Maietti's modification of the propositions-as-types doctrine to bring it into line with the set-theoretic interpretation of formulas is, in essence, to take propositions (or formulas) to correspond to mono-objects, that is, singletons, rather than to arbitrary sets.
Interpretation of formulas and programs by mutual recursion.
Note that these clauses still are not purely structural conditions but conditions on the interpretation of formulas.
On such an interpretation the formula H2O represents two atoms of hydrogen combined with one of oxygen.
Secondly, we introduce novel notions of (bi simulations and prove that they indeed preserve the interpretation of formulas in (the universal fragment of) SOPML.
Now in the propositions-as-monotypes interpretation, wherein formulas correspond to singletons, \(\forall i \in I\) continues to correspond to \(prod_{i \in I}\), since the product of singletons is still a singleton.
Although there have been modifications to this theory, this has predominantly been in the realm of alternative interpretations of formulae and models rather than the simplification or reduction of levels (Cuvalay, 1995; Hannay and Bolkestein, 1998).
In particular, if f and m live in the domain of M, then M ⊨ "f : ω → m is a bijection" if and only if f really is a bijection between the natural numbers and m.[22] Under its ordinary English interpretation, this formula says that the set-theoretic universe doesn't contain any bijections between the natural numbers and x.
The interpretation of such formulae is not unproblematic, however.
Using different and more complex problems dealing with the interpretation of chemical formulae, we studied the Brain Mapping Hypothesis' robustness towards a different domain, multidimensional strategy spaces, and modeling errors.
