"Many" quantifier in first order logic

Question

Suppose I want to express this sentence in first-order logic:

"Many printers are broken" 

If $p(x)$ is true if the term $x$ is a printer and $b(x)$ is true if $x$ is broken, how can I express this sentence?

Answer 1

First-order logic does not have a notion of "many" - this, like "most," "almost all," etc. is an example of a generalized quantifier, and handling them takes us to extensions of first-order logic.

We can express each of the following in first-order logic:

• "Some printer is broken." ($\exists x(p(x)\wedge b(x))$)

• "Multiple printers are broken." ($\exists x,y(x\not=y \wedge p(x)\wedge p(y)\wedge b(x)\wedge b(y))$)

• "At least $n$ printers are broken." ($\exists x_1,...,x_n((\bigwedge_{1\le i<j\le n}x_i\not=x_j)\wedge (\bigwedge_{1\le k\le n}p(x_k)\wedge b(x_k))$)

• "All printers are broken." ($\forall x(p(x)\implies b(x))$)