Editing Plural quantification (section)

=== 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]].