[FOM] Inclusive and Exclusive Disjunction

[Richard Heck]

As several people pointed out, exclusive disjunction is of course
associative. I'll blame my statement to the contrary on the Iraqi
insurgency.

As several people have also mentioned, Boole treated disjunction as
exclusive. He seems not to have been widely followed in this.  Frege
mentions in a footnote to a paper written in the early 1880s (Posthumous
Writings, p. 10) that Stanley, Jevons, Schr"oder, et al, "have not
followed" Boole, and he suggests that Boole's treatment was a departure
from the Leibnizian tradition. One would suppose, however, that Boole's
choice was not simply rejected out of hand but that people had their
reasons.

My hunch is that the failure of certain algebraic analogies was what
impressed the Booleans, the whole point of the Boolean approach being to
highlight those very analogies. My intention in mentioning the
associative law was supposed to have been to highlight one such failure.
That was a poor choice, but there are other such failures. For example,
the distribution law for 'and' over 'xor' fails. Letting 'xor' be +, we
do not have:
    A & (B + C) = (A + B) & (A + C)
Just take the case where A is false and B and C are both true.

Richard

-- 
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Richard G Heck, Jr
Professor of Philosophy
Brown University
http://bobjweil.com/heck/
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