I've been reading through De Morgan's Formal Logic. The book seems more of a combination of an introduction to logic and a grab bag of De Morgan's ideas than a formal system, but I can see why some of them are influential, especially his introduction of quantification as a kind of operator.

What I can't find in there are the laws De Morgan is most famous for, the laws that are named for him (even though he wasn't the first to observe them):

~(P & Q) <=> ~P | ~Q
~(P | Q) <=> ~P & ~Q

Surprisingly, neither Wikipedia nor the Stanford Encyclopedia of Philosophy states where De Morgan wrote this, or even if it is in Formal Logic or some other work. It seems that he would have have to have read Boole to be thinking of AND and OR like operations this way, but Boole's The Mathematical Analysis of Logic came out the same year, so in that case it would have to have been in a later work.

Alternately, are what we call De Morgan's laws a symbolic interpretation of an idea he expressed very differently in Formal Logic as rules about quantification?


1 Answer 1


See Formal Logic (1847), page 116:

The contrary of PQR is p,q,r. [...] In contraries, conjunction and disjunction change places.

And see A.De Morgan's On the Syllogism III (1858), page 182:

The contrary of an aggregate is the compound of the contraries of the aggregants: the contrary of a compound is the aggregate of the contraries of the components. Thus (A,B) and AB have ab and (a.b) for contraries.

A more "modern" symbolic formulation is present into C.S.Perice, On the Algebra of Logic: A Contribution to the Philosophy of Notation (1885), page 191:

"overline"(x+y)= "overline (x) "overline"(y).

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