Logical paradoxes by Robert L. Baber

Logical paradoxes:
Fundamental difficulties in mathematics
or just ordinary word problems?
(Draft)

by Robert L. Baber, Bob@RLBaber.de
2004 January 18, modified 2004 May 8

Abstract: It is the thesis of this paper that several well known paradoxes in logic can be viewed as ordinary word problems in mathematics. From the translation of the natural language statement of each paradox into a simple mathematical formulation of the problem, it can be straightforwardly deduced that the problem has no solution. Associated with these paradoxes are extensive mathematical analyses and investigations into mathematical logic, set theory, etc., over the past hundred years or more. While those investigations may have been valuable for other purposes, they were not really necessary to resolve the particular paradoxes in question, and they can lead to unnecessarily complicated ways to solve these and other problems in mathematical logic. This observation may have implications for teaching mathematics and logic, especially to students who do not expect to become theoretical mathematicians, but who will need only to use and apply mathematics and logic in other disciplines.

The Liar's Paradox: Consider the sentence

This sentence is false. [1]

and the question whether this sentence is true or false. The noun phrase "This sentence" is interpreted to refer to the above sentence as a whole. [Wikipedia, Liar paradox]

Mathematically, sentence [1] above can be viewed (modeled) as a Boolean expression (an expression that evaluates to "true" or to "false"). The phrase "This sentence" can be viewed as a Boolean variable which we will call s below. Sentence [1] above can be interpreted to mean that the Boolean value of s is equal to the Boolean constant "false":

s = false [2]

The mathematical expression [2] is a translation of sentence [1] into the language of mathematics.

The statement of the problem restricts the value of the variable s to be the Boolean value of sentence [1], i.e. to be the value of the expression [2] above, leading to the requirement (condition on the value of s) that

s = (s = false) [3]

The answer to the question whether sentence [1] is true or false is, then, the value of s that satisfies expression [3], i.e. the value of s for which expression [3] has the value true. Expression [3] can be simplified to (s = ¬s) and further to the logical constant "false". Thus, no Boolean value of s satisfies the requirement. The problem has no solution. These simplifications assume the definitions of the functions = and ¬ only. Alternatively, one can simplify expression [3] directly to the constant "false" assuming only the definition of =.

Often this result is viewed as a fundamental problem or paradox, but it need not be so viewed. Many equations in mathematics have no solution, e.g. x=x+4 in the numbers, sin(x)=2, etc. Mathematics abounds with examples of problems that have no solution, one solution, or two or more solutions (e.g. x²-3x+2=0). None of these situations is considered to be a fundamental problem for mathematics or mathematical logic.

A major causal factor of such a paradox is sometimes considered to be the recursive reference within the sentence to the sentence itself, but such a recursive reference is neither a necessary nor a sufficient condition for a paradox, i.e. a requirement that is always false.

Consider a variation of the liar's paradox, which one might call the truth teller's paradox (although it does not strictly satisfy the definition of "paradox"):

This sentence is true. [4]

with the question whether this sentence is true or false.

Translating sentence [4] into the language of mathematics as above, we obtain the expression

s = true [5]

as the expression corresponding to sentence [4] above and

s = (s = true) [6]

as the requirement which the value of s must satisfy. Expression [6] can be simplified to (s = s) and further to the logical constant "true". Thus, both "true" and "false" as values of s are consistent with the requirement [6].

Russell's Paradox: Consider the set S to be the set of all sets that do not contain themselves as members. More formally, R is an element of S if and only if R is not an element of R. The question to be answered is, "Is S an element of S?". [Wikipedia, Russell's paradox]

The above condition defining whether or not R is an element of S (R∈S) can be translated into the mathematical expression R∉R. Thus,

(R∈S) = (R∉R)

or, equivalently,

(R∈S) = ¬(R∈R) [7]

To determine whether or not R is in S for a given R, one need only evaluate the expression ¬(R∈R). Its value is also the value of (R∈S).

In order to answer the question "Is S an element of S?", we must consider the special case in which R=S. Expression [7] becomes

(S∈S) = ¬(S∈S) [8]

To seek the answer to the question whether or not S∈S, we must find a value of the expression S∈S for which the value of expression [8] is true, that is, find a value v for which the Boolean expression (v = ¬v) is true. But this expression is never true; the problem has no solution. In other words, the "definition" of S above is inadequate, in particular, both S∈S and S∉S fail to satisfy the "definition".

Consider the variation of this paradox in which the set S is the set of all sets that do contain themselves as members, i.e., R is an element of S if and only if R is an element of R. The condition corresponding to [7] is, then,

(R∈S) = (R∈R)

and the condition corresponding to [8] is

(S∈S) = (S∈S)

which reduces to the logical constant true. Again in this case, the "definition" of S above is inadequate to determine whether or not S is an element of S, but now both S∈S and S∉S satisfy the "definition".

The Barber's Paradox: Figaro, a barber in Seville, is said to have placed a sign in his window stating "I shave all those men in town, and only those men, who do not shave themselves." Does Figaro shave himself? [Wikipedia, Barber paradox]

We define s as infix operator between two names of men in Seville so that X s Y if and only if X shaves Y. X and Y may be the same, i.e. we allow X s X, which means that X shaves himself.

The condition stating whether or not Figaro (F) shaves a man (e.g. X) in Seville, when translated into the language of mathematics, becomes

(F s X) = ¬(X s X) [9]

and the condition which must be "solved" for the answer to the question whether or not Figaro shaves himself becomes

(F s F) = ¬(F s F) [10]

which, as in the cases of the paradoxes above, has no solution. Again, the "definition" of whom Figaro shaves is inadequate to determine whether or not he shaves himself. Both the case in which he does shave himself and the case in which he does not shave himself are inconsistent with the "definition". The above "definition" is insuitable.

Reformulating the condition to state that Figaro shaves all those men in town, and only those men, who do shave themselves (unrealistic but logically meaningful) is also an inadequate "definition" in that no unique answer to the question can be determined. Both the case that Figaro shaves himself and the case that he does not shave himself are consistent with the statement; the "definition" still leaves the answer to the question open.

Conclusion: The above three famous paradoxes in logic do not, really, represent a fundamental problem in mathematics or in set theory. Instead, they provide additional examples that not every problem that can be stated in natural language -- or even in mathematical language -- has a solution. Apparent definitions are not always really definitions. As already previously known in mathematics, some problems have no solution, some have one (a unique) solution, and others have many solutions. In this context, a solution means particular values for variables appearing in a Boolean expression such that the value of the expression is true. A Boolean expression that is equivalent to the logical constant false has no solutions. A Boolean expression that is not equivalent to the logical constant false, i.e. which is true for some value(s) of the variable(s) appearing therein, will have one or more solutions. The logical paradoxes examined above all have the characteristic that they seek solutions to Boolean expressions that can be reduced to the form (x = ¬x) and, in turn, to the logical constant false. No additional mathematical theory is needed to explain this phenomenon arising in these closely related paradoxes. Each of the logical paradoxes above is nothing other than an ordinary word problem for which no solution exists.

References:
Wikipedia, The Free Encyclopedia, "Barber paradox", http://en.wikipedia.org/wiki/Barber_paradox.
Wikipedia, The Free Encyclopedia, "Liar paradox", http://en2.wikipedia.org/wiki/Liar_paradox.
Wikipedia, The Free Encyclopedia, "Russell's paradox", http://en2.wikipedia.org/wiki/Russell's_paradox.

See also:
Alfred North Whitehead, Bertrand Russell, F.R.S., Principia Mathematica, Cambridge University Press, 1910 and later editions and reprints.
and the great number of books, articles, etc. published on this and closely related subjects in the last century or so.