[FOM] Why inclusive disjunction? A conclusive summary

Arnon Avron aa at tau.ac.il
Sun Jan 14 03:40:51 EST 2007

Ignoring the historical question, I would like to summarize in this
message all the reasons I know (and read in this thread) why the choosing the
inclusive or as the default interpretation of "or" was an extremely
clever choice for *mathematical* logic. The reasons are three:

A) Because this is the interpretation actually used in mathematical
   texts and by people practising mathematics (at all levels)

B) Because using inclusive or in mathematics is much more convenient 
   than using xor (This, I guess, is related to A).

C) Because from a theoretical point of view the use of inclusive or
   (henceforth: "or") has a lot of technical advantages over
   the use of xor.

Let me elaborate with Examples.

A) In order not to go too far, I have checked several textbooks for 
*highschool" trigonometry (written for under-undergraduates). In all of them
I found the following exercise:

 "Prove that a triangle in which  Sin(2A)=Sin(2B) (where A and B are
 two of the triangle's angles) is either isosceles
 or has a right angle."

On the other hand in no book did I find the exercise formulated as:

 "Prove that a triangle in which  Sin(2A)=Sin(2B) (where A and B are
 two of the triangle's angles) is either isosceles
 or it has a right angle, or it is a right isosceles triangle"

Another example from the same textbooks: in order to solve exercises
as above, and for solving trigonometric equations, the following
principle is formulated:
 (*)  Sin(A)=sin(B) iff either A=B+360K or A+B=180+360K

Needless to say, the "or" here is inclusive, not xor.

B) Let us examine the last example. It is written in the language
used in mathmatical textbooks for highschool. In this language
the explicit use of quantifiers is almost completely avoided.
Thus the intended meaning of (*) is:

  Sin(A)=sin(B) iff 
     either \exists K s.t. A=B+360K or \exists K s.t A+B=180+360K

This is equivalent to:

 Sin(A)=sin(B) iff 
     \exists K s.t. A=B+360K or  A+B=180+360K

(the two formulations are equivalent only for inclusive or, of course).

Because of the close connection between \exists and inclusive or,
it is possible to omit the existential quantifier in the less
precise language of mathematical textbooks, and write only (*),
without too much risk of confusion (even though it cannot
be said that it causes no difficulties for students!).

Now try to do the same with Xor! Writing:

  Sin(A)=sin(B) iff either A=B+360K xor A+B=180+360K xor both

will *certainly* be misleadsing. The natural understanding of it would be:

Sin(A)=sin(B) iff 
     either \exists K s.t. A=B+360K xor \exists K s.t A+B=180+360K
        xor \exists K s.t. A=B+360K and A+B=180+360K

Which is simply wrong. The correct formulation would be:

Sin(A)=sin(B) iff
     either \exists K s.t. A=B+360K xor \exists K s.t A+B=180+360K
        xor \exists K,L s.t. A=B+360K and A+B=180+360L

Now compare how complicated is this formulation of the 
principle using xor in comparison to its formulation using or!
And this is what happens when we use *precise* language.
As for formulating this principle using exclusive interpretation
of "or" in the langauge actually used for highschool mathematics - it is 
not even clear to me how to do it at all in an unambiguous way (but
maybe native English speakers can do better).

C) Here is a list of technical advantages of inclusive or (most of which
   were noted in various previous messages on the subject):

  1) The two most important operations on sets are intersection
     and union. The first corresponds to conjunction. The
     second - to inclusive or. Therefore the properties of union 
     and inclusive or are easily understood in terms of each other.

  2) It is helpful to concieve of the quantifiers as infinitary
     analogues of connectives: the universal quantifier as an
     infinite conjunction, the existential quantifier as an
     infinite disjunction. The disjunction here is of course
     the inclusive one (the connection is reflected e.g. by the
     distribution of \exists over or, already noted above).

  3) The duality between conjunction and inclusive or is extremely
     useful. Among other things, this duality is reflected in 
     De-Morgan laws, which turns the task of negating a proposition 
     into a mechanical task (with xor we lose this duality
     as well as De-Morgan laws).

  4) The set consisting of negation and or is functionally complete.
     The set consisting of negation and xor is not. In particular:
     material implication is very easily defined in terms of 
     negation and or, while it cannot be defined in terms
     of negation and xor.

Arnon Avron

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