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A previous question, Principal ultrafilters, asked for clarification of Markus Kracht's assertion that "In a standard model (where we allow quantifying over all subsets) there is a biunique correspondence between the individuals of the domain and the set of all subsets of the domain containing that individual (such sets are also called principal ultrafilters)" (p.12 here: https://user.phil-fak.uni-duesseldorf.de/~rumpf/SS2010/ComSem/Kra08.pdf).

The OP wished to know whether the qualification "where we allow quantifying over all subsets" was necessary, and why a standard model was necessary for the generalisation to hold.

Noah Schweber responded that "defining the map "Send x to the set of all sets containing x" requires us to quantify over sets of sets. It's not that this bijection would fail to 'work' in too weak a logic, but that $\textit{it would fail to exist."$\hspace{0.2cm}($my italics})$

Would such a function fail to exist in a second order logic with the henkin semantics simply for the banal reason that there may be subsets that $x$ belongs to but which are not part of the model since Henkin models are restricted to a proper subset of all sets?

Just what was meant by fail to 'exist' in this sense?

A further question: given it is standard for people working in the Montagovian tradition of formal semantics for natural language to treat names of individuals such as $\textit{John}$ as denoting the principal ultrafilters they generate, is this practice at all put into question if we are using a Henkin semantics and not the full semantics (and so can't gather together all subsets of the domain to which $John$ belongs)?

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  • It seems to me that the map would no longer be injective, in general, in Henkin models. Consider a simple domain with individuals $a$ and $b$ and only one set, ${a,b}$. On the other hand, the map would seem to always exist, because at worst an element $x$ would map to $\emptyset$ if no sets in the model contain $x$. – Carl Mummert Oct 17 '16 at 23:44

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The issue is not really the quantification over sets. The map that is being constructed is between individuals ("first order" or "type 0" objects) and sets of sets of individuals ("third order" or "type 2" objects). So the map itself can't be constructed in second-order logic regardless of the semantics.

The issue is that, in non-full models, there might be two distinct individuals which cannot be distinguished by any set in the model - each set in the model contains both, or neither, of the individuals. Then the map that is being constructed from individuals to sets of sets of individuals will no longer be injective.

Carl Mummert
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  • Montague formulated his theory in type theory, so the function could presumably be defined. But my question was also this: those natural language semanticists in linguistics and computer science working in weaker theories often treat names of individuals such as $John$ as denoting the principal ultrafilters they generate, and so given the problem of distinguishing two distinct individuals that you mentioned, they would have to restrict admissible models to those in which every individual has some unique property that singles them out. Can this be done elegantly? –  Oct 18 '16 at 10:19
  • Also, you write "So the map itself can't be constructed in second-order logic regardless of the semantics." I take this to mean that the usual practice of natural language semanticists in linguistics and computer science, in which $John$ denotes the function $\lambda P. \thinspace P(john)$, from arbitrary sets $P$ to the true iff John is a member of those sets $\lambda P. \thinspace P(john)$, REQUIRES a logic of higher order than full second order logic. Is that correct? –  Oct 18 '16 at 10:33
  • It is hard to say what is elegant. The usual construction seems to be in type theory (I am no expert) and could also be done in set theory, which is typeless, but the particular nature of second-order logic prevents the map from being defined there. I think that any worry about second-order logic is misplaced; I mentioned it only because it had come up already in the question. The sources linked in the question are already working in type theory, it seems. – Carl Mummert Oct 18 '16 at 12:05