I saw this question earlier today and was trying to figure out how to determine one way or the other whether a most quantifier can be implemented using plural quantifiers. Then I read the SEP article on plural quantifiers and now I'm really confused as to what these things mean.
If we allow sets of truth values, then there's a straightforward way to encode statements in ordinary first order logic as expressions involving sets. What is the right way to do this for plural quantifiers?
I've attempted to include below what I think plural quantifiers mean based on the article but there are few things I'm unsure about.
- Treatment of singleton plural individuals
- Extending singleton-taking predicates to plural individuals
I'm specifically going after the formal language called LPFO+ in the article.
The article goes into some detail about using plural individuals as a type of non-set collection to avoid Russell's paradox. I'm not using them for this purpose; I'm just looking at the cases where a domain of discourse that is a set has already been nailed down.
This is my attempt to define run-of-the-mill higher order quantifiers (of which first-order quantifiers are a special case) by constructing sets of truth values. I'm then attempting to do the same thing for plural quantifiers. I'm thinking of first-order logic as a special case of higher-order logic where the set quantified over is always $D$.
First a word on notation.
Let $\varepsilon$ be the empty set.
Let $\prec$ be a logical relation insisting that the left argument is contained in the right argument.
Let $[\cdot]$ be the Iverson bracket, which sends true sentences to $1$ and false sentences to $0$.
Let $\phi[x:=v]$ refer to the well-formed formula formed by replacing $x$ with $v$ in $\theta$.
Let $\text{Set}[xx]$ refer to the set corresponding to value of the plural variable $xx$.
Let $D$ be a set, the sometimes-implicit domain or universe of discourse of first-order logic.
Let $\phi$ be a well-formed formula, possibly with free variable $x$. Let $\Phi$ be a well-formed formula, possibly with free variable $xx$.
Following the notation in the SEP article, let $x$ be a singular variable and $xx$ be a plural variable.
If we have the language of set theory, we use that to define the meaning of $\forall$ and $\exists$ in classical higher order logic.
$$ \forall x : D \mathop. \phi \stackrel{\text{def}}{\iff} \{[\phi[x:=t]] \mathop. t \in D\} \in \{\varepsilon, \{1\}\} $$
$$ \exists x : D \mathop. \phi \stackrel{\text{def}}{\iff} \{[\phi[x:=t]] \mathop. t \in D \} \in \{\{1\}, \{0,1\}\} $$
In LPFO+, some predicates treat plural entities collectively and others do not.
let $\Phi$ be an arbitrary wff where $xx$ is a free variable. We can define existential and universal plural quantification in terms of our earlier definitions.
$$ \exists xx \mathop. \Phi \stackrel{\text{def}}{\iff} \exists t : D \to 2 \mathop. t \neq \varepsilon \land \Phi[xx := t] $$
$$ \forall xx \mathop. \Phi \stackrel{\text{def}}{\iff} \forall t : D \to 2 \mathop. t \neq \varepsilon \to \Phi[xx:=t] $$
However, we have an implicit conversion of sorts between predicates taking singular arguments and predicates taking plural arguments. Predicates taking singular argument can also be applied distributively.
If $P$ is a predicate taking a singular argument, then
$$ P(xx) \stackrel{\text{def}}{\iff} \forall t : \text{Set}[xx] \mathop. P(t) $$
Let $ww$ be a free plural variable in the below expressions. If $\theta$ is a wff with a singular free variable $x$, then
$$ \theta[x:=ww] \;\;\text{is equivalent to}\;\; (\forall x \mathop. x \prec ww \to \theta) $$
I think this is sufficient to nail down the truth conditions of a formula, open or closed, involving plural quantifiers.
