Monday, August 18, 2014

Mereological Syllogisms

Mereology is the logic of parts and wholes, although in practice people mostly stick to parts. The two major mereological operators are Part and Overlap. They are pretty much what they sound like. Overlap is symmetric; if a overlaps b, b overlaps a. Part is asymmetric; if a is part of b, it does not follow that b is part of a. (One tricky point is that parthood as usually understood requires that everything is a part of itself, and allows for the possibility that a may be part of b and b part of a, in which case a and b are equivalent.) Part directly implies Overlap. In principle you can do everything with Part than you can with Overlap and vice versa, but it is sometimes convenient to have both.

One of the things you can do with Part and Overlap is to do syllogisms with them, quite literally. For instance, here is a mereological Barbara syllogism:

b is part of c
a is part of b
Therefore a is part of c.

We could summarize this as Pbc + Pab = Pac. (If we wanted to say that something was not a part of something else, we would use ~P instead of P.) Add O for overlap and all the other traditional syllogisms, including the weakened ones, follow with valid mereological arguments.

Barbara: Pbc +Pab =Pac
Celarent: ~Obc + Pab = ~Oac
Darii: Pbc + Oab = Oac
Ferio: ~Obc + Oab = ~Pac
Barbari: Pbc + Pab = Oac
Celaront: ~Obc + Pab = ~Pac

Baroco: Pcb + ~Pab = ~Pac
Cesare: ~Ocb + Pab = ~Oac
Camestres: Pcb + ~Oab = ~Oac
Festino: ~Ocb + Oab = ~Pac
Cesaro: ~Ocb + Pab = ~Pac
Camestros: Pcb + ~Oab = ~Pac

Bocardo: ~Pbc + Pba = ~Pac
Darapti: Pbc + Pba = Oac
Datisi: Pbc + Oba = Oac
Disamis: Obc + Pba = Oac
Ferison: ~Obc + Oba = ~Pac
Felapton: ~Obc + Pba = ~Pac

Bramantip: Pcb + Pba = Oac
Camenes: Pcb + ~Oba = ~Oac
Dimaris: Ocb + Pba = Oac
Fesapo: ~Ocb + Pba = ~Pac
Fresison: ~Ocb + Oba = ~Pac

In addition, invalid categorical syllogisms correspond to invalid mereological syllogisms. Any logical method adequate for categorical syllogisms -- Venn diagrams, for instance, or literal diagrams, or distribution rules, can be directly applied to mereological syllogisms. You can also do the ordinary traditional operations on the mereological syllogisms. For instance, to transform a Cesare to a Celarent, you do exactly the same perfectly legitimate operation, conversion:

~Ocb + Pab = ~Oac
becomes, through conversion of the major,
~Obc + Pab = ~Oac

None of this is in any way surprising or new, for the simple reason that the original reasoning went the other way. Aristotle thought out the rules governing categorical syllogisms in part by thinking them out mereologically, and the discovery of logical quantity is due to recognition that you could have in some sense terms as parts of other terms (giving us A propositions) and terms as overlapping other terms (giving us I propositions). Universal and particular quantity are just Parthood and Overlap.

Since mereological operators are just modal operators, this is one reason, albeit not a definitive one, for thinking that logical quantifiers are just a kind of modal operator, Box and Diamond for a particular kind of domain.

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