# [FOM] A historical question about notation

Alasdair Urquhart urquhart at cs.toronto.edu
Thu Aug 22 14:39:59 EDT 2013

```> Over lunch today with one of the members of this list, the question of the
> notation for the empty set came up. I am wondering if listmembers can confirm
> the following allegations which i find floating around inside my head..
>
> 1   The capital \$\Lambda\$ notation for the empty set (used by Russell and
> Quine) is a capital `L' connoting German `leer';

This symbol for the empty set is used by Peano in his 1889 pamphlet
"Arithmetices principia, nova methodo exposita."  Peano doesn't explain
the origin of the notation.  However, we can be pretty sure  that the
sign was produced by the typesetter by turning a capital "V" upside down.
Peano uses this device repeatedly, for example, his symbol for implication
is an upside-down "C" (the origin of the horseshoe).

2   The `v' for disjunction comes from Latin `vel' meaning `or';

3   `\$\cup\$' for set union derives from vel;
4 `\$\cap\$' derives from `\$\cup\$' and `\$\wedge\$' analogously from `\$\vee\$'
turning symbols upside-down connotes duality.

Peano uses \$\cap\$ and \$\cup\$ in his 1889 pamphlet indifferently
as set operators and propositional connectives.  Whitehead and Russell,
under the influence of Frege, distinguishes the two syntactical
categories strictly.  So, I am inclined to think that "v" derives
from Peano's \$\cup\$ and not the other way around.

> 5   `\$V\$' for the universe is an upside-down `\$\Lambda\$'.

I think that, following 1 above, the derivation is the other
way around.

```