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CBD.GOEDEL AND LIAR'S PARADOX
Liar's "Paradox" has been formulated in Greece about 600 BC so it's reasonable to push it first through the filter of Aristotelian Logic. As we shall see below, it's no paradox at all, but simple error in negating a proposition. Aristoteles would get rid of it in under a minute if he considered it worth bothering. Let's recall some elementary concepts of Aristotelian Logic. Four types of propositions A, E, I, O: A: a universal affirmative proposition All S are P E: a universal negative proposition No S is P I: a particular affirmative proposition Some S is P O: a particular negative proposition Some S is not P SYLLOGISM: a logical deductive structure consisting of three propositions: First and Second Premises and Conclusion. NOTE: Strictly speaking Syllogisms were not "logical operations", but a genial anticipation of naive Set-Theoretical operations as they use Universal and Existential quantifiers and anticipated Contain operator "is/are". Depending on proposition types one may distinguish: AAA or Barbara (see below) for instance: P1: All humans are mortal (A) P2: All Greeks are humans (A) P3: All Greeks are mortal (A) EAE or Celarent (see below) for instance: P1; No human is immortal (E) P2: All Greeks are humans (A) P3: No Greek is immortal (E) etc. For mnemotechnic reasons all Syllogism types ordered in four groups have been enumerated in form of hexameter: 1. Barbara, Celarent, Darii, Ferioque, prioris: 2. Cesare, Camestres, Festino, Baroco, secundae: 3. Tertia, Darapti, Disamis, Datisi, Felapton, Bocardo, Ferison, habet: 4. Quarta insuper addit Bramantip, Camenes, Dimaris, Fesapo, Fresison. With this light baggage we may easily find that Liar is no paradox, but a simple deduction error. In syllogistic form the Liar becomes: Let P1="All my_statements are false" (A) giving not_P1: "None of my_statements is false" (E) (under the ASSUMPTION: not_A --> E) Then starting with assuming P1 to be true: First loop (Barbara): P1: (All my_statements are false) is true (A) P2: P1 is my_statement (A) Thus P3: P1 is false (A) (thus not_P1 is true (E)) Second loop (Celarent): not_P1: (None of my_statements is false) is true (E) P2: P1 is my_statement (A) thus P3': P1 is not false (E) thus P1 is true (A) and etc. da capo al fine. We negated First Premise of Barbara getting Celarent, returned to Barbara, etc. However, our ASSUMPTION was wrong: negation of A results not in E but in I: not_A --> I. Indeed, "All cars are red" (A) negated does not give "No car is red" (E) but "Some cars are not-red" (I) In Liar's case not_P1 (not(All my_statements are false)) not_(A) results in "Some of my_statements are true" (I) implying that some are false or: "Any of my_statements is true or false". or "Any of my_statements is logically indeterminate" The proper deduction becomes: First loop: P1: All my statements are false (A) P2: P1 is my_statement (A) Thus P3: P1 is false (A) Second loop: not_P1: Some of my statements are true (or false) (I) P2: P1 is my_statement (A) thus P3': P1 is true or false i.e. indeterminate. (I) So, by proper negation of Barbara's First Premise we don't get Celarent, but Disamis which says that we cannot conclude anything about P1. In other terms we don't know anything about P1 and "Liar" does not loop infinitely but peters out after the second correct loop.
Liar and Contemporary Established Logic
In CBA.LOST PARADISE we have said that 19th Century Logic was dominated by the Predicate Logic (PL) and shown that PL is a noumenal, ill-founded, naive instance of Naive View. Russell's Paradox shuttered PL with its feeble foundations, but Logicians did not dare to abandon the Naive View and tried vainly to save and rebuild the collapsed structure upon its rotten base rather than construct a new edifice upon new rational foundations. As result, the established Logic became an extraordinary proliferation of competitive remedes of PL, all noumenal, ill-founded and meaningless. We have counted about 100 of them. There are also about as many non-PL systems, mainly based upon corrupted Boolean Algebra (CA1.INTRODUCTION TO PROPOSITIONAL CALCULUS) as meaningless and useless as PL remedes, but we shall consider here only the latter, as on the one hand they constitute the mainstream founding the official Set Theory and, on the other hand, they reserve an eminent place for Liar's Paradox. Indeed, after 2000 years of quiet rest at the cemetery of deduction errors it resurrected, donned the old moth eaten dress of Paradox and made a triumphal come-back as a principal hinge of most, if not all PL remedes. We don't intend to discuss all 100+ of them and shall have a short look on two leading and typical ones, those of Russell and Goedel.
Russell
In his Types Theory Russell introduces the notion of first-order, second-order and higher order logics in this way: ...We may define an individual as something destitute of complexity; it is then obviously not a proposition, since propositions are essentially complex. Hence in applying the process of generalization to individuals we run no risk of incurring reflexive fallacies. Elementary propositions together with such as contain only individuals as apparent variables we will call first-order propositions. We can thus form new propositions in which first-order propositions occur as apparent variables. These we will call second-order propositions; these form the third logical type. The super-naive noumenal Ontology underlying these assertions is discussed in CA1.INTRODUCTION TO PROPOSITIONAL CALCULUS . Here we are mainly interested in surprising resurgence of Liar's "Paradox" in the monumental Principia Mathematica often considered as the principal contribution to foundations of Logic and Mathematics: -Thus, for example, if Epimenides asserts "all first-order propositions affirmed by me are false," he asserts a second-order proposition; he may assert this truly, without asserting truly any first-order proposition, and thus no contradiction arises.- If Russell studied Aristoteles, he would have noticed that the "contradiction" disappeared 2000 years ago, reduced to confusion of Disamis with Celarent, which would have saved him from writing thick treaties about it and us from being muddled. But that's not the worst. Liar had many shades, one of them, the Eubulides paradox: "This statement is false, thus it's true, thus it's false", etc. Russell treated it seriously and even went to the trouble of creating its two statement version: "The following statement is true. The preceding statement is false". Recalling it one feels embarrassed like listening to a friend who had one over the eight and laughs heartily at his own silly jokes. Symbolizing "This statement is false" with "R", R is neither false, nor true for the simple reason that it is NO statement at all. By the standards of Russell's own Predicate Logic a statement is a predication, an assignment of a property to a subject. "Truth/Falsity" qualifies the predication itself and not the subject of predication. "S" (All cars are red") is a valid predication or statement which may be true or false and by virtue of observations turns out to be false. Now, R does not assign any property to any subject, thus is not a predication, not a statement at all, nothing that may possibly be true or false. Let's now pass to Goedel.
Goedel
-For any consistent formal theory that proves basic arithmetical truths, it is possible to construct an arithmetical statement that is true but not provable in the theory. That is, any theory capable of expressing elementary arithmetic cannot be both consistent and complete.- Whatever the meaning, if any, of "basic arithmetical truths" may possibly be, we shall recall how Goedel describes his "theory" and the famous "G" (Goedel Sentence) true but not provable in the theory. "Theory" refers to an (infinite) set of statements, some of which are taken as true without proof (these are called axioms), and others (the theorems) are taken as true because they are provable from the axioms. "Provable in the theory" means "derivable from the axioms and primitive notions(?) of the theory, using standard first order logic." A theory is "consistent" if it never proves a contradiction. "It is possible to construct" means that there exists some mechanical(?) procedure which can construct the statement, given the axioms, primitives, and first order logic. The resulting true but unprovable statement is often referred to as "the Goedel sentence" for that theory. In fact, there are infinitely many statements in the theory that share with the Goedel sentence the property of being true but not provable from the theory. "Elementary arithmetic" consists merely(?) of addition and multiplication over the natural numbers. Shalving the rather clear upshot that it's probably the most inane pile of garbage ever dumped in "Logic", let's go to brass tacks, to the famous G: "This sentence is not provable" and to the olympian theorem: "G is true but not provable in the theory": (G cannot be false, because then it would be provable and all provable sentences are true; so it's necessarily true and unprovable). Now, as Russell's "R", Goedel's "G" is a shade of the exhumated Liar's "Paradox" and no "sentence" at all. Why did Goedel play amateur gravedigger and disinter the poor Liar, when he had any number of true and unprovable axioms at hand, will stay for ever a closed book.