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Law of excluded middle

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In logic, the law of excluded middle, or the principle of tertium non datur (Latin "a third is not given", that is, "[next to the two given positions] no third position is available") is formulated in traditional logic as "A is B or A is not B", in which statement A is any subject and B any meaningful predicate to be asserted or denied for A, as in: "Socrates is mortal or Socrates is not mortal". It can be reformulated equivalently as "B(A) or not B(A)". It is conventional in contemporary logical systems to give the same name to the axiom or theorem of propositional logic that typically takes the syntactic form P ∨ ¬P, that is, P or not P, where P is any proposition.

Contents

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Examples

For example, if P is the proposition:

Joe is bald

then the logical disjunction:

It is the case that Joe is bald or it is not the case that Joe is bald

is true by virtue of its form alone.

This is not quite the same as the principle of bivalence, which states that P must be either true or false. It also differs from the law of noncontradiction, which states that ¬(P ∧ ¬P) is true. The law of excluded middle only says that the total (P ∨ ¬P) is true, but does not comment on what truth values P itself may take. In a bivalent logic, P and ¬P will have opposite truth-values (that is, one is true and one is false), and the law of excluded middle necessarily follows. However, the same cannot be said about non-bivalent logics, or many-valued logics.

Certain systems of logic may reject bivalence by allowing more than two truth values (e.g.; true, false, and indeterminate; true, false, neither, both), but accept the law of excluded middle. In such logics, (P ∨ ¬P) may be true while P and ¬P are not assigned opposite truth-values like true and false, respectively.

Some logics do not accept the law of excluded middle, most notably intuitionistic logic. The article "Bivalence and related laws" discusses this issue in greater detail.

The law of excluded middle can be misapplied, leading to the logical fallacy of the excluded middle, also known as a false dilemma.

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The long-arm of the law

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Precedents

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Aristotle

Aristotle wrote that ambiguity can arise from the use of ambiguous names, but cannot exist in the "facts" themselves:

It is impossible, then, that 'being a man' should mean precisely 'not being a man', if 'man' not only signifies something about one subject but also has one significance …

And it will not be possible to be and not to be the same thing, except in virtue of an ambiguity, just as if one whom we call 'man', and others were to call 'not-man'; but the point in question is not this, whether the same thing can at the same time be and not be a man in name, but whether it can be in fact. (Metaphysics 4.4, W.D. Ross (trans.), GBWW 8, 525–526).

We must note, however, that Aristotle's assertion that "...it will not be possible to be and not to be the same thing" is in the form of

~(P & ~P)

and not in the form (P V ~P).

Aristotle asserts that "since it is impossible that contradictories should be at the same time true of the same thing, obviously contraries also cannot belong at the same time to the same thing" (Book IV, CH 6, p. 531). He then proposes that "there cannot be an intermediate beween contradictories, but of one subject we must either affirm or deny any one predicate" (Book IV, CH 7, p. 531, italics added).

Aristotle's words in italics seem to assert the more familiar form of the law of excluded middle (P V ~P), but his additional words "of one subject" and "any one predicate" qualifies his assertion to the "one predicate" under consideration (a "predicate" is: "something which is affirmed or denied of the subject in a proposition in logic" (Websters)). Thus:

I assert that: (Q exist) AND (P V ~P)
I assert that: "Pigs" exist AND ("Pigs do fly" or "pigs do NOT fly")

Whether Aristotle means that ALL pigs are under consideration, or just the ones we see in the pen before us, is unclear. This is the source of the so-called "intuitionist" objections to the "use" of the law of excluded middle in when we are asserting statements about ALL pigs seen and unseen; see more below and at intuitionism.

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Leibniz

" Its usual form, "Every judgment is either true or false" [footnote 9]..."(from Kolmogorov in von Heijenoort, p. 421)
footnote 9: "This is Leibniz's very simple formulation (see Nouveaux Essais, IV,2)....(ibid p 421)
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Bertrand Russell and Principia Mathematica

Bertrand Russell asserted a distinction between the "law of excluded middle" and the "law of contradiction". He cites three "Laws of Thought" as more or less "self evident" or "a priori" in the sense of Aristotle:

"(1) The law of identity: 'Whatever is, is.'
(2) The law of contradiction: 'Nothing can both be and not be.'
(3) The law of excluded middle: 'Everything must either be or not be.'
These three laws are samples of self-evident logical principles..." (p. 72)

[The following needs a reference, or discard it]: It is correct, at least for bivalent logic -- i.e. it can be seen with a Karnaugh map -- that Russell's Law (2) removes "the middle" of the inclusive-OR used in his law (3). And this is the point of Reichenbach's demonstration that some believe the exclusive-OR should take the place of the inclusive-OR; see below.

What Aristotle and Russell believed is characteristic of traditional logic, but this view implicitly depends on a particular notion of truth in which every statement is either true or false. Examples of logical systems that contradict this idea include ternary logic, in which statements may be true, false, or unknown; fuzzy logic, in which statements may be true, false, or somewhere in between; and intuitionistic logic, in which the notion of truth is despensed with entirely, and provability considered in its place.

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A Formal definition from Principia Mathematica

Principia Mathematica (PM) defines the “law of excluded middle” formally:

*2.1 : ~p ∨ p ( PM p. 101 )
Example: Either “this is red” is true or “this is not red” is true or both “this is red” and “this is not red” is true. (See below for more about how this is derived from the primitive axioms).

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

Truth-values. The “truth-values” 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 v 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...” (p. 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. 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”:

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” (p. 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):

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).

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Principia Mathematica's theorems derived from the law of excluded middle

From the law of the excluded middle, formula *2.1 in Principia Mathematica, Whitehead and Russell derive the most powerful tools in the logician’s argumentation tool-kit.

*2.1 ~p V p “This is the Law of excluded middle” (PM, p. 101):
Either “pigs fly” is true or “pigs don’t fly” or both “pigs fly” and “pigs don’t fly” are true assertions.
*2.11 p V ~p [Permutation of the assertions is allowed by axiom 1.4]
*2.12 p -> ~(~p) ["Princple 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”.]
*2.13 p v ~{ ~(~p) [Lemma together with *2.12 used to derive *2.14]
*2.14 ~(~p) -> p [“Principle of double negation, part 2 ]
*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.]
*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.”]
*2.17 ( ~p -> ~q ) -> ( p -> q) [Another of the 'Principles of transposition']
*2.18 (~p -> p) -> p [Called “The complement of reductio ad absurdum. It states that a proposition which follows from the hypothesis of its own falsehood is true” (p. 103-104).

These "tools" -- in particular *2.1 and *2.11, *2.12, *2.14 -- are those objected to by the so-called Intuitionists. 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. 335).

Propositions *2.1 and *2.11, "the law of excluded middle": These are the law of the excluded middle. PM derives “the law" *2.1 p V ~p from Theorem 2.08 ( p -> p) and “primitive idea” *1.08 p -> q = ~p V q. First we substitute p for q in 1.08 to get p -> p. Then we use definition so that ~p V p. QED

Propositions *2.12 and *2.14, "double negation": In the Intuitionist writings of Brouwer in particular the reader will see references 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. 335)".

This "principle" is commonly called "the principle of double negation" (cf PM p. 101-102). From the law of excluded middle *2.1 and *2.11 PM derives princple *2.12 immediately. We subsitute ~p for p in *2.11 to yield ~p V ~(~p), and by the definition of implication (i.e. *1.01 p -> q = ~p V q) then ~p V ~(~p)= p -> ~(~p). QED (The derivation of 2.14 is a bit more involved.)

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The Logicists versus the Intuitionists

In late 1800's through the 1930's a bitter, persistent debate raged between Hilbert and his followers versus Weyl and Brouwer. Brouwer's philosophy-- called Intuitionism-- started in earnest with Kronecker in the late 1800's.

Kronecker's ideas were intensely disliked by Hilbert:

"...Kronecker insisted that there could be no existence without construction. For him, as for Gordan [another elderly mathematician], Hilbert's proof of the finiteness of the basis of the invariant system was simply not mathematics. Hilbert, on the other hand, throughout his life was to insist that if one can prove that the attributes assigned to a concept will never lead to a contradiction, the mathematical existence of the concept is thereby established" (Reid p. 34)
"It was his [Kronecker's] contention that nothing could be said to have mathematical existence unless it could actually be constructed with a finite number of positive integers" (Reid p. 26)

The debate had a profound effect on Hilbert. Reid indicates that his "Second Problem" (cf Hilbert's problems from the Second International Conference in Paris in 1900) evolved from this debate [italics in the original]:

"In his second problem [Hilbert] had asked for a mathematical proof of the consistency of the axioms of the arithmetic of real numbers.
" To show the significance of this problem, he added the following observation:
" 'If contradictory attributes be assigned to a concept, I say that mathematically the concept does not exist '..."(Reid p. 71)

Thus Hilbert was saying: "If "p" and "~p" exist together, then "p" does not exist", and he was thereby invoking The law of excluded middle cast into the form of the law of contradiction.

"And finally constructivists ... restricted mathematics to the study of concrete operations on finite or potentially (but not actually) infinite structures; completed infinite totalities ... were rejected, as were indirect proof based on the Law of Excluded Middle. Most radical among the constructivists were the intuitionists, led by the erstwhile topologist L. E. J. Brouwer ..."(Dawson p. 49)

The rancorous debate continued through the early 1900's into the 1920's-- in 1927 Brouwer complained about "polemicizing against it [intuitionism] in sneering tones"(Brouwer in van Heijenoort, p. 492). But the debate had been fertile: it had resulted in PM (1910-1913), and PM gave a precise definition to the "Law of Excluded Middle", and all this provided an intellectual setting and the tools necessary for the mathematicians of the early twentieth century:

"Out of the rancor, and spawned in part by it, there arose several important logical developments...Zermelo’s axiomatization of set theory (1908a) ... that was followed two years later by the first volume of Principia Mathematica ... in which Russell and Whitehead showed how, via the theory of types, much of arithmetic could be developed by logicist means" (Dawson p. 49).

Brouwer reduced the debate to the use of proofs designed from "negative" or "non-existence" versus "constructive" proof:

"According to Brouwer, a statement that an object exists having a given property means that, and is only proved, when a method is known which in principle at least will enable such an object to be found or constructed...
"Hilbert naturally disagreed.
" '...pure existence proofs have been the most important landmarks in the historical development of our science," he maintained." (Reid p. 155)
"Brouwer ... refused to accept the Logical Principle of the Excluded Middle... His argument was the following:
"Suppose that A is the statement 'There exists a member of the set S having the property P.' If the set is finite, it is possible -- in principle -- to examine each member of S and determine whether there is a member of S with the property P or that every member of S lacks the property P. For finite sets, therefore, Brouwer accepted the Principle of the Excluded Middle as valid. He refused to accept it for infinite sets because if the set S is infinite, we cannot -- even in principle -- examine each member of the set. If, during the course of our examination, we find a member of the set with the property P, the first alternative is substantiated; but if we never find such a member, the second alternative is still not substantiated -- perhaps we have just not persisted long enough!
"Since mathematical theorems are often proved by establishing that the negation would involve us in a contradiction, this third possibility which Brouwer suggested would throw into question many of the mathematical statements currently accepted.
" 'Taking the Principle of the Excluded Middle from the mathematician," Hilbert said, 'is the same as ... prohibiting the boxer the use of his fists.'
"The possible loss did not seem to bother Weyl... Brouwer's program was the coming thing, he insisted to his friends in Zürich." (Reid, p. 149)

In his lecture in 1941 at Yale and the subsequent paper Gödel proposed a solution:

"...that the negation of a universal proposition was to be understood as asserting the existence ... of a counterexample" (Dawson, p. 157))

Gödel's approach to the law of excluded middle was to assert that objections against "the use of 'impredicative definitions'" "carried more weight" than "the law of excluded middle and related theorems of the propositional calculus'" (Dawson p. 156). He proposed his "system Σ ... and he concluded by mentioning several applications of his interpretation. Among them were a proof of the consistency with intuitionistic logic of the principle ~(For all A)(A V ~A) (despite the inconsistency of the assumption (There Exists an A)~(A V ~A)..."(Dawson, p. 157)

The debate seemed to weaken: mathematicians, logicians and engineers continue to use the law of excluded middle (and double negation) in their daily work.

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Intuitionist Definitions of Law (Principle) of Excluded Middle

The following high-lights the deep mathematical and philosophic problem behind what it means to "know", and also helps elucidate what the "law" implies (i.e. what the law really means). Their difficulties with the law emerges: that they do not want to accept as true, implications drawn from that which is unverifiable (untestable, unknowable) or from the impossible or the false. (All quotes are from van Heijenoort, boldface added).

Brouwer offers his definition of "principle of excluded middle"; we see here also the issue of "testability":

"On the basis of the testability just mentioned, there hold, for properties conceived within a specific finite main system, the 'principle of excluded middle', that is, the principle that for every system every property is either correct [richtig] or impossible, and in particular the principle of the reciprocity of the complementary species, that is, the principle that for every system the correctness of a property follows from the impossibility of the impossibility of this property." (335).

Kolmogorov's definition cites Hilbert's two Axioms of Negation

5. A -> (~A -> B)
6. (A -> B) -> {(~A -> B) -> B}
"Hilbert's first axiom of negation, "Anything follows from the false", made its appearance only with the rise of symbolic logic, as did also, incidentaly, the first axiom of implication.... while... the axiom under consideration [axiom 5] asserts something about the consequences of something impossible: we have to accept B if the true judgement A is regarded as false"(p. 421)
" Hilbert's second axiom of negation expresses the principle of excluded middle. The principle is expressed here in the form in which is it used for derivations: if B follows from A as well as from ~A, then B is true. Its usual form, "Every judgment is either true or false" [footnote 9] is equivalent to that given above [footnote 10]" (p. 421)
footnote 9: "This is Leibniz's very simple formulation (see Nouveaux Essais, IV,2). The formulation "A is either B or not-B" has nothing to do with the logic of judgments.
footnote 10: "Symbolically the second form is expessed thus
"A V ~A "where V means "or". The equivalence of the two forms is easily proved..." [etc]" (p. 421)
"Clearly from the first interpretation of negation, that is, the interdiction from regarding the judment as true, it is impossible to obtain the certitude that the principle of excluded middle is true... Brouwer showed that in the case of such transfinite judgments the principle of exdluded middle cannot...be considered obvious" (p. 421)
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Do pigs fly? An example of the Intuitionist objections

The section above stated that the intuitionists believe that only if we can investigate all instances of a finite set { a, b, c, d, e, f, g, ... z } can we assert a truth or its negation. But when we evoke NOT-(FOR ALL x) where "ALL x" covers all cases known and unknown, we can "never be sure",

e.g. Not true that "Pigs fly" can be written:
~((Ax)Ey Ez:(IF x is a z THEN x & y)) ["A" here the universal quantifier "For all...", "E" is the existential quantifier "There exists at least one instance of..."
Thus e.g. IT IS NOT TRUE IN ANY INSTANCE WHATEVER THAT:
"For all objects x of the entire universe known and all universes unknown, given that at least one prototype (concept, description, attribute, sensation) 'pig' exists and at least one prototype (concept, description, attribute, sensation) 'fly' exists: IF object x is a member of the set (class) of objects 'pig' THEN object 'pig' possesses attribute 'fly'."

Here's an example using finite logic: Let's examine n objects (say n=142), O1, O2, O3, up to On=O142 and decide if they are flying pigs. Let p1, p2, p3,...pi up to p142 stand for "object Oi is a pig that flies" (e.g. we examine O42 and decide whether or not O42 is a flying pig, and then we assign p42 a "truth value" of 1 if O42 is a flying pig and 0 if it is not). Let "P" stand for the proposition that "Some of these objects are flying pigs". And let "Q" be the question P V ~P: "Some are flying pigs", or not-"some are flying pigs"

The first proposition P we write as "the disjunction" (fancy word for logical OR) of all 142 of our investigations: P = (p1 V p2 V p3 ... V p142), e.g. "Either p1 or p2 or p3 or ... pn indicates a flying pig". And we know that this statement is either "true" or "false".

The second proposition ~P starts out simply as the negation of P, but then we do some fancy footwork to convert it into a conjunction (fancy word for logical AND). Thus

Q = P V ~P = (p1 V p2 V p3 ... V p142) V ~(p1 V p2 V p3 V p4 ... V p142)
The next step is the fancy footwork: it comes about from the definition in PM of logical AND i.e. (a & b) = ~(~a V ~b), and the theorem a = ~(~a).
Q = P V ~P = (p1 V p2 V p3 V ... V p142) V (~p1 & ~p2 & ~p3 & ~p4 & ... ~p142)

Now suppose that we want to determine the truth-value of our question Q. We examine our objects-as-possible-flying-pigs O1 to O142 one after another. We are done as soon as we find a flying pig: e.g. if p1 is not a flying pig = 0 (false), but if p2 is a flying pig = 1 (true), then we're done:

Q = P V ~P = (0 V 1 V doesn't matter) V (doesn't matter)

Thus even a single instance of a flying pig answers the question:

(0 V 1 V any truth-values whatever) = 1

But suppose we aren't having much luck, and we have to go "all the way to the end", i.e. examine all 142 objects. When we hit #141 we have:

Q = P V ~P = (0 V 0 V 0 V ... V 0 V p142) V (~0 & ~0 & ~0 V ... & ~0 & ~p142)

AS we approach object O142 there's only two possible otucomes: either (1) O142 is a flying pig and p142=1 or (2) O142 is not a flying pig and p142=0:

Case 1-- p142 is a flying pig:
Q = (0 V 0 V 0 ... 0 V 1) V (~0 & ~0 & ~0 ... & ~0 & ~1)
Q = (0 V 0 V 0 ... 0 V 1) V (1 & 1 & 1 ... & 1 & 0)
Q = (P=1 V ~P=0) = (1) V (0) = 1
It's true: Pigs fly!
Case 2-- p142 is not a flying pig:
Q = (0 V 0 V 0 ... 0 V 0) V (~0 & ~0 & ~0 ... & ~0 & ~0)
Q = (0 V 0 V 0 ... 0 V 1) V (1 & 1 & 1 ... & 1 & 1)
Q = (P=0 V ~P=1) = (0) V (1) = 1
It's true: Pigs don't fly! [More accurately we now know that: Some pigs don't fly. ]

Observe, however, that to reach our understanding that ~P is the true case (not a single instance of flying pigs in our 142 objects), we had to go all the way to the end. Every pig-object had to be examined-- and this is because of the logical AND in the construction.

This is okay when we're dealing with finite numbers of objects, but what happens when we have infinite numbers of objects? How can we be sure there isn't a lone instance of a flying pig somewhere e.g. a freak hiding under a rock on planet Org in sun-system R2D2, galaxy C3PO, universe Chewbacca? We can't be sure. One thus would argue that we have to rely on inductive reasoning-- the probability that after examining 1.32 x 10^132 pig-like objects and not finding a single flying pig, that there really and truly aren't any. (cf Russell 2)


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Use of exclusive-or in place of inclusive-or

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

The tertium non datur
(x)[f(x)V ~f(x)] (29)
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'
(x)[f(x)^~f(x)] (30) [the "^" signifies exclusive-or]
in which form it would be fully exhaustive and therefore nomological in the narrower sense. (Reichenbach, p. 376)

In the line (30) the "(x)" means "For ALL" or "For EVERY", thus an example of the expression would look like this:

(For ALL Q): (P ^ ~P)
(For ALL instances of "pig" seen and unseen): ("Pig does fly" or "Pig does not fly" but not both simultaneously)
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References

  • Dawson, J., Logical Dilemmas, The Life and Work of Kurt Gödel, A.K. Peters, Wellesley, MA, 1997.
  • van Heijenoort, J., From Frege to Gödel, A Source Book in Mathematical Logic, 1879-1931, Harvard University Press, Cambridge, MA, 1967. Reprinted with corrections, 1977.
* Luitzen Egbertus Jan Brouwer, 1923, On the significance of the principle of excluded middle in mathematics, espcecially in function theory [reprinted with commentary, p. 334, van Heijenoort]
* Andrei Nikolaevich Kolmogorov, 1925, On the princple of excluded middle, [reprinted with commentary, p. 414, van Heijenoort]
* Luitzen Egbertus Jan Brouwer, 1927, On the domains of definitons of functions,[reprinted with commentary, p. 446, van Heijenoort] Although not directly germane, in his (1923) Brouwer uses certain words defined in this paper.
* Luitzen Egbertus Jan Brouwer, 1927(2), Intuitionistic reflections on formalism,[reprinted with commentary, p. 490, van Heijenoort]
  • Kneale, W. and Kneale, M., The Development of Logic, Oxford University Press, Oxford, UK, 1962. Reprinted with corrections, 1975.
  • Alfred North Whitehead and Bertrand Russell, Principia Mathematica to *56, Cambridge at the University Press 1962 (Second Edition of 1927, reprinted). Extremely difficult because of arcane symbolism, but a must-have for serious logicians.
  • Bertrand Russell, The Problems of Philosophy, With a New Introduction by John Perry, Oxford University Press, New York, 1997 edition (first published 1912). Very easy to read: Russell was a wonderful writer.
  • Bertrand Russell, The Art of Philosophizing and Other Essays, Littlefield, Adams & Co., Totowa, NJ, 1974 edition (first published 1968). Includes a wonderful essay on "The Art of drawing Inferences".
  • Donald Kreider, et. al., An Introduction to Linear Analysis, Addison-Wesley, Reading Mass., 1966.
  • Webster's Ninth New Collegiate Dictionary, Merriam-Webster, 1990. Also: Webster's New World Dictionary, College Edition, World Publishing Co. 1966. This entry for Hermaphroditus mentions the nymph by name: Salmacis.
  • Robert Graves The Greek Myths Volume One, Penguin Books, First published 1955, Reprinted 1975.
  • Constance Reid, Hilbert, Copernicus: Springer-Verlag New York, Inc. 1996, first published 1969. Contains a wealth of biographical information, much derived from interviews.
  • Bart Kusko, Fuzzy Thinking: The New Science of Fuzzy Logic, Hyperion, New York, 1993. Fuzzy thinking at its finest. But a good introduction to the concepts.
  • Gregory Chaitin, "The Limits of Reason", Scientific American, March 2006, p.74-81. His book does not address "the law" directly but for more about algorithic complexity see: Meta math!: The Quest for Omega. Gregory chaitin. Pantehon Books, 2005.
  • David Hume, An Inquiry Concerning Human Understanding, reprinted in Great Books of the Western World Encyclopedia Britannica, Volume 35, 1952, p.449ff. This work was published by Hume in 1758 as his rewrite of his "juvenile" Treatise of Human Nature: Being An attempt to introduce the experimental method of Reasoning into Moral Subjects Vol. I, Of The Understanding first published 1739, reprinted as: David Hume, A Treatise of Human Nature, Penguin Classics, 1985. Also see: David Applebaum, The Vision of Hume, Vega, London, 2001: a reprint of a portion of An Inquiry starts on p. 94ff
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See also

Retrieved from "http://en.wikipedia.org/wiki/Law_of_excluded_middle"
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