Editing Law of excluded middle (section)

=== Bertrand Russell and ''Principia Mathematica'' ===
The principle was stated as a [[theorem]] of [[propositional logic]] by [[Bertrand Russell|Russell]] and [[Alfred North Whitehead|Whitehead]] in ''[[Principia Mathematica]]'' as:

<math>\mathbf{*2\cdot11}. \ \  \vdash . \ p \ \vee \thicksim p</math>.<ref>{{citation|author=[[Alfred North Whitehead]], [[Bertrand Russell]]|title=Principia Mathematica|publisher=[[Cambridge]]|year=1910|pages=105 |url=http://name.umdl.umich.edu/aat3201.0001.001}}</ref>

So just what is "truth" and "falsehood"? At the opening ''PM'' quickly announces some definitions:

{{quote|''Truth-values''. The "truth-value" of a proposition is ''truth'' if it is true and ''falsehood'' if it is false* [*This phrase is due to Frege] … the truth-value of "p ∨ q" is truth if the truth-value of either p or q is truth, and is falsehood otherwise … that of "~ p" is the opposite of that of p …" (pp. 7–8)}}

This is not much help. But later, in a much deeper discussion ("Definition and systematic ambiguity of Truth and Falsehood" Chapter II part III, p.&nbsp;41 ff), ''PM'' defines truth and falsehood in terms of a relationship between the "a" and the "b" and the "percipient". For example "This 'a' is 'b{{'"}} (e.g. "This 'object a' is 'red{{'"}}) really means {{"'}}object a' is a sense-datum" and {{"'}}red' is a sense-datum", and they "stand in relation" to one another and in relation to "I". Thus what we really mean is: "I perceive that 'This object a is red{{'"}} and this is an undeniable-by-3rd-party "truth".

''PM'' further defines a distinction between a "sense-datum" and a "sensation":
{{quote|That is, when we judge (say) "this is red", what occurs is a relation of three terms, the mind, and "this", and "red". On the other hand, when we perceive "the redness of this", there is a relation of two terms, namely the mind and the complex object "the redness of this" (pp. 43–44).}}

Russell reiterated his distinction between "sense-datum" and "sensation" in his book ''The Problems of Philosophy'' (1912), published at the same time as ''PM'' (1910–1913):
{{quote|Let us give the name of "sense-data" to the things that are immediately known in sensation: such things as colours, sounds, smells, hardnesses, roughnesses, and so on. We shall give the name "sensation" to the experience of being immediately aware of these things … The colour itself is a sense-datum, not a sensation. (p. 12)}}

Russell further described his reasoning behind his definitions of "truth" and "falsehood" in the same book (Chapter XII, ''Truth and Falsehood'').

==== Consequences of the law of excluded middle in ''Principia Mathematica'' ====
From the law of excluded middle, formula ✸2.1 in ''[[Principia Mathematica]],'' Whitehead and Russell derive some of the most powerful tools in the logician's argumentation toolkit. (In ''Principia Mathematica,'' formulas and propositions are identified by a leading asterisk and two numbers, such as "✸2.1".)

✸2.1 ~''p'' ∨ ''p'' "This is the Law of excluded middle" (''PM'', p.&nbsp;101).

The proof of ✸2.1 is roughly as follows: "primitive idea" 1.08 defines ''p'' → ''q'' = ~''p'' ∨ ''q''. Substituting ''p'' for ''q'' in this rule yields ''p'' → ''p'' = ~''p'' ∨ ''p''. Since ''p'' → ''p'' is true (this is Theorem 2.08, which is proved separately), then ~''p'' ∨ ''p'' must be true.

✸2.11 ''p'' ∨ ~''p'' (Permutation of the assertions is allowed by axiom 1.4)<br />
✸2.12 ''p'' → ~(~''p'') (Principle of double negation, part 1: if "this rose is red" is true then it's not true that {{"'}}this rose is not-red' is true".)<br />
✸2.13 ''p'' ∨ ~{~(~''p'')} (Lemma together with 2.12 used to derive 2.14)<br />
✸2.14 ~(~''p'') → ''p'' (Principle of double negation, part 2)<br />
✸2.15 (~''p'' → ''q'') → (~''q'' → ''p'') (One of the four "Principles of transposition". Similar to 1.03, 1.16 and 1.17. A very long demonstration was required here.)<br />
✸2.16 (''p'' → ''q'') → (~''q'' → ~''p'') (If it's true that "If this rose is red then this pig flies" then it's true that "If this pig doesn't fly then this rose isn't red.")<br />
✸2.17 ( ~''p'' → ~''q'' ) → (''q'' → ''p'') (Another of the "Principles of transposition".)<br />
✸2.18 (~''p'' → ''p'') → ''p'' (Called "The complement of ''reductio ad absurdum''. It states that a proposition which [[Logical consequence|follows from]] the hypothesis of its own falsehood is true" (''PM'', pp.&nbsp;103–104).)

Most of these theorems—in particular ✸2.1, ✸2.11, and ✸2.14—are rejected by intuitionism. These tools are recast into another form that Kolmogorov cites as "Hilbert's four axioms of implication" and "Hilbert's two axioms of negation" (Kolmogorov in van Heijenoort, p.&nbsp;335).

Propositions ✸2.12 and ✸2.14, "double negation":
The [[Intuitionism|intuitionist]] writings of [[L. E. J. Brouwer]] refer to what he calls "the ''principle of the reciprocity of the multiple species'', that is, the principle that for every system the correctness of a property follows from the impossibility of the impossibility of this property" (Brouwer, ibid, p.&nbsp;335).

This principle is commonly called "the principle of double negation" (''PM'', pp.&nbsp;101–102). From the law of excluded middle (✸2.1 and ✸2.11), ''PM'' derives principle ✸2.12 immediately. We substitute ~''p'' for ''p'' in 2.11 to yield ~''p'' ∨ ~(~''p''), and by the definition of implication (i.e. 1.01 p → q = ~p ∨ q) then ~p ∨ ~(~p)= p → ~(~p). QED (The derivation of 2.14 is a bit more involved.)