What is the dual of a logical proposition?
I’m reading a paper on the geometric implications of analytical logic and it discusses the concept of the “dual” of a proposition. I would normally be inclined to think this means some sort of equivalent proposition (e.g., p AND (NOT q) = NOT[(NOT p) OR q], but it appears to refer to, say, p AND q versus p OR q. I’m familiar with the duality principle of projective geometry, but this appears to be very different. Can anybody illuminate this relationship for me? I’m not immediately seeing it.
You’re right, this logical duality is different from the duality in projective geometry.
But the dual of a proposition is not an equivalent proposition. As Michel N said, the dual of a proposition is obtained by interchanging AND with OR, and interchanging 0 (FALSE) with 1 (TRUE). NOT is left unchanged. (I’m not sure what Michel meant by bringing in DeMorgan’s Laws, but this idea is called duality, not DeMorgan’s Laws.)
Thus the dual of
P AND (Q OR NOT R)
is
P OR (Q AND NOT R)
The idea of duality gets more complicated if you try to bring in IMPLIES, so that is usually omitted.
If you regard propositions as formulas of Boolean Algebra, then some equations are valid in Boolean Algebra, and others are not.
P AND Q = Q AND P (valid)
P AND Q = NOT Q (not valid)
The principle of duality says that if an equation is valid, then its dual is also valid.
There is a sense in which projective geometry can be reformulated in terms of lattice theory, and when that is done, then the duality of projective geometry turns out to be the same as the duality of Boolean Algebra.
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