Inclusion (logic)

In logic and mathematics, inclusion is the concept that all the contents of one object are also contained within a second object.[1]

For example, if m and n are two logical matrices, then

${\displaystyle m\subset n\quad {\text{when}}\quad \forall i,j\quad m_{ij}=1\implies n_{ij}=1.}$

The modern symbol for inclusion first appears in Gergonne (1816), who defines it as one idea 'containing' or being 'contained' by another, using the backward letter 'C' to express this. Peirce articulated this clearly in 1870, arguing also that inclusion was a wider concept than equality, and hence a logically simpler one.[2] Schröder (also Frege) calls the same concept 'subordination'.[3]

References

1. Quine, W. V. (December 1937). "Logic based on inclusion and abstraction". The Journal of Symbolic Logic. 2 (4): 145–152. doi:10.2307/2268279. JSTOR 2268279.
2. "Descr. of a notation", CP III 28.
3. Vorlesungen I., 127.