The Analytic Atavar

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Saturday, April 28, 2007

Liar, Liar, Logic Afire

Possibly the oldest logical paradox is what is today known as the Liar Paradox. The first appearance of the paradox is attributed by some to Epimenides of Knossos, a Cretan philosopher and poet, circa the 6th century B.C.E.. All of his works are lost and known only from quotations by other authors. The paradox is said to arise from a fragment of his poem Cretica, where Minos addresses Zeus thusly:

"They fashioned a tomb for thee, O holy and high one—
The Cretans, always liars, evil beasts, idle bellies!
But thou art not dead: thou livest and abidest forever,
For in thee we live and move and have our being."
From the bolded phrase, the question is whether the statement 'Cretans are always liars', when said by a Cretan, is true or false? If assumed true, then he, as a Cretan, is lying, and so his statement is false (because a lie), producing a contradiction. If assumed false, then he or some other Cretan has told the truth at some time, perhaps with this very statement, and there is no contradiction – hence, no paradox. It seems that neither Epimenides nor subsequent commentators regarded this fragment as paradoxical. In the context of the poem it is clear that Epimenides is referring to Cretans other than himself, so there is no self-reference and hence no logical problem. The bolded phrase was quoted by the poet Callimachus (ca. 305 - 240 B.C.E.) in his Hymn to Zeus, and there are references to this fragment in the New Testament. [Paul in Titus 1:12-13 : "12 One of themselves, even a prophet of their own, said, 'The Cretans are always liars, evil beasts, slow bellies.' 13 This witness is true." KJV. Paul does not appear to consider it paradoxical but instead believes it to be true.] None of the subsequent classical or medieval discussions of the Liar make reference to Epimenides, Callimachus, or Titus. An oblique reference to Epimenides appears in "The Logical Calculus", W. E. Johnson, Mind (New Series), vol. 1, no. 2 (April, 1892), pages 235-250, and the 'Epimenides Paradox' is explicitly so named in "Mathematical Logic as Based on the Theory of Types", Bertrand Russell, American Journal of Mathematics, vol. 30, no. 3 (July, 1908), pages 222-262. The association of Epimenides and the paradox seems of fairly recent origin.

The discovery of the Lair Paradox is generally attributed to Eubulides of Miletus, a Greek philosopher circa the 4th Century B.C.E. and student of Euclid of Megara, founder of the Megarian school of philosophy, based on a remark by Diogenes Laertius (c. 200 C.E.) in Lives of the Philosophers II.108. Eubulides reportedly asked: "A man says that he is lying. Is what he says true or false?". If assumed true, then the man is lying and therefore his statement is false (because a lie). If assumed false, then the man is not lying and his statement is therefore true. In either case a contradiction results, so this is truly a paradoxical statement, and accounts for the name. However Diogenes merely stated that Eubulides discussed the Liar, not that he discovered it. Aristotle does not mention it. Whenever and wherever its origin, the Liar Paradox was widely known in later antiquity. The Stoic logician Chrysippus (c. 279-206 B.C.E.), whose works are lost, wrote no less than 14 books on the Liar. Seneca (ca. 4 B.C.E. – 65 C.E.) in Epistle 45.10 mentioned the Liar Paradox. St. Augustine (354 - 430 C.E.) probably had the Liar in mind in Against the Academicians (III.13.29) when he referred to "… the most lying calumny, 'if it is true [it is] false, if it is false it is true'." Aulus Gellius's (c. 125 - after 180 C.E.) Attic Nights (XVIII.ii.10) and Cicero's (c.106 B.C.E. – 43 B.C.E.) Academica priora, II.xxix.95-xxx.97 both discuss it.

This paradox was widely discussed by medieval philosophers, who considered it but one of a number of the insolubilia (insolubles). The two most prominent schools of thought on the Liar were that it was a variant of an Aristotelian fallacy secundum quid et simpliciter [i.e., true in some limited respect but false on the whole], or that this paradox made necessary certain rules to restrict the possibility of self-reference, though there was no agreement on the set of rules [the restrictors]. In the Renaissance the paradox began to take its modern form, reformulated by eliminating any reference to 'lying', viz., the statement: "This statement is false." or explicitly labeling the statement:

X :    "The statement 'X' is false."(1)

It is not sufficient to restrict such self-reference to eliminate the Liar paradox, for it can also be reproduced by a series of non-self-referential statements such as:
X :    "The statement 'Y' is true."(2a)
Y :    "The statement 'X' is false."(2b)

These are perfectly reasonable statements, discussing the truth or falsity of other statements, of a type which we must make continuously in logic, but certain combinations or loops result in the equivalent of the Liar.

The modern resolution of the Liar came in the first part of the 20th Century, a form of restriction due to Tarski. Tarski "proved" that there cannot be a general 'truth' predicate in the logic itself, hence the concept of truth must lie outside the logic -- truth is a meta-concept, so that statements involving the truth of other statements are then meta-statements, on a higher level than the statement itself and discussable only in a meta-language. Separating logical propositions and statements involving their truth (or falsity) unto different levels prohibits any formulation of the Liar, avoiding the paradox. Unfortunately, this separation also results in a vicious infinite regress in truth value; for suppose there is some statement, P, assumed to be at level 0 and it is asserted that:

P1 :    "The statement 'P' is true."(3a)

(where P1 labels the meta-statement on the next level, i.e. level 1), specifying a relation between P and 'truth' in the meta-language. But to determine if P is true, we must determine if meta-statement P1 is true, that is, we must evaluate the meta-meta-statement (level 2 meta-statement):
P2 :    "The statement 'P1' is true."(3b)

To determine its truth, we must evaluate the meta-meta-meta-statement (level 3 meta-statement):
P3 :    "The statement 'P2' is true."(3c)

and so on, ad infinitum, ad nauseum, ad absurdum. Either truth is completely abandoned because it becomes infinitely remote, or we arbitrarily stop at some level, typically the very first. To avoid this infinite regress Tarski introduced his material criterion for the coordination of truth values, viz.:
" 'P' is true." if and only if P.    i.e.,    " 'P' is true." ≡ P(4a)
" 'P' is false." ≡ " '¬P' is true." ≡ ¬P(4b)

Note that both (4a) and (4b) violate the rule for separation of levels since the left side of the equivalence is a level 1 meta-statement whilst the right side is a normal, level 0 proposition. The material criterion dictates that all the meta-statements (3a et. al.), on different levels, have precisely the same logical value, collapsing the regress by violating the very rule of level separation which makes it necessary to begin with. Tarski's meta-level solution to the Liar Paradox has a distinctly ad hoc flavor: mixing levels is prohibited except when it is necessary as an application of the material criterion. This license might perhaps be acceptable if it truly resolved the Liar Paradox in all its forms, but it does not. Unfortunately Tarski's meta-separation fails in the case of multiple statements like (2a) and (2b) because a confusion of levels results. Statement X, asserting the truth of Y, must be on a higher level than Y, while Y, asserting the falsity of X, must be on a higher level than X, so there is no way to consistently assign levels to the two statements. The level of each statement is undeterminable, and the concept of meta-levels is unsatisfactory for this formulation of the Liar.

In an analysis in the complex logic, begin by restricting the values of distinction symbols to the formal constants {U, U~, U~V, V}, interpreted as true, false, undecidable, and imaginary respectively. This restriction of values reduces the complex algebra to a complex propositional logic. The statement " 'X' is true." can be represented by the expression [X = U] and " 'X' is false." by [X = U~], so the Liar paradox can be represented by the implicit definition X = [X = U~]. By a consequence (C10.5), [X = U~] = X~ (compare this result with (4b)), resulting in the reduced equation X = X~, having the solution set X = {U~V, V} as can be verified by substitution. It is also straightforward to demonstrate there are no other possible solutions. There is no contradiction, hence no paradox

So far, this appears to follow Bochvar's approach, with his 'paradoxical' logical value denotated by either U~V or V, to which Haack [1] raises the 'Strengthened Liar' paradox:

P :    'P' is false or paradoxical.(5a)

If the statement is true, then it is false or paradoxical, while if it is false or paradoxical, then it is true – a contradiction in either case, seeming to revive the paradox. This may be variously interpreted as one of the following statements:
P = [P = U~V](5b)
P = [P = U~][P = U~V](5c)
P = [P = U~][P = V](5d)

The first two (5b & 5c) both have the solution P = V, while (5d) has the solution P = U~V; thus the 'Strengthened Liar' paradox does not present any problem under any of the possible interpretations of statement (5a). Extending the spirit of Haack's criticism (as we must, to be thorough), consider the statement:
P :    'P' is imaginary.    or    P = [P = V](6a)

It could at this point be argued that (6a) is prima facia self-contradictory, since it is manifestly non-imaginary, being represented by real symbol tokens. Continuing the analysis however, by Theorem 4.1 any solution must have the form P = V or P = R1(R2V)~. Substituting:
[V = [V = V]] = [V = U] = V(6b)
[R1(R2V)~ = [R1(R2V)~ = V]] = [R1(R2V)~ = V] = V(6c)

There is no solution to (6a), seeming to revive Haack's criticism. However, this statement is not paradoxical, it simply has no solution which would make it true. A more thorough analysis is necessary to explain why this statement alone has no solution.

Begin by distinguishing between the logical value of a statement, P, and the logical value of its implicit definition, [P = [P = …]] – they are not necessarily the same. The self-reference introduces an additional relation between the statement and its asserted value. For (1), the value of the statement is P = {U~V, V}, while the value of its definition is [P = [P = U~]] = U for either solution. Similarly, for (5b-5d), the value of the statement is P = U~V or P = V, while the value of each of the definitions is U. We assume a definition true, solve for the value of the statement, and the existence of a solution justifies our initial assumption, rendering the definition true for those values. Consider the analogous but seldom mentioned case involving the other primitive formal constant, U:

P :    'P' is true.    or    P = [P = U](7)

This definition is true for any value of P, so the value of the statement is unrestricted and undetermined. Statement (7) is true as a definition, though the value of P is undetermined because any value is possible; statement (6a) is imaginary as a definition (see 6b & 6c above), and the value of P is undeterminable because no value is possible. We may regard both statements as definitions having the claimed value, i.e.:
[[P = [P = U]] = U] = U(8a)
[[P = [P = V]] = V] = U(8b)

but elsewise of no effect since they do not determine a value for the statement P. The two primitive formal constants have special status in the formal system, and this status is reflected in the symmetry of these self-referential statements and the fact that they both can be interpreted in the same manner.

The simple paradoxes can thus be regarded quite differently in the complex logic: because of the larger set of formal constants available, they are no longer paradoxical but instead have solutions which may be consistently interpreted as 'undecidable' or 'imaginary'. Because representation is possible directly in the logic, there is no need for meta-levels to avoid paradox. Because all definitions have solutions (except (6a)), there is no need to restrict self-reference, and the formalism is equally effective in dealing with multiple statements lacking self-reference. The fact that multiple solutions are possible in some cases simply means that these implicit definitions are incomplete, i.e., insufficient to completely determine a value. This viewpoint also clarifies some of the medieval conceptions and misconceptions. The secundum quid et simpliciter criticism is seen to have some basis, but in exactly the reverse sense from its medieval application: the statements are true (or imaginary in the sole case of (6a)), absolutely, as definitions, but not in certain respects, i.e., the value of the statements themselves. The notion that a statement can somehow be both true and false perhaps reflected a misunderstood and ill-expressed distinction between the value of an implicit definition and the value of the statement implicitly defined. A general criterion is provided for the restrictors, i.e., a non-real solution, but the need for any restriction is simultaneously removed. The Theory of Cassation is likewise given a formal criterion, that a statement is meaningless if its solution is 'imaginary' (because it then has no possible real import), but there is no need to classify all statements as meaningless to avoid paradox.

[1]    Haack, Susan, Philosophy of Logics, p. 140.

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