Model-theoretical semantics, composotionality princeple and truth definition

2015 ◽  
Keyword(s):  
Truth Definition

Certain predicates defined by induction schemata

1953 ◽  
Vol 18 (1) ◽  
pp. 49-59 ◽  
Author(s):  
Hao Wang
Keyword(s):  
Number Theory ◽  
Set Theory ◽  
Formal System ◽  
General Notion ◽  
Constant Number ◽  
Natural Numbers ◽  
System L ◽  
Truth Definition ◽  
Consistency Proofs ◽  
Special Case

It is known that we can introduce in number theory (for example, the system Z of Hilbert-Bernays) by induction schemata certain predicates of natural numbers which cannot be expressed explicitly within the framework of number theory. The question arises how we can define these predicates in some richer system, without employing induction schemata. In this paper a general notion of definability by induction (relative to number theory), which seems to apply to all the known predicates of this kind, is introduced; and it is proved that in a system L1 which forms an extension of number theory all predicates which are definable by induction (hereafter to be abbreviated d.i.) according to the definition are explicitly expressible.In order to define such predicates and prove theorems answering to their induction schemata, we have to allow certain impredicative classes in L1. However, if we want merely to prove that for each constant number the special case of the induction schema for a predicate d.i. is provable, we do not have to assume the existence of impredicative classes. A certain weaker system L2, in which only predicative classes of natural numbers are allowed, is sufficient for the purpose. It is noted that a truth definition for number theory can be obtained in L2. Consistency proofs for number theory do not seem to be formalizable in L2, although they can, it is observed, be formalized in L1.In general, given any ordinary formal system (say Zermelo set theory), it is possible to define by induction schemata, in the same manner as in number theory, certain predicates which are not explicitly definable in the system. Here again, by extending the system in an analogous fashion, these predicates become expressible in the resulting system. The crucial predicate instrumental to obtaining a truth definition for a given system is taken as an example.

Fuzzy truth definition of possibility measure for decision classification

1979 ◽  
Vol 11 (4) ◽  
pp. 447-463 ◽  
Author(s):  
J.F. Baldwin ◽  
B.W. Pilsworth
Keyword(s):  
Possibility Measure ◽  
Truth Definition ◽  
Definition Of

scholarly journals Bounded arithmetic and truth definition

1988 ◽  
Vol 39 (1) ◽  
pp. 75-104 ◽  
Author(s):  
Gaisi Takeuti
Keyword(s):  
Bounded Arithmetic ◽  
Truth Definition

Truth definitions without exponentiation and the Σ1 collection scheme

2012 ◽  
Vol 77 (2) ◽  
pp. 649-655 ◽  
Author(s):  
Zofia Adamowicz ◽  
Leszek Aleksander Kołodziejczyk ◽  
J. Paris
Keyword(s):  
Size Parameter ◽  
Universal Formula ◽  
Truth Definition ◽  
Definable Elements

AbstractWe prove that: • if there is a model of IΔ0 + ¬exp with cofinal Σ1-definable elements and a Σ1 truth definition for Σ1 sentences, then IΔ0 + ¬exp + ¬BΣ1 is consistent,• there is a model of IΔ0 + Ω1 + ¬exp with cofinal Σ1-definable elements, both a Σ2 and a Π2 truth definition for Σ1 sentences, and for each n ≥ 2, a Σn truth definition for Σn sentences.The latter result is obtained by constructing a model with a recursive truth-preserving translation of Σ1 sentences into boolean combinations of sentences.We also present an old but previously unpublished proof of the consistency of IΔ0 + ¬exp + ¬BΣ1 under the assumption that the size parameter in Lessan's Δ0 universal formula is optimal. We then discuss a possible reason why proving the consistency of IΔ0 + ¬exp + ¬BΣ1 unconditionally has turned out to be so difficult.

scholarly journals Naturalist Structuralism's Aporia? Essentialism, Indeterminacy, and Nostalgia – a response to Paul Livingston

Konturen ◽  
2010 ◽  
Vol 2 (1) ◽  
pp. 43
Author(s):  
Samuel C. Wheeler III
Keyword(s):  
External Approach ◽  
Stimulus Meaning ◽  
Indeterminacy Thesis ◽  
Truth Definition

This essay argues that what Livingston calls the “structuralist” project, combined with a naturalistic, external approach to language, does not in fact lead to a paradoxical failure to match lived language. Quine’s indeterminacy argument is not a consequence of naturalism and structuralism, but is rather a consequence of thorough anti-essentialism, a thesis he shares with Derrida and Davidson. Contemporary naturalism is in fact not committed to Quine’s thesis. Davidson’s views are a purification of the views of Quine, removing Quine’s empiricist appeal to stimulus meaning and Quine’s scientism. Davidson abandons the conventionalist conception of language but retains the “structuralist” conception of language, as captured by a truth-definition. The indeterminacy thesis is a consequence of anti-essentialism applied to semantics, that is, the denial of transcendental signifieds. The essay concludes by arguing that Quine’s aporia (which is also Davidson’s and Derrida’s aporia) is a discovery rather than a paradox.

On generalized quantifiers in arithmetic

1982 ◽  
Vol 47 (1) ◽  
pp. 187-190 ◽  
Author(s):  
Carl Morgenstern
Keyword(s):  
Axiom System ◽  
Peano Arithmetic ◽  
Order Logic ◽  
First Order Logic ◽  
Generalized Quantifiers ◽  
First Order ◽  
Order Language ◽  
Truth Definition ◽  
Order Arithmetic ◽  
Infinite Set

In this note we investigate an extension of Peano arithmetic which arises from adjoining generalized quantifiers to first-order logic. Markwald [2] first studied the definability properties of L1, the language of first-order arithmetic, L, with the additional quantifer Ux which denotes “there are infinitely many x such that…. Note that Ux is the same thing as the Keisler quantifier Qx in the ℵ0 interpretation.We consider L2, which is L together with the ℵ0 interpretation of the Magidor-Malitz quantifier Q2xy which denotes “there is an infinite set X such that for distinct x, y ∈ X …”. In [1] Magidor and Malitz presented an axiom system for languages which arise from adding Q2 to a first-order language. They proved that the axioms are valid in every regular interpretation, and, assuming ◊ω1, that the axioms are complete in the ℵ1 interpretation.If we let denote Peano arithmetic in L2 with induction for L2 formulas and the Magidor-Malitz axioms as logical axioms, we show that in we can give a truth definition for first-order Peano arithmetic, . Consequently we can prove in that is Πn sound for every n, thus in we can prove the Paris-Harrington combinatorial principle and the higher-order analogues due to Schlipf.

Truth, the Liar, and Tarskian Truth Definition

2007 ◽  
pp. 164-176
Author(s):  
Greg Ray
Keyword(s):  
Truth Definition ◽  
The Liar

scholarly journals How Tarski Defined the Undefinable

2015 ◽  
Vol 23 (1) ◽  
pp. 139-149
Author(s):  
Cezary Cieśliński
Keyword(s):  
Truth Definition

This paper describes Tarski’s project of rehabilitating the notion of truth, previously considered dubious by many philosophers. The project was realized by providing a formal truth definition, which does not employ any problematic concept.

A note on modal formulae and relational properties

1975 ◽  
Vol 40 (1) ◽  
pp. 55-58 ◽  
Author(s):  
J. F. A. K. van Benthem
Keyword(s):  
Interesting Case ◽  
Order Logic ◽  
Modal Formula ◽  
First Order ◽  
Modal Operators ◽  
Relational Properties ◽  
Truth Definition ◽  
Image Position ◽  
Second Order Logic ◽  
Monadic Second Order Logic

Consider modal propositional formulae, constructed using proposition-letters, connectives and the modal operators □ and ⋄. The semantic structures are frames, i.e., pairs <W, R> with R ⊆ W2. Let F, V be variables ranging respectively over frames and functions from the set of proposition-letters into the powerset of W. Then the relationmay be defined, for arbitrary formulae α, following the Kripke truth-definition. From this relation we may further defineNow, to every modal formula α there corresponds some property Pα of R. A particular example is obtained by considering the well-known translation of modal formulae into formulae of monadic second-order logic with a single binary first-order predicate. For these particular Pα we havefor all F and w ∈ W. These formulae Pα are, however, rather intractable and more convenient ones can often be found. An especially interesting case occurs when Pα may be taken to be some first-order formula. For example, it can be seen thatfor all F and w ∈ W. It is customary to talk about a related correspondence, namely when for all F we haveNote that this correspondence holds whenever the first one above holds.

Sign in / Sign up

Export Citation Format

Share Document