In we expand our semantic theory to capture cases of intra-sentential anaphora.
In, we combine ’s perspective-sensitive semantic theoryPerspective-sensitive semantic theory with a hyperintensional situation semantics, HYPE, using monadsMonads from category theor圜ategory theory in order to ‘upgrade’ an ordinary intensional semanticsIntensional semantics to a possible hyperintensional counterpart. P-HYPE is a hyperintensional situation semanticsSituation semantics in which hyperintensionality is modelled as a ‘side effect’, as this term has been understood in natural language semanticsNatural language semantics, and in functional programmingFunctional programming. Moreover, if we keep only loops and double edges and discard single edges, the resulting graph contains countably many connected components isomorphic to any given finite connected graph with loops. We show that, in ZFA, if we keep the loops and ignore the multiple edges, we obtain the "random loopy graph" (which is $\aleph_0$-categorical and homogeneous), but if we keep multiple edges, the resulting graph is not $\aleph_0$-categorical, but has infinitely many $1$-types. The resulting graph may fail to be simple, it may have loops (if $x\in x$ for some $x$) or multiple edges (if $x\in y$ and $y\in x$ for some $x,y$). The crucial axiom in the proof of this is the Axiom of Foundation, so it is natural to wonder what happens if we delete this axiom, or replace it by an alternative (such as Aczel's Anti-Foundation Axiom). It is known that, if we take a countable model of Zermelo-Fraenkel set theory ZFC and "undirect" the membership relation (that is, make a graph by joining $x$ to $y$ if either $x\in y$ or $y\in x$), we obtain the Erd\Hs-R\'enyi random graph.
We show that, in ZFA, if we keep the loops and ignore the multiple edges, we obtain the “random loopy graph” (which is ℵ0-categorical and homogeneous), but if we keep multiple edges, the resulting graph is not ℵ0-categorical, but has infinitely many 1-types. The resulting graph may fail to be simple it may have loops (if x∈x for some x) or multiple edges (if x∈y and y∈x for some distinct x, y). The crucial axiom in the proof of this is the Axiom of Foundation so it is natural to wonder what happens if we delete this axiom, or replace it by an alternative (such as Aczel’s Anti-Foundation Axiom). It is known that, if we take a countable model of Zermelo–Fraenkel set theory ZFC and “undirect” the membership relation (that is, make a graph by joining x to y if either x∈y or y∈x), we obtain the Erdős–Rényi random graph.
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