scholarly journals A GRAPH-THEORETIC ANALYSIS OF THE SEMANTIC PARADOXES

2017 ◽  
Vol 23 (4) ◽  
pp. 442-492 ◽  
Author(s):  
TIMO BERINGER ◽  
THOMAS SCHINDLER
Keyword(s):  
Truth Predicate ◽  
Theoretic Analysis ◽  
Semantic Paradoxes ◽  
Truth Values ◽  
First Order ◽  
Graph Theoretic ◽  
Fine Grained ◽  
Stage Process ◽  
Order Arithmetic ◽  
The Liar

AbstractWe introduce a framework for a graph-theoretic analysis of the semantic paradoxes. Similar frameworks have been recently developed for infinitary propositional languages by Cook [5, 6] and Rabern, Rabern, and Macauley [16]. Our focus, however, will be on the language of first-order arithmetic augmented with a primitive truth predicate. Using Leitgeb’s [14] notion of semantic dependence, we assign reference graphs (rfgs) to the sentences of this language and define a notion of paradoxicality in terms of acceptable decorations of rfgs with truth values. It is shown that this notion of paradoxicality coincides with that of Kripke [13]. In order to track down the structural components of an rfg that are responsible for paradoxicality, we show that any decoration can be obtained in a three-stage process: first, the rfg is unfolded into a tree, second, the tree is decorated with truth values (yielding a dependence tree in the sense of Yablo [21]), and third, the decorated tree is re-collapsed onto the rfg. We show that paradoxicality enters the picture only at stage three. Due to this we can isolate two basic patterns necessary for paradoxicality. Moreover, we conjecture a solution to the characterization problem for dangerous rfgs that amounts to the claim that basically the Liar- and the Yablo graph are the only paradoxical rfgs. Furthermore, we develop signed rfgs that allow us to distinguish between ‘positive’ and ‘negative’ reference and obtain more fine-grained versions of our results for unsigned rfgs.

scholarly journals INCOMPLETENESS VIA PARADOX AND COMPLETENESS

2019 ◽  
Vol 13 (3) ◽  
pp. 541-592
Author(s):  
WALTER DEAN
Keyword(s):  
Set Theory ◽  
Completeness Theorem ◽  
Axiomatic Theories Of Truth ◽  
Semantic Paradoxes ◽  
First Order ◽  
Incompleteness Theorems ◽  
Order Arithmetic ◽  
The Relationship ◽  
The Liar ◽  
Central Tool

AbstractThis paper explores the relationship borne by the traditional paradoxes of set theory and semantics to formal incompleteness phenomena. A central tool is the application of the Arithmetized Completeness Theorem to systems of second-order arithmetic and set theory in which various “paradoxical notions” for first-order languages can be formalized. I will first discuss the setting in which this result was originally presented by Hilbert & Bernays (1939) and also how it was later adapted by Kreisel (1950) and Wang (1955) in order to obtain formal undecidability results. A generalization of this method will then be presented whereby Russell’s paradox, a variant of Mirimanoff’s paradox, the Liar, and the Grelling–Nelson paradox may be uniformly transformed into incompleteness theorems. Some additional observations are then framed relating these results to the unification of the set theoretic and semantic paradoxes, the intensionality of arithmetization (in the sense of Feferman, 1960), and axiomatic theories of truth.

SUPER LIARS

2010 ◽  
Vol 3 (3) ◽  
pp. 374-414 ◽  
Author(s):  
PHILIPPE SCHLENKER
Keyword(s):  
Fixed Points ◽  
Detailed Analysis ◽  
Expressive Power ◽  
Truth Predicate ◽  
Truth Values ◽  
Kripke’S Theory Of Truth ◽  
Domain Restrictions ◽  
The Liar ◽  
Self Reference ◽  
Linguistic Means

Kripke’s theory of truth offered a trivalent semantics for a language which, like English, contains a truth predicate and means of self-reference; but it did so by severely restricting the expressive power of the logic. In Kripke’s analysis, the Liar (e.g., This very sentence is not true) receives the indeterminate truth value, but this fact cannot be expressed in the language; by contrast, it is straightforward to say in English that the Liar is something other than true. Kripke’s theory also fails to handle the Strengthened Liar, which can be expressed in English as: This very sentence is something other than true. We develop a theory which seeks to overcome these difficulties, and is based on a detailed analysis of some of the linguistic means by which the Strengthened Liar can be expressed in English. In particular, we propose to take literally the quantificational form of the negative expression something other than true. Like other quantifiers, it may have different implicit domain restrictions, which give rise to a variety of negations of different strengths (e.g., something other than true among the values {0, 1}, or among {0, 1, 2}, etc). This analysis naturally leads to a logic with as many truth values as there are ordinals—a conclusion reached independently by Cook (2008a). We develop the theory within a generalization of the Strong Kleene Logic, augmented with negations that each have a nonmonotonic semantics. We show that fixed points can be constructed for our logic, and that it enjoys a limited form of ‘expressive completeness’. Finally, we discuss the relation between our theory and various alternatives, including one in which the word true (rather than negation) is semantically ambiguous, and gives rise to a hierarchy of truth predicates of increasing strength.

scholarly journals Interpreting the compositional truth predicate in models of arithmetic

2021 ◽  
Author(s):  
Cezary Cieśliński
Keyword(s):  
Proof Theory ◽  
Formal Proof ◽  
Truth Predicate ◽  
Saturated Model ◽  
First Order ◽  
Order Arithmetic ◽  
Recursively Saturated ◽  
Recursively Saturated Model ◽  
Models Of Arithmetic

AbstractWe present a construction of a truth class (an interpretation of a compositional truth predicate) in an arbitrary countable recursively saturated model of first-order arithmetic. The construction is fully classical in that it employs nothing more than the classical techniques of formal proof theory.

Yablo’s Paradox: Is the Infinite Liar Lying to Us?

2019 ◽  
Vol 56 (3) ◽  
pp. 88-102
Author(s):  
Andrei V. Nekhaev ◽  
Keyword(s):  
Liar Paradox ◽  
Semantic Content ◽  
Basic Structure ◽  
Truth Predicate ◽  
Semantic Paradoxes ◽  
Yablo’S Paradox ◽  
Truth Values ◽  
Traditional Notion ◽  
Self Reference

In 1993, the American logic S. Yablo was proposed an original infinitive formulation of the classical ≪Liar≫ paradox. It questioned the traditional notion of self-reference as the basic structure for semantic paradoxes. The article considers the arguments underlying two different approaches to analysis of proposals of the ≪Infinite Liar≫ and understanding of the genuine sources for semantic paradoxes. The first approach (V. Valpola, G.-H. von Wright, T. Bolander, etc.) imposes responsibility for the emergence of semantic paradoxes on the negation of the truth predicate. It deprives the ≪Infinite Liar≫ sentences of consistent truth values. The second approach is based on a modified version of anaphoric prosententialism (D. Grover, R. Brandom, etc.). The concepts of truth and falsehood are treated as special anaphoric operators. Logical constructs similar to the ≪Infinite Liar≫ do not attribute any definite truth values to sentences from which they are composed, but only state certain types of relations between the semantic content of such sentences.

Syntactical truth predicates for second order arithmetic

2001 ◽  
Vol 66 (1) ◽  
pp. 225-256 ◽  
Author(s):  
Loïc Colson ◽  
Serge Grigorieff
Keyword(s):  
Order Term ◽  
Second Order ◽  
Truth Predicate ◽  
Standard Interpretation ◽  
Closed Set ◽  
Second Order Arithmetic ◽  
First Order ◽  
Order Arithmetic ◽  
Main Model

AbstractWe introduce a notion of syntactical truth predicate (s.t.p.) for the second order arithmetic PA2. An s.t.p. is a set T of closed formulas such that:(i) T(t = u) if and only if the closed first order terms t and u are convertible, i.e., have the same value in the standard interpretation(ii) T(A → B) if and only if (T(A) ⇒ T(B))(iii) T(∀xA) if and only if (T(A[x ← t]) for any closed first order term t)(iv) T(∀X A) if and only if (T(A[X ← ∆]) for any closed set definition ∆ = {x ∣ D(x)}).S.t.p.'s can be seen as a counterpart to Tarski's notion of (model-theoretical) validity and have main model properties. In particular, their existence is equivalent to the existence of an ω-model of PA2, this fact being provable in PA2 with arithmetical comprehension only.

Relativized realizability in intuitionistic arithmetic of all finite types

1978 ◽  
Vol 43 (1) ◽  
pp. 23-44 ◽  
Author(s):  
Nicolas D. Goodman
Keyword(s):  
Extension Theorem ◽  
General Notion ◽  
Conservative Extension ◽  
First Order ◽  
New Information ◽  
Reflection Theorem ◽  
Order Arithmetic ◽  
Special Case ◽  
Dependent Choice ◽  
Intuitionistic Arithmetic

In this paper we introduce a new notion of realizability for intuitionistic arithmetic in all finite types. The notion seems to us to capture some of the intuition underlying both the recursive realizability of Kjeene [5] and the semantics of Kripke [7]. After some preliminaries of a syntactic and recursion-theoretic character in §1, we motivate and define our notion of realizability in §2. In §3 we prove a soundness theorem, and in §4 we apply that theorem to obtain new information about provability in some extensions of intuitionistic arithmetic in all finite types. In §5 we consider a special case of our general notion and prove a kind of reflection theorem for it. Finally, in §6, we consider a formalized version of our realizability notion and use it to give a new proof of the conservative extension theorem discussed in Goodman and Myhill [4] and proved in our [3]. (Apparently, a form of this result is also proved in Mine [13]. We have not seen this paper, but are relying on [12].) As a corollary, we obtain the following somewhat strengthened result: Let Σ be any extension of first-order intuitionistic arithmetic (HA) formalized in the language of HA. Let Σω be the theory obtained from Σ by adding functionals of finite type with intuitionistic logic, intensional identity, and axioms of choice and dependent choice at all types. Then Σω is a conservative extension of Σ. An interesting example of this theorem is obtained by taking Σ to be classical first-order arithmetic.

scholarly journals The Liar Without Relativism

Erkenntnis ◽  
2021 ◽  
Author(s):  
Poppy Mankowitz
Keyword(s):  
Natural Language ◽  
Recent Literature ◽  
Liar Paradox ◽  
Full Understanding ◽  
Truth Values ◽  
The Liar ◽  
The Liar Paradox ◽  
Semantic Relativism

AbstractSome in the recent literature have claimed that a connection exists between the Liar paradox and semantic relativism: the view that the truth values of certain occurrences of sentences depend on the contexts at which they are assessed. Sagi (Erkenntnis 82(4):913–928, 2017) argues that contextualist accounts of the Liar paradox are committed to relativism, and Rudnicki and Łukowski (Synthese 1–20, 2019) propose a new account that they classify as relativist. I argue that a full understanding of how relativism is conceived within theories of natural language shows that neither of the purported connections can be maintained. There is no reason why a solution to the Liar paradox needs to accept relativism.

Graph-Theoretic Analysis of Power Grid Robustness

2021 ◽  
pp. 175-194
Author(s):  
Dorcas Ofori-Boateng ◽  
Asim Kumer Dey ◽  
R. Gel Yulia ◽  
H. Vincent Poor
Keyword(s):  
Power Grid ◽  
Theoretic Analysis ◽  
Graph Theoretic

scholarly journals Integrative statistical analyses of multiple liquid biopsy analytes in metastatic breast cancer

2021 ◽  
Vol 13 (1) ◽  
Author(s):  
Corinna Keup ◽  
Vinay Suryaprakash ◽  
Siegfried Hauch ◽  
Markus Storbeck ◽  
Peter Hahn ◽  
...  
Keyword(s):  
Breast Cancer ◽  
Overall Survival ◽  
Metastatic Breast Cancer ◽  
Mutual Information ◽  
Hierarchical Clustering ◽  
Liquid Biopsy ◽  
Metastatic Breast ◽  
The Other ◽  
Theoretic Analysis ◽  
Graph Theoretic

Abstract Background Single liquid biopsy analytes (LBAs) have been utilized for therapy selection in metastatic breast cancer (MBC). We performed integrative statistical analyses to examine the clinical relevance of using multiple LBAs: matched circulating tumor cell (CTC) mRNA, CTC genomic DNA (gDNA), extracellular vesicle (EV) mRNA, and cell-free DNA (cfDNA). Methods Blood was drawn from 26 hormone receptor-positive, HER2-negative MBC patients. CTC mRNA and EV mRNA were analyzed using a multi-marker qPCR. Plasma from CTC-depleted blood was utilized for cfDNA isolation. gDNA from CTCs was isolated from mRNA-depleted CTC lysates. CTC gDNA and cfDNA were analyzed by targeted sequencing. Hierarchical clustering was performed within each analyte, and its results were combined into a score termed Evaluation of multiple Liquid biopsy analytes In Metastatic breast cancer patients All from one blood sample (ELIMA.score), which calculates the contribution of each analyte to the overall survival prediction. Singular value decomposition (SVD), mutual information calculation, k-means clustering, and graph-theoretic analysis were conducted to elucidate the dependence between individual analytes. Results A combination of two/three/four LBAs increased the prevalence of patients with actionable signals. Aggregating the results of hierarchical clustering of individual LBAs into the ELIMA.score resulted in a highly significant correlation with overall survival, thereby bolstering evidence for the additive value of using multiple LBAs. Computation of mutual information indicated that none of the LBAs is independent of the others, but the ability of a single LBA to describe the others is rather limited—only CTC gDNA could partially describe the other three LBAs. SVD revealed that the strongest singular vectors originate from all four LBAs, but a majority originated from CTC gDNA. After k-means clustering of patients based on parameters of all four LBAs, the graph-theoretic analysis revealed CTC ERBB2 variants only in patients belonging to one particular cluster. Conclusions The additional benefits of using all four LBAs were objectively demonstrated in this pilot study, which also indicated a relative dominance of CTC gDNA over the other LBAs. Consequently, a multi-parametric liquid biopsy approach deconvolutes the genomic and transcriptomic complexity and should be considered in clinical practice.

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