On notation for ordinal numbers

Kleene, S. C.

doi:10.2307/2267778

Cited by 570 publications

(276 citation statements)

References 5 publications

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“…In order to see that syntax cannot go all the way up to indefinite extensibility, we can also consider that syntax can take us as far as automata can and, under Church's thesis, this is no farther than the least non Church-Kleene-constructive ordinal  1 CK (Church 1936;Church and Kleene 1936;Kleene 1938). On the contrary, definable ordinals and language extensibility levels go indefinitely beyond: to see this, simply note that we can name  1 CK and this enables us to ride on our diagonalization procedures beyond it.…”

Section: Semantics Quantifiers and The Cardinality Problemmentioning

confidence: 99%

Indefinite Extensibility in Natural Language

Luna¹

2013

Monist

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The Monist's call for papers for this issue ended: "if formalism is true, then it must be possible in principle to mechanize meaning in a conscious thinking and language-using machine; if intentionalism is true, no such project is intelligible". We use the GrellingNelson paradox to show that natural language is indefinitely extensible, which has two important consequences: it cannot be formalized and model theoretic semantics, standard for formal languages, is not suitable for it. We also point out that object-object mapping theories of semantics, the usual account for the possibility of non intentional semantics, doesn't seem able to account for the indefinitely extensible productivity of natural language. Keywords: natural language; formalization; indefinite extensibility; universe of discourse; semantics. Introduction.There exist at least three published papers (Cook 2007(Cook , 2009Schlenker 2010), arguing that natural language is indefinitely extensible in some sense. More concretely, all the three papers start from the Liar paradox and by means of Strengthened and Strengthened Strengthened Liars construct a hierarchy of truth values with as many of them as there are ordinal numbers. Both authors renounce the principle of Bivalence in order to solve the Liar and its revenge forms.Bivalence is, however, a fundamental law of classical logic with a considerable intuitive appeal; it is the logical counterpart of the ontological principle that any well-defined situation either obtains or does not obtain. As we see it, trading such a fundamental principle for a solution of the Liar is no bargain. The price could be too high for many people. So, we offer a path to the indefinite extensibility of natural language that is compatible with all principles commonly held as laws of logic. Instead of the Liar, we will use the Grelling-Nelson paradox.We will first address the issue of formalizability, just to show that natural language is not formalizable. Then we will interpret the result in terms of indefinite extensibility as defined in due time. Vagueness.Some readers may contemplate natural language as trivially non formalizable because of its inherent vagueness. We do not wish to go into the discussion whether vagueness can be formalized or not. We just wish to show that, leaving vagueness aside, natural language turns out to be non formalizable on quite another account, namely, indefinite extensibility.So we ask the reader to assume we are dealing with a definite segment of Englishwhich we will call -English-so that all metalinguistic predicates here used are definite and that the corresponding sets, when they exist, are ordinary nonfuzzy sets. The reader may well not believe such a crisp core of English to exist but this should not prevent him from following us: we only purport to show that on the assumption that -English exists we can all the same prove that it cannot be formalized; this will reveal at least that

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Section: Semantics Quantifiers and The Cardinality Problemmentioning

confidence: 99%

Indefinite Extensibility in Natural Language

Luna¹

2013

Monist

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“…We have also not investigated the possible connections between the lim u -computable characteristic functions and the hierarchies of [Ers68]. We have further not considered the possible dependencies of the lim u -computable functions or of the classes Lim u Mex a b on the choice of notation system [Kle38,Rog67].…”

Section: Constructive Ordinal Bounds On Limitsmentioning

confidence: 99%

Not-so-nearly-minimal-size program inference (preliminary report)

Case

Suraj

Jain

1995

Algorithmic Learning for Knowledge-Based Systems

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Freivalds defined an acceptable programming system independent criterion for learning programs for functions in which the final programs were required to be both correct and "nearly" minimal size, i.e, within a computable function of being purely minimal size. Kinber showed that this parsimony requirement on final programs severely limits learning power. Nonetheless, in, for example, scientific inference, parsimony is considered highly desirable. A lim-computable function is (by definition) one computable by a procedure allowed to change its mind finitely many times about its output. Investigated is the possibility of assuaging somewhat the limitation on learning power resulting from requiring parsimonious final programs by use of criteria which require the final, correct programs to be "not-sonearly" minimal size, e.g., to be within a lim-computable function of actual minimal size. It is interestingly shown that some parsimony in the final program is thereby retained, yet learning power strictly increases. Also considered are lim-computable functions as above but for which notations for constructive ordinals are used to bound the number of mind changes allowed regarding the output. This is a variant of an idea introduced by Freivalds and Smith. For this ordinal complexity bounded version of lim-computability, the power of the resultant learning criteria form strict infinite hierarchies intermediate between the computable and the lim-computable cases. Many open questions are also presented.

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“…First, the identity function (ϕ) = ϕ conservatively maps the Logic of Paradox (LP ) [14] into classical propositional logic, because LP has exactly the same valid formulas as classical propositional logic. Second, the function (ϕ) = p ∧ ¬p (where p is some atomic formula) conservatively maps Strong Three-valued Logic (K 3 ) [10,11] into classical propositional logic, because K 3 does not have any valid formulas. For our present purposes we therefore need a notion of translation that makes sharper distinctions.…”

Section: Introductionmentioning

confidence: 99%

Three-valued Logics in Modal Logic

Kooi

Tamminga

2012

Stud Logica

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Abstract.Every truth-functional three-valued propositional logic can be conservatively translated into the modal logic S5. We prove this claim constructively in two steps. First, we define a Translation Manual that converts any propositional formula of any threevalued logic into a modal formula. Second, we show that for every S5-model there is an equivalent three-valued valuation and vice versa. In general, our Translation Manual gives rise to translations that are exponentially longer than their originals. This fact raises the question whether there are three-valued logics for which there is a shorter translation into S5. The answer is affirmative: we present an elegant linear translation of the Logic of Paradox and of Strong Three-valued Logic into S5.

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On notation for ordinal numbers

Cited by 570 publications

References 5 publications

Indefinite Extensibility in Natural Language

Indefinite Extensibility in Natural Language

Not-so-nearly-minimal-size program inference (preliminary report)

Three-valued Logics in Modal Logic

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