By Dr. Peter C. Lugten
Prepared for the 11th International Zoom Conference on the Philosophy of Sir Karl Popper.
Solving the Epimenides Paradox
Epimenides, a Cretan, was famed for the paradox he created by saying “All Cretans are liars”. It is easily resolved by abandoning the absolutist “all”, allowing us to attribute untruthfulness to only some Cretans. Now, Epimenides can make his false claim without paradox. The Inversion Theory of Truth maintains that certainty, completeness and self-consistency are traits found only in the objective state of the world, and that subjective experience results from an evolved biological mechanism that improves upon objectivity, by introducing an essential degree of falsehood. The same applies to statements that we make, excepting introspective claims about our feelings.
A key distinction of the Inversion Theory of Truth is the division of falsehood into “not yet disproven” statements, which contain essential falsehood, and “already disproven” statements, which contain superessential falsehood, colloqually, mistakes or lies. The essential falsehood is what makes not yet disproven statements interesting, compared to tautologies and definitions. Once disproven, they still have knowledge content, but they are no longer “best knowledge”. Consider this version of the Epimenides paradox:
- The next sentence is false.
- The previous sentence is true.
It can be resolved by foregoing the absolutist assumption that what is not false is true.
Instead, we will assume that what is not false is knowledge, defined as “meaningful subjective ideas which make falsifiable predictions and which have not yet been disproved, but which are uncertain”. We may also deny absolute certainty to allegations of falsehood, or disproof, since these, too, may be mistaken. Reconstructing the paradox to accord with the properties of knowledge, we begin:
l) “The next sentence has been disproved, though not with certainty.”
2) “The previous sentence has not been disproved, but is uncertain.”
What this couplet is saying about itself is that it is uncertain whether or not it has been disproved, and hence, that it is undecided. This, unlike being simultaneously true and false, is logically permissible.
With this couplet, we have jettisoned the conventional concept of truth, which gave an inconsistent answer, and relied instead on our concept of knowledge. In so doing, we have gained both consistency and the result, already put forward, that knowledge is uncertain.
Alfred Tarski, in Logic, Semantics and Metamathematics, Chapter VIII, the Concept of Truth in Formalized Languages, considers the following couplet:
l) The next sentence is c.
2) C is not a true sentence.
He infers, by substitution, that “C is true if, and only if, C is not a true sentence,” creating paradox. This he could have avoided by foregoing the absolutist view that “not true” means “false”. A thing “not true” can still mean “best knowledge”. “Not true” can mean either “not disproven yet”, or “already disproven.” Reconstructing the couplet and its inference, we conclude that “C is best knowledge if, and only if, C is uncertain knowledge.” Far from being paradoxical, this is in full agreement with the conclusions presented so far.
Nevertheless, Tarski persisted with the concept of truth, attempting to produce a definition consistent with logical usage. Borrowing mathematical symbols from an earlier, unsatisfactory attempt to eliminate the Epimenides paradox (in Whitehead and Russell’s “Principia Mathematica”), he created formal languages within languages: these were called “metalanguages”. Ideas could be represented within a Formal language by strictly formulated strings of (mathematical) symbols, and these could be manipulated by means of strictly defined logical operations to produce logically valid consequences of inferences. By translating the symbols into those of a symbolically richer (but fully inclusive) metalanguage, sentences of metalanguage referring to themselves in their original linguistic formulation could be constructed. Using this hierarchical “language of the calculus of classes”, Tarski hoped to create a formally correct and materially adequate semantic definition of the expression “true sentence”. What he described as a “special peculiarity” of the calculus of classes, with respect to colloquial languages, was responsible for what he considered to be his success. Two such features that I can identify include, firstly, a complete absence in the former of any appeal to empirical evidence, and secondly, a recursive definition of correspondence (or “satisfaction” of a sentential function by a sequence of objects). A recursive definition is one which seems to define a thing in terms of a simpler version of itself. These are problematic.
In the summary concluding Chapter VIII, Tarski attempted to outwit the paradox in the following way (using this particular form of symbolic mathematical language): We can construct a sentence X of an object science, satisfying “it is not true that X is an element of the class of provable statements, if, and only if, ‘p’”, where p represents the whole sentence X. Translated, this means we can construct the sentence: “This statement (X) is unprovable if, and only if, it is true”. Using the language of the calculus of classes, Tarski was able to reduce this to: “X is an element of the class of true statements” (XeTr), “X is not an element of the class of Provable statements” (XePr) and “the negation of X is not an element of the class of Provable statements” (XePr), in order to conclude that X, while actually undecidable, is, at the same time, true.
Tarski continued: since X is true, it is also true in metatheory, the theory constructed in metalanguage. Since its expression in metatheory contains no specific terms of the metatheory, it can revert to the theory as a decidable sentence: hence truth can be defined provided the science is enriched by the introduction of variables of a higher order. But in making this assertion, Tarski serves only to re-introduce Epimenides’ Paradox: for now he has a decidable sentence that nonetheless contains both (XePr) & (XePr) – (meaning that neither X nor its negation are provable), i.e., the sentence is undecidable.
It is left for me to recall Tarski’s earlier warning that “should we succeed in construction in a metalanguage of a correct definition of truth, then the metalanguage would acquire that universal character which was the primary source of the semantical antinomies in colloquial language”.
However, by discarding absolutist assumptions and denying that any true statement can be known, a satisfactory statement can be produced. In particular, note that “X is an element of Disproved Statements”, is included in but not identical to “X is an element of Unprovable Statements” (XePr), nor is it identical to “X is not an element of Provable Statements”, (XePr). Indeed, the only elements of Provable Statements are tautologies, together with definitions and their logical extensions, which form a tautologous network of “if…then” statements contingent upon our accepting the definition. For X not to be an element of Provable Statements, it may either be a self-contradictory or an empirical observation, and if the latter, it is not necessarily an element of Disproved Statements. Also, we have seen that disproved statements can be falsely disproved, hence disproof isn’t certain. The statement:
“This statement is false if, and only if, it is true”, must therefore be rephrased:
“This statement has been uncertainly disproved if, and only if, it hasn’t yet been disproved”.
This is equivalent to saying: “This statement asserts about itself that it can’t be decided certainly (disproven) until it is disproven”; in other words, our final decision in this case must be that it is undecided.
Likewise, consider the statement:
“This statement is the negation of itself”, or, “This statement is false.”
We can say of the statement:
1) If false, it is still knowledge (since its predicate informs us of its premise)
2) If knowledge, it is tentatively disproved (its predicate denies its own premise) and
3) If tentatively disproved, it is not yet disproved (tentatively denying its self-denial)
Again, we have determined that the statement cannot be decided certainly, and must remain undecided.
This sort of result can be expected from the analysis of linguistic form conducted by Douglas Hofstadter, in his “Godel, Escher, Bach: An Eternal Golden Braid( Basic, 1980, chap 7). He defined two terms:
1) Syntax: predictably terminating tests to show if a statement is well-formed, and
2) Semantics: open ended tests approaching but never reaching meaning.
He noted that the act of interpreting statements involves establishing the implications of all its connections to other formed statements: an object’s meaning is not localized within itself. Aspects of its meaning will be hidden arbitrarily long. Thus Epimenides Paradox can be true if, and only if, it is false. (Hofstadler uses “true” and “false” where I would use “decided” and “undecided”). In the mental processing of symbols by which the brain classifies statements as true or false, the act of classification seems to disrupt the processing of the symbols. Hofstadter suggests that the brain can never provide a fully accurate representation for the (classical) notion of truth; that it would require physically incompatible events to occur within the circuitry of the brain.
Rejecting the Law of the Excluded Middle for the Excluded Affirmation
The absolutist assumption which has to be dropped to allow the resolution of the various versions of Epimenides’ Paradox has a name within the study of logic. It is called the Law of the Excluded Middle, which asserts that every statement is either true or false. This is one of two logical suppositions that Tarski maintained were essential inclusions within the language of calculus of classes. Falsehood is treated by Tarski as being the mutual opposite of Truth, presumably irrespective of whether the nature of any given statement’s status is known to humankind or not. A definition of falsehood as that which though subjectively experienced does not exist does not alone guarantee mutual opposition to truth. For instance, a falsehood that does not objectively exist can no more be paired with its opposite than can be the concept of nothingness. Additionally, because lies refer to conditions that don’t exist, or objective nothingness, attempts to win corroborative support for subjective, superessential falsehoods eventually fail, and it is this that provides the basis for Popper’s theory of falsifiability.
The utility of Karl Popper’s Principle of Falsifiability depends on knowledge of falsehood and not falsehood per se. Thus for practical purposes we may treat subjective falsehood as equivalent to “known” falsehood, or the already disproven, with anything else being “possibly false”. But even if the concept of “false” is expanded beyond that of “already disproven”, to include all statements which will ever in the infinite future prove to be false, one cannot currently label Falsehood to be the mutual opposite of truth. This is because two types of falsehood – the known false and the unknown false- cannot be balanced in opposition to a single kind of truth. Overriding this fundamental asymmetry of status creates the type of error-in-logic formally known as a disjunctive syllogism, whereby one proclaims the truth of one of a number of theories on the basis of the falsehood of another. In this case, we would be ignoring the unknowable percentage of statements that will remain undecidable even when resolved to within infinite limits.
The principle of subjective uncertainty would suggest 3 possibilities for any proposition:
l) that it is considered disproven
2) that it is not yet considered disproven
3) that it is undecidable
This could be called the Law of the Excluded Affirmation.
This technique works for any self-referential statement where there is an asymmetry between the two sides of the paradox. For instance, in the original version of the Epimenides paradox, the symmetry of “All Cretans are liars” is broken by denying that liars have to lie all the time. In 1902, Bertrand Russell was stumped by his discovery of the following paradox: “Let R be the set of all sets that are not members of themselves”. Sets that aren’t members of themselves include the alphabet (which isn’t a letter), the set of primes under a hundred (not itself a prime number) and humanity (not itself a person). If R is a member of itself, then, by definition, it must not be a member of itself. If R is not a member of itself, then it must be a member of itself. This superficially resembles Groucho Marx’s quip that he would never join a club that would allow him as a member. In the case of Russell’s Paradox, we have to resolve the contradiction that
“R is a member of itself”,
“R is not a member of itself”
can both be read as true. The Law of Excluded Affirmation intervenes to resolve this contradiction by determining that it cannot be decided whether R is a member of itself. As for Groucho, however, his case is best resolved simply by not applying to join any clubs.
Saul Kripke, in his “Outline of a Theory of Truth”(1975) offered a solution to the paradox that contrasted with Tarski’s hierarchy of metalanguages. It allowed the predicate “is true” to be used of statements in the base language (type-free semantics) that built to a fixed point of stabilized definitions by adding truth-definitions in stages. Crucially, he allowed “partial”, or “three-valued” logic instead of Tarski’s true-false dichotomy. He assigned “truth-value gaps” rather arbitrarily to paradoxical sentences, to create sentences either “grounded”, with definite truth (true/ false) values, or “ungrounded”, that fall into a truth-value gap. This left disjunctive couplets as being true, false, or, otherwise, undefined. In effect, he tried to replace the Law of the Excluded Middle with the Law of Excluded Affirmation, to give undecidable statements, while claiming true and false statements to be “grounded”, presumably on the strength of the flawed Correspondence theory of truth or its alethic alternatives.
Contemporary mathematicians recognise the problem posed by Russell’s Paradox, and consider it solved without rejection of the Law of the Excluded Middle. Instead, as theoretical physicist Manon Bischoff has explained, they define sets so self-contradictions aren’t allowed: “Sets must be composed of already existing sets and must not refer to themselves” (“Three of the Strangest Paradoxes in Mathematics” www.scientificamerican.com, 16 April, 2024). She acknowledges Kurt Gödel’s warning that unresolvable contradictions are inevitable, and presents a solution: “The best we can do is hope that an unsolvable contradiction never arises”.
The Law of Logical Consequence
The other law critical to Tarski’s project was his Law of Logical Consequences: sentence A is a logical consequence of a set of sentences B iff it is impossible for any sentences in B to be false while A is true. Ensuring a true conclusion where all premises are true, it was infected with Tarski’s concept of provability, as the following illustration from the language of the calculus of classes demonstrates (op. cit., Chapter XVI, On the Consequence of Logical Consequences). Imagine the sequence:
Ao. “O possesses the property p”
Ai. “1 possesses the property p”
An. “n possesses the property p”
Correctly, he points out that the sequence does not prove:
A. “every natural number possesses the property p”.
This is because An+1 may not possess property p.
Therefore, Tarski proposed sentence “B”. B states that Ao, Ai,…, An are provable, not that they have been proved. He reasoned that if B is proved, then A is proved. He then said that if sentence B is replaced by sentence B’, the arithmetical interpretation of B, then a rule is formulated which doesn’t deviate from the rules of inference with respect to
l) conditions of applicability
2) the nature of conceptions involved in its formulation, and
3) its “intuitive infallibility”
Tarski said that the rule B’ is equivalent to an extension of the concept “a sentence proved on the basis of rules hitherto used”, which, thus extended, can be extended ad infinitum. But Tarski was wrong. By asserting that “Ao, Ai,…,An” are provable, one is asserting that they may be disproved. Therefore, all one has to do is falsify the next term in the term sequence, An+1, (by means of the discovery that “n+1 does not possess the property p”) in order to negate the notion that “if B is proved, then A is proved”. No conclusions, in fact, can be ‘true’ because no premises can be ‘true’. As statements, they all admit a degree of essential falsehood, self-contradiction or incompleteness.
Conclusion
To conclude this discussion of Truth, I bear in mind another quotation from Tarski: “Whoever wishes to pursue the semantics of colloquial language with the help of exact methods will be driven first to undertake the thankless task of a reform of this language”. I would like to claim the presumption that by reforming Tarski’s conception of truth with a definition of knowledge that we cannot know “all”, we can eliminate glaring and unresolved inconsistencies in favor of a theoretically desirable uncertainty.
In “The Logic of Scientific Discovery”(Routledge, London, 1954, p.273), Popper footnotes that Tarski’s idea of truth is that which corresponds to the facts. But, he observed, “Once we realize that this correspondence cannot be one of structural similarity, the task of elucidating the correspondence seems hopeless; and as a consequence, we may become suspicious of the whole concept of truth, and prefer not to use it. Tarski solved (with respect to formalized languages) this apparently hopeless problem by making use of a semantic metalanguage, reducing the idea of correspondence to that of ‘satisfaction’ or ‘fulfilment’.” He continued:
“As a result of Tarski’s teaching, I no longer hesitate to speak of ‘truth’ and ‘falsity’. And like everybody else’s views (unless he is a pragmatist), my views turned out, as a matter of course, to be consistent with Tarski’s theory of absolute truth.”
As a result of Tarski’s teaching, I’m afraid, Popper was unable to develop his principle of falsifiability as far as he might have, and was stuck with an absolute truth that he knew to be unattainable, a Correspondence Theory he knew could never exactly correspond to the truth, and a susceptibility to being deceived by the absolutist “all”.
Scholarship of Fools Philosophy 