Editing Law of excluded middle (section)

===Reichenbach===
It is correct, at least for bivalent logic—i.e. it can be seen with a [[Karnaugh map]]—that this law removes "the middle" of the [[Logical disjunction|inclusive-or]] used in his law (3). And this is the point of Reichenbach's demonstration that some believe the [[Exclusive or|''exclusive''-or]] should take the place of the [[Logical disjunction|''inclusive''-or]].

About this issue (in admittedly very technical terms) Reichenbach observes:

::The tertium non datur
::29. (''x'')[''f''(''x'') ∨ ~''f''(''x'')]
::is not exhaustive in its major terms and is therefore an inflated formula. This fact may perhaps explain why some people consider it unreasonable to write (29) with the inclusive-'or', and want to have it written with the sign of the ''exclusive''-'or'

::30. (''x'')[''f''(''x'') ⊕ ~''f''(''x'')], where the symbol "⊕" signifies [[exclusive-or]]<ref>The original symbol as used by Reichenbach is an upside down V, nowadays used for AND. The AND for Reichenbach is the same as that used in Principia Mathematica – a "dot" cf p. 27 where he shows a truth table where he defines "a.b". Reichenbach defines the exclusive-or on p. 35 as "the negation of the equivalence". One sign used nowadays is a circle with a + in it, i.e. ⊕ (because in binary, a ⊕ b yields modulo-2 addition – addition without carry). Other signs are ≢ (not identical to), or ≠ (not equal to).</ref>
::in which form it would be fully exhaustive and therefore [[nomological]] in the narrower sense. (Reichenbach, p. 376)

In line (30) the "(x)" means "for all" or "for every", a form used by Russell and Reichenbach; today the symbolism is usually <math>\forall</math> ''x''. Thus an example of the expression would look like this:

* (''pig''): (''Flies''(''pig'') ⊕ ~''Flies''(''pig''))
* (For all instances of "pig" seen and unseen): ("Pig does fly" or "Pig does not fly" but not both simultaneously)