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In logic, the law of excluded middle, or the principle of tertium non datur, is formulated in traditional logic as "A is B or A is not B ". 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, where p is a propositional variable, "∨" means "or", and "¬" means "not".

For example, if P is

Joe is bald

then the inclusive disjunction

Joe is bald, or Joe is not bald

is true.

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 any case, the semantics of any bivalent logic will assign opposite truth values to P and ¬P (i.e., if P is true, then ¬P is false), so the law of excluded middle will be equivalent to the principle of bivalence in a bivalent logic. 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.

1 Historical background
2 A precise definition, and historical importance
3 Kronecker's, and Brouwer's objection
- 3.1 A detour: an example of the intuitionist objection
- 3.2 Back to the debate
4 Recent debate
5 Notes
6 References
7 See also

Historical background

Aristotle discussed the question and, to some minds, nailed it precisely: that when the "excluded middle" is approached, the issue resolves into ambiguity in the mind of the beholder, not in the "fact" itself:

..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, Book IV, Chapter 4, translated by W.D Ross, GBWW volume 8 p. 525-6)

Nevertheless, the ancient Greeks tolerated their ambiguities well: in their mythology Hermaphroditus was the son of Hermes and Aphrodite, rendered by the gods half-man and half-woman while bathing with the nymph Salmacis (Webster's; a different tale is told by Graves)."d. Flattered by Hermes's frank confession of his love for her, Aphrodite presently spent a night with him, the fruit of which was Hermaphroditus, a double-sexed being.." (Graves, Volume I, p. 68)

"8. Aphrodite's son Hermaphroditus was a youth with womanish breasts and long hair .. the hermaphrodite had, of course, its freakish physcial counterpart, but as religious concepts.. originated in the transition from matriarcy to patriarchy .." (Graves, Volume I, p. 73)Bertrand Russell echoed the Greek philosophy as well. 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. Law (2) removes "the middle" of the inclusive-OR used in law (3). For more about this, see below.)However, what Aristotle and Russell believed may not be necessarily be true. Because we ask ourselves: what is "the fact"? Here's an example: Suppose we define a rather typical "truth function" as a "unit step function":

Y=0

when our "input"

X

ranges from

X=0

X=0.5^-

(i.e. 0.4999..) and

Y=1

from

X=0.5^+

(i.e. 0.50.0 plus a hair at infinity) to 1.00. What happens "in the middle" when

X=0.5

exactly? Does

Y=0.5

exactly? As we approach from "underneath" maybe our "truth function" hops suddenly from 0 to 1? Or maybe it shoots to plus infinity and then drops back down, settling out 1. Who's to say?During the 1800's this question was a serious pest to mathematicians studying "analysis". The Aristotelian "fact" is this: we can give examples of mathematical equations that will produce exactly

Y=0.5

when

X=0.5

exactly. So how exactly do we treat this special place

X=0.5

on the number line? One example of such a function is the "sigmoid" function (Cf Mitchell p. 97, so-called because its X-Y plot looks like an S-shape. Mitchell also calls it the "squashing function" or the "logistic function"):

Y = \frac11 + e^k(0.5-X)

When

X \leq 0

Y \geq 0

, but approaches 0 as

X

approaches

-\infty

. When

X \geq 1

then

Y < 1

, but

Y

approaches 1 as

X

goes to

\infty

. We can make our sigmoid as "sharp" a decision-maker as we want to by making

k

larger and larger (

k > 0

). But what happens "in the middle"? Here we see:

Y = \frac11 + e^0 = \frac11 + 1 = 0.5

When

X = 0.5

exactly,

Y = 0.5

exactly. There is no Aristotelian "ambiguity" here. Like the case of Hermaphorditus, we truly have a third case (the "middle") that we have to confront.We see a similar case when we approximate our "step-change truth function" with the coefficients of the Fourier Transform (cf Kreider): a Fourier Transform converts a mathematical function, or a series of points considered "sort of continuous", into a sum of sines and cosines with coefficients we can readily determine). When we're done "transforming" our "step-change truth-function", we will be left with sines only, and their coefficients will look something like this in "the middle" when

X = 0.5

Y = 0.5 - 1/3 + 1/5 -1/7 + 1/9 - 1/11 + \dots

What does this series converge to?

Y = 0.5

exactly. If we are confronted with something like

Y=0.5

when

X=0.5

"in the real world" (and yes this does happen: at many team sporting events) how do we deal with it? We can add a little "noise" (randomness) to our "step-change truth-function" and force the decision (1 OR 0) to be made one way or the other "with the flip of a coin". (See Dither for more). But, philosophically, such a solution is not very satisfying.

A precise definition, and historical importance

Principia Mathematica (PM) defines the “law of excluded middle” formally; in the following we will adopt the more traditional symbols of PM "~" for NOT and "V" OR (inclusive OR, from Latin "vel"):There is no end to the confusion of symbols. Principia Mathematica uses a complex heirarchy of symbols related to how sentences are written (i.e. ending with "periods"). But our confusion becomes intense when PM begins to use "." for logical AND and the horse-shoe shape for "implication", a symbol written "backwards" that is commonly used by set theorists to indicate "set inclusion" (Suppes, Halmos, etc). . Reichenbach (Elements of Symbolic Logic, Dover, 1947, 1975) uses the forward horse-shoe for implication, "." for AND, "V" for inclusive-OR, "^" for exclusive-or (not-equivalence), and "bar" over an expression or symbol for NOT

(negation). Some logicians and mathematicians use inverted V (A without the cross-bar) for AND. Suppes (Axiomatic Set Theory, Dover, 1962, 1970) uses "-" (minus-sign) for NOT, "&" for AND, "V" for OR, "->" for implication, "<->" for "If and only IF" (logical equivalence); Halmos (Naive Set Theory, Springer-Verlag 1970) adds some spice to the sauce with both forward and backward horse-shoes to indicate set-inclusion in both directions. Engineers use horse-shoe or "->" for implication, bar for NOT, V for OR, "+" for OR, elevated dot (multiplication dot or star) for AND or no symbol at all i.e.

C+(A*B) = C V (A & B) = C+A*B+A = (C V A) & (B V A)

C(A+B) = C & (A V B), etc.

Observe that in this first example we have "distributed" a logical OR ("+" sign), something that we cannot do in arithmetic-- caution is advised when using these signs in logic. The historical engineering symbols for AND and OR have been drawn as side-ways "hats" with little input lines on the flat (AND) or curved (OR) side and an output line on the round side, with circles for NOT (negation). But Texas Instruments in the 1970's pushed an abortive attempt of the IEEE to replace the engineering symbols with blocks with symbols drawn inside them to indicate the various logic functions.

Most authors use the commom "=" for equals in the numerical sense. Some use three bars in the equal-sign to indicate "defined as".

Precedence (power) of the symbol is similar to that used in arithmetic: logical AND has precidence over logical OR; parentheses are useful.

Some philosophers have adopted the square and the diamond for use in modal logic. Usually their use of the bent bar indicates NOT (negation). Unless the symbols are defined precisely at the outset, we can assume nothing at all about their meaning.

*2.1 : ~p V 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).

The last part of the sentence “or both ‘this is red’ and ‘this is not red’ is true” is of serious philosophic interest. The precision of the theorem (formula ~p V p) is very important, because as noted in PM (p. 37-38, pp. 59 ff) we can create an antinomy (a sentence that asserts its own falsehood) from the law. The antinomy goes by the name of “Russell’s Paradox”:

.. Take, for example, the law of excluded middle, in the form 'all propositions are true or false.' If from this law we argue that, because the law of excluded middle is a proposition, therefore the law of excluded middle is true or false, we incur a vicious circle fallacy” (p. 38)

PM tries to walk around Russell’s Paradox--disappointingly and unconvincingly and ultimately failing--with its “theory of types” (e.g. instances are categorized in classes, then classes are categorized into classes of classes, ad infinitum). Thus it was in 1930-1931, by the use of a similar antinomy ("This statement is false"), that Gödel demonstrated the undecidability of Principia Mathematica (i.e. logic with arithemtic, logic with the Peano 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* ..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).

These definitions derived directly from Bertrand Russell's philosophy. At the beginning of the 1900's a debate raged between the so-called “Logicists” (e.g. Russell, Frege) and the “constructivists” (“.. strict finitism, intuitionism, and predicativism..” (Dawson p. 49 in footnote 5). Russell presented 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).

This is of interest here because the debate between the “logicists” and the so-called “constructivists” resulted in PM, and PM gave the precise definition the "Law of Excluded Middle", and all this provided some of the tools required for Gödel's undecidability proof:

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 ..

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)

Dawson observes that Brouwer "..campaigned against the use of the Law of Excluded Middle in proofs" (p. 321). (See next section for more about Brouwer.) About this issue (in admittedly very technical terms) Reichenbach observes:

The tertium non datur

(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) (30)

in which form it would be fully exhaustive and therefore nomological in the narrower sense. (Reichenbach, p. 376)The definition of "p exclusive-or q" ( p XOR q, p^q), excludes the third term (the "middle") of the inclusive OR i.e. "or both p is true and q is true"; as shown below where p=1 and q=1 yields p^q=0:

p q ^

0 0 0

0 1 1

1 0 1

1 1 0

Engineers know this as the "half-adder"--it adds in a binary (finite) field, i.e. 1+1=0 without a "carry". We can derive "carry" from "p AND q", i.e. when both are 1, the carry results. "^" is also equivalent to "not-equals", as in "p is not equal to q".

Exclusive-or "^" (XOR) together with AND, "&", can be used to form OR, "V", so these two (exclusive-or "^" with AND) can form the complete set of all 16 possible logical operations (with two "inputs" and one "output"). How is this possible? Simply seen: if we "freeze" p=1 in the truth table above, the "^" operation turns q into ~q, i.e. the "^" symbol becomes NOT with p=1. AND and NOT are sufficient to form all

16 of the logic operations (we can derive OR from these two). If we were equipped with only NOT and AND and XOR we could build an OR as follows:

INCLUSIVE-OR V = ^(p&q)

This simplifies to:

INCLUSIVE-OR V = (p^q)^(p&q)

Interestingly, this is our half-adder XOR'd with its carry term! Unfortunately the "XOR" is difficult to build as its own tiny machine; the NOT-OR and NOT-AND in particular are very easy to build and form the basis of all digital machines (e.g. computers).

In PM all logic is derived from “primitive ideas” (cf PM p. 91 ff) defined as follows:As numbered in PM, the following exist:

(1) Elementary propositions:

(2) Elementary propositional functions:

(3) Assertion:

(4) Assertion of a propositional function:

(5) Negation:

(6) Disjunction (logical sum): p V q:

PM’s definition of p V q immediately above is not particularly satisfactory, as it subtly invokes a notion of “mutual exclusion”. Usually stated: p is true or q is true or both are true simultaneously. But this is no better: “simultaneously” invokes the logical “AND” which is not primitive in PM. The precise definition of “logical OR” comes from Emil Post’s doctoral dissertation of 1920 (et. al., cf Dawson p. 51) and Post’s use of “Truth Tables” to derive the formulas (theorems) of PM. In the examples below we can use “True” and “False” or “1” and “0”; if we use numerals the idea of “logical sum” is more apparent (and also strange, that 1 V 1 = 1). This is the “minterm” that “not mutually exclusive” is dealing with:

a b V

0 0 0

1 0 1

0 1 1

1 1 1

And in this form, “negation” is defined simply as:

a ~ a

0 1

1 0

From the “Law of the excluded middle” PM derives the most powerful tools in the logician’s argumentation tool-kit. Observe the strange and powerful outcome of *2.18:

*2.1 ~p V p

*2.11 p V ~p

*2.12 p -> ~(~p)

*2.13 p v ~ ~(~p)

*2.14 ~(~p) -> p

*2.15 (~p -> q) -> (~q -> p)

*2.16 (p -> q) -> (~q -> ~p)

*2.17 ( ~p -> ~q ) -> ( p -> q)

*2.18 (~p -> p) -> p , 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 :

"In his second problem 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.

> Godel was to later point out that problem came in two parts, the first less serious than the second. The second resides/evolves from, when designing proofs, the use of the universal quantifier "for all" versus the existential "there exists at least one". Issue is described well in Reid's book .

"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)

A detour: an example of the intuitionist objection

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))

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 induction-- 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)

-- work in progress here --

This sub-section will attempt to give meaning to Reichenbach's difficult statement above re the possible use of "exclusive-OR" for tertium non datur. Reichenbach has given the following meaning with regard to so-called nomological formulas:

"The term 'nomological', derived from the Greek word 'nomos' meaning 'law', is chosen to express the idea that the formulas are either laws of nature or logical laws. Analytic nomological formulas are tautological formulas , or logical laws; synthetic nomological formulas are laws of nature. The term 'nomological' is therefore a generalization of therm 'tautological' (Reichenbach, p.)(see Natural Law for more)

In our example above, "Q = P V ~P" is called a "tautology" because it is always true-- either (1) some of the objects are flying pigs or (2) not true that some of the objects are flying pigs or (3) both statements are true. In fact Reichenbach defines "tautology" as the tertium non datur P V ~P:

"All tautologies have the same shortest disjunctive normal form namely (cf Reichenbach p 52). An explanation of "disjunctive normal form" will follow shortly ("disjunction" is the fancy name for logical inclusive OR)

Here's Reichenbach again:

"A true statement is exhaustive in its major terms if none of its residual statements in major terms is true." (p. 362)

A "residual statement" is what is left if we cancel/remove one or more terms of our expression, e.g. if we start with the tertium non datur Q = "P V ~P" we can cancel "P" or "~P". The residual statement would then be "~P" if we cancel "P", and vice versa.

"An inflated form we shall understand a one-scope statement that is not exhaustive in its major terms .." (p. 365)

What is a "term"? Suppose we have two "variables" (sentences identifying sensations for example sentence "a" is "This is a pig" and sentence "b" is "this is flying". We can form 4 possible combinations of the "truths" of our sentences (here abbreviated0: "not-pig", "pig", and "not-flying, "flying". Now what happens when we look for these "characteristics" simultaneously in an object? We can create what known as "the power set" of 2 x 2 = 4 "entities" describes this logical AND of two "variables": (not-pig & not-flying), (not-pig & flying), (pig & not-flying), (pig & flying) .

Reichenbach invokes Russell (p. 357) and David Hume in particular

"We agree with Hume that physical necessity is translatable into statements about repeated occurrences, including the prediction that the same combination will occur in the future, without exception. 'Physcially necessary' is expressible in terms of 'always'." (p. 356)

Back to the debate

Hilbert believed that, if accepted, the Intuitionist would eliminate the mathematics of Cantor et. al. with regard to "the infinite" and "trans-finite", have profound effects on set theory, etc.

> Godel's PhD thesis invoked the Law. An important connection because after he finished his thesis he proceeded on to his "First Incompleteness Theorem" that (arguably) answered Hilbert's Second Question (some argue: in the negative):

"In any case, Godel was certainly well acquainted with the tenets of intuitionism. Toward the end of his introductory remarks he defended the "essential use" he had made of the Law of Excluded Middle, arguing that "from the intuitionistic point of view, the entire problem would be a different one"(Dawson p. 54-56]

> The question reduced to the use of proofs designed from 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)

> Late in his professional life 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))

> Turing's second proof (that no method (algorithm, machine) exists that can determine in general whether or not a given Turing machine M will print a particular symbol e.g. 0) in particular uses the universal quantifier and an "existence" (reductio ad absurdum) proof. The philosophic issues that still surround "the infinite" (a Turing machine has an infinite tape, but a finite "program") are not trivial.

Recent debate

-- under construction --

> So-called fuzzy logic attempts to address the "third" or "middle" value(cf Kusko):

"Aristotle's binary logic came down to one law: A or not-A. Either this or not this. The sky is blue or not blue" (p. 6)

> Chaitin invokes axioms that are susceptible to debate and substitution with others, including the law of excluded middle, "..intuitionist logic or constructivist mathematics" (Chaitin, p. 81)

> Multivalued logics:An entry in Encyclopedia Britannica (Encyclopedia Britannica 2002 Delux Edition CD) cites examples of a 3-valued logic where the value 1/2 is added to the set 0, 1 of truth values to yield the "ternary" logic with truth-values 0, 1/2, 1. They note that various logics can be constructed and are satisfactory with respect to completeness. However, their example is not intuitive (e.g. 1/2 & 1/2 = 1/2).

What happens if we construct our logic from sigmoids with gains "g" of e.g. +8 (for negation), or -8 and a threshold of 0.5 as above? We first begin with the OR "V" as follows:

Logical negation NOT "~":

We simply reverse the sign of the gain "g" so it is positive, and when we do so our sigmoid "inverts" (negates) the input:

~a = 1/(1+exp(+g*( (a) - 0.5)))

Logical OR "V":

We numerically add our "inputs" a and b inside the sigmoid:

c = a V b = 1/(1+exp(-g*((a)+(b)-0.5)))

c = a V b = 1/(1+exp(-g*((1/2)+(1/2)-0.5)))

We see with this formula that (1/2 V 1/2) = 1, which agrees with the usage-approximation of "V" as algebraic "plus" in a finite field.

Logical AND "&":

We use the definition from Principia Mathematica for the logical AND (observe the +sign of the gain g; as noted above, this causes the sigmoid to "invert" its output):

c = (a & b) = ~(~a V ~b)

c = 1/(1+exp(+g*((~a)+(~b) - 0.5))), i.e. where (1-1/2) =def ~a:

c= /(1+exp(+g*((1-1/2)+(1-1/2) - 0.5))),

We see in this case that when (a=0.5 & b=0.5) the output = 0.02 approximately when +g is e.g. +8. This agrees with the usage-approximation of "&" as algebraic "multiplication" -- thus 0.5 x 0.5 = 0.25 which, with enough "gain" in the sigmoid, yields approximately 0.

> If "the law" were "repealed", it is unknown what impact this would have on mathematics and philosophy today . The antinomies (cf antinomy) would still exist (we can create them with the exclusive-OR). Logic would still exist, but "existence" proofs such as reductio ad absurdum might no longer be allowed, and much of modern mathematics (e.g. Turing's second proof in particular(cf Turing's proof)-- because it relies on "not-all" (i.e. the construction ~((Ax):(..)) would be rendered null and void.

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.

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.

1 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.

2 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".

Hans Reichenbach, Elements of Symbolic Logic, Dover, New York, 1947, 1975.

Aristotle, Metaphysics, translated by W.D. Ross, printed in Volume 8 of Great Books of the Western World (GBWW), Encyclopedia Britannica, Chicago 1952, (reprinted from The Works of Aristotle, Oxford University Press).

Donald Kreider, et. al., An Introduction to Linear Analysis, Addison-Wesley, Reading Mass., 1966.

Tom Mitchell, Machine Learning, WCB McGraw-Hill, 1997.

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