[FOM] Why inclusive disjunction? A conclusive summary

[Kenny Easwaran]

I believe at least one reason was mentioned in this discussion that
was left out of this list, namely, the Gricean explanation.  There are
clear reasons why a connective that semantically expresses inclusive
"or" might often pragmatically implicate an exclusive "or" (which is a
stronger claim), while there are very few conditions under which a
connective semantically expressing exclusive "or" would somehow only
pragmatically commit one to the weaker inclusive "or".

In fact, it would seem very surprising if any language had a
connective that clearly indicated exclusive "or", for precisely this
reason.  (The supposed counterexample of Latin is discussed in the
Stanford Encyclopedia page on disjunction, together with some good
clarification of a lot of issues brought up in this thread.)

http://plato.stanford.edu/entries/disjunction/

Kenny Easwaran

On 1/14/07, Arnon Avron <aa at tau.ac.il> wrote:
> 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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