique mathematics. It is then necessary to consider the context as a Axiome. The conclusion is then a theorem relative to axiomatic necessarily containing the conjunction of the contexts.
Thus, it is deduced here that, when volume and density of cheese are constant, one has well “more there is cheese, less there is cheese”. What can be reduced to a idiotic Truisme: “as much there is cheese, as much there is cheese”.

Ultimately, the conclusion must be considered invalid rather than false because the two occurrences of the word “cheese” are related to two distinct concepts:

In the first (that of major), the vacuums belong to cheese. The cheese is defined then by its external appearance.
In the second (that of minor), the cheese is defined as a matter, and consequently, excludes the vacuum.

Thus these two antagonistic concepts should not be amalgamated in the same sentence. It is an underhand form of Polysémie.

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