Editing Plural quantification (section)

== Formal definition ==

This section presents a simple formulation of plural logic/quantification approximately the same as given by Boolos in ''Nominalist Platonism'' (Boolos 1985).

=== Syntax ===

Sub-sentential units are defined as

* Predicate symbols <math>F</math>, <math>G</math>, etc. (with appropriate arities, which are left implicit)
* Singular variable symbols <math>x</math>, <math>y</math>, etc.
* Plural variable symbols <math>\bar{x}</math>, <math>\bar{y}</math>, etc.

Full [[sentence (mathematical logic)|sentences]] are defined as

* If <math>F</math> is an ''n''-ary predicate symbol, and <math>x_0, \ldots, x_n</math> are singular variable symbols, then <math>F(x_0, \ldots, x_n)</math> is a sentence.
* If <math>P</math> is a sentence, then so is <math>\neg P</math>
* If <math>P</math> and <math>Q</math> are sentences, then so is <math>P \land Q</math>
* If <math>P</math> is a sentence and <math>x</math> is a singular variable symbol, then <math>\exists x.P</math> is a sentence
* If <math>x</math> is a singular variable symbol and <math>\bar{y}</math> is a plural variable symbol, then <math>x \prec \bar{y}</math> is a sentence (where ≺ is usually interpreted as the relation "is one of")
* If <math>P</math> is a sentence and <math>\bar{x}</math> is a plural variable symbol, then <math>\exists \bar{x}.P</math> is a sentence

The last two lines are the only essentially new component to the syntax for plural logic. Other logical symbols definable in terms of these can be used freely as notational shorthands.

This logic turns out to be equi-interpretable with [[monadic second-order logic]].

=== Model theory ===

Plural logic's model theory/semantics is where the logic's lack of sets is cashed out. A model is defined as a tuple <math>(D,V,s,R)</math> where <math>D</math> is the domain, <math>V</math> is a collection of valuations <math>V_F</math> for each predicate name <math>F</math> in the usual sense, and <math>s</math> is a Tarskian sequence (assignment of values to variables) in the usual sense (i.e. a map from singular variable symbols to elements of <math>D</math>). The new component <math>R</math> is a binary relation relating values in the domain to plural variable symbols.

Satisfaction is given as

* <math>(D,V,s,R) \models F(x_0, \ldots, x_n)</math> iff <math>(s_{x_0}, \ldots, s_{x_n}) \in V_F</math>
* <math>(D,V,s,R) \models \neg P</math> iff <math>(D,V,s,R) \nvDash P</math>
* <math>(D,V,s,R) \models P \land Q</math> iff <math>(D,V,s,R) \models P</math> and <math>(D,V,s,R) \models Q</math>
* <math>(D,V,s,R) \models \exists x.P</math> iff there is an <math>s' \approx_x s</math> such that <math>(D,V,s',R) \models P</math>
* <math>(D,V,s,R) \models x \prec \bar{y}</math> iff <math>s_xR\bar{y}</math>
* <math>(D,V,s,R) \models \exists \bar{x}.P</math> iff there is an <math>R' \approx_\bar{x} R</math> such that <math>(D,V,s,R') \models P</math>

Where for singular variable symbols, <math>s \approx_x s'</math> means that for all singular variable symbols <math>y</math> other than <math>x</math>, it holds that <math>s_y = s'_y</math>, and for plural variable symbols, <math>R \approx_\bar{x} R'</math> means that for all plural variable symbols <math>\bar{y}</math> other than <math>\bar{x}</math>, and for all objects of the domain <math>d</math>, it holds that <math>dR\bar{y} = dR'\bar{y}</math>.

As in the syntax, only the last two are truly new in plural logic. Boolos observes that by using assignment ''relations'' <math>R</math>, the domain does not have to include sets, and therefore plural logic achieves ontological innocence while still retaining the ability to talk about the extensions of a predicate. Thus, the plural logic comprehension schema <math>\exists \bar{x}. \forall y. y \prec \bar{x} \leftrightarrow F(y)</math> does not yield Russell's paradox because the quantification of plural variables does not quantify over the domain. Another aspect of the logic as Boolos defines it, crucial to this bypassing of Russell's paradox, is the fact that sentences of the form <math>F(\bar{x})</math> are not well-formed: predicate names can only combine with singular variable symbols, not plural variable symbols.

This can be taken as the simplest, and most obvious argument that plural logic as Boolos defined it is ontologically innocent.