# There exist less than 3 in predicate logic

Consider a predicate logical language with predicate symbol A that takes one argument, and predicate symbol R that takes two arguments:

• $a$ denotes Alice
• $b$ denotes Bob
• $Ax$ means $x$ is a woman
• $Rxy$ means $x$ is a child of $y$

I want to translate the following statement in predicate logic:
Alice and Bob have less than 3 sons

I interpret this sentence as Anna having less than 3 sons and Bob having less than 3 sons.

Here is my try:

$$\forall y \Big( (y=a\lor y=b)\to \big[\exists k_{ 0 },k_{ 1 },k_{ 2 }(Rk_{ 0 }y\land Rk_{ 1 }y\land Rk_{ 2 }y\land \lnot Ak_{ 0 }\land \lnot Ak_{ 1 }\land \lnot Ak_{ 2 })\to (k_{ 0 }=k_{ 1 }\lor k_{ 0 }=k_{ 2 } \lor k_1 = k_2) \big] \Big)$$

• Is there a reason to use so many colors, except testing out the \color function? – Asaf Karagila Jul 11 '13 at 18:56
• Because I have a problem spotting at least one pair of parentheses, and another problem in discerning the color of two pairs (which I know is different by looking at the code). I consider this to be an excessive use of the color features. Please read this meta thread and the comments here. – Asaf Karagila Jul 11 '13 at 18:59
• I think a more natural reading of the given sentence is that there are less than three people who are sons of both Anna and Bob – Chris Eagle Jul 11 '13 at 18:59
• @AsafKaragila I was getting confused of all the parenthesis, this was my way of checking if the parenthesis are correct. – Kasper Jul 11 '13 at 19:01
• You can use \Bigg,\bigg,\Big,\big as well. It's much easier to see, understand, and everyone can check that. Also you should program for a few months in Common Lisp and then parentheses won't be any real problem anymore. – Asaf Karagila Jul 11 '13 at 19:02

The most natural reading of the sentence Anna and Bob have less than $3$ sons is that Anna and Bob are a couple, and that they are the parents of at most two boys (and some unknown number of girls, possibly none). This does not rule out the possibility that Bob, say, has more sons by some other woman. I would interpret it, then, as the verbal equivalent of this:

$$\neg\exists x,y,z(x\ne y\land x\ne z\land y\ne z\land\neg Ax\land\neg Ay\land\neg Az\land Rxa\land Rxb\land Rya\land Ryb\land Rza\land Rzb)$$

‘There do not exist $x,y$, and $z$ that are distinct, not women, and children of both Anna and Bob.’

You can of course replace

$$\neg\exists x,y,z\big(x\ne y\land x\ne z\land y\ne z\land\varphi(x,y,z)\big)$$

by

$$\forall x,y,z\Big(\varphi(x,y,z)\to x=y\lor x=z\lor y=z\Big)$$

if you prefer.