Inclusion (logic)

In logic and mathematics, inclusion is the concept that all the contents of one object are also contained within a second object.{{cite journal| doi=10.2307/2268279 | journal=The Journal of Symbolic Logic | volume=2 | number=4 | date=December 1937 | pages=145–152 | title=Logic based on inclusion and abstraction | first=W. V. | last=Quine | jstor=2268279 }}

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

:$m\; \backslash subset\; n\; \backslash quad\; \backslash text\{when\}\; \backslash quad\; \backslash forall\; i,j\; \backslash quad\; m\_\{ij\}\; =\; 1\; \backslash implies\; n\_\{ij\}\; =\; 1\; .$

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."Descr. of a notation", CP III 28. Schröder (also Frege) calls the same concept 'subordination'.Vorlesungen I., 127.