The document discusses formalizing a theory of truth degrees in the framework of axiomatic truth theory PAŁTr2 over Łukasiewicz infinite-valued predicate logic. It defines a degree theoretic ordering ≤ and ordering of truthhood ≺ based on the truth predicate Tr. However, PAŁTr2 provides a counterexample showing that ≤ and ≺ are not identical due to ω-inconsistency, failing the formalized truth degree theory. This means axiomatic and semantic analyses can differ on relating degrees to truth.
1. The document discusses three computational modes of the mind: ordinary thought (conscious classical computation and unconscious quantum computation), and meta-thought (unconscious and non-algorithmic).
2. It proposes a quantum meta-language (QML) and probabilistic identity axiom to describe aspects of human reasoning and the disintegration of the self in conditions like schizophrenia.
3. The document introduces the concept of quantum coherent states of the mind, drawing an analogy to coherent states in quantum field theory which are eigenstates of the annihilation operator.
The document discusses various topics relating to knowledge representation in artificial intelligence, including:
1) Different types of knowledge that need representation including declarative, procedural, commonsense, and scientific knowledge.
2) Ontologies define terminology and objects/relationships in a systematic way to enable knowledge sharing between agents.
3) Semantic networks represent knowledge graphically with nodes for objects/events and arcs for relationships, enabling reasoning through inheritance and matching.
4) Conceptual graphs also represent knowledge graphically as a bipartite graph with concepts and relations, and can represent logical expressions.
T4 Introduction to the modelling and verification of, and reasoning about mul...EASSS 2012
This document provides an overview of a course on modelling, verification, and reasoning in multi-agent systems. The course is divided into 6 lectures covering topics such as linear and branching time, cooperative agents, comparing semantics of alternating-time temporal logic, reasoning with examples, and complexity analysis of verification and reasoning problems. It also lists required background knowledge and recommended reading materials on related logics, automata theory, and specification and verification of multi-agent systems.
Objective Bayes Factors for Informed Hypotheses Presentation SIDIM 2009David Torres
Objective Bayes Factors for Informed Hypotheses: "Completing" The Informed Hypothesis and "Splitting" the Bayes Factors.
Lecture presentation in The Seminario Interuniversitario de Investigación en Ciencias Matemáticas (Interuniversity Seminar on Mathematical Sciences Research, SIDIM) at University of Puerto Rico - Rio Piedras Campus on February 2009. http://sidim2009.uprr.pr/
A General Principle of Learning and its Application for Reconciling Einstein’...Jeffrey Huang
This document proposes a general principle of learning based on discovering intrinsic constraint models. It defines intelligence as the ability to understand the world by discovering intrinsic variables and constraints that have minimum entropy. The key goals of learning are to map observations to intrinsic variables and detect constraints to minimize a model entropy objective function. Discovering intrinsic variables is critical for maximizing prediction accuracy, learning efficiency, and generalization power. The principle provides a theoretical foundation for explaining and developing artificial general intelligence.
1. The document provides definitions and examples of logical connectives like conjunction, disjunction, negation, implication, biconditional, and translations between symbolic logic and English sentences.
2. It discusses translating statements involving conditionals, conjunctions, disjunctions, and negations into symbolic logic using abbreviations for terms.
3. The document also addresses identifying the logical form of statements, such as identifying conditionals, conjunctions, and determining if a statement is a valid sentence of symbolic logic.
The document provides an overview of Truth Maintenance Systems (TMS) in artificial intelligence. It discusses key aspects of TMS including:
1. Enforcing logical relations among beliefs by maintaining and updating relations when assumptions change.
2. Generating explanations for conclusions by using cached inferences to avoid re-deriving inferences.
3. Finding solutions to search problems by representing problems as sets of variables, domains, and constraints.
The document also covers justification-based and assumption-based TMS, and how a TMS interacts with a problem solver to add and retract assumptions, detect contradictions, and perform belief revision.
Machine mediated meaning for semantic interoperability pvn 120109 pdfPutcha Narasimham
1) The document proposes a definition of meaning as "the property of valid expression in Natural Language Text (NLT) along with the context, which is capable of generating “Predefined Responses (PreRes)” from a “Specified Class of Recipients (R)” and the “meaning of NLT for R is PreRes”."
2) It examines existing definitions of meaning from dictionaries and scholars and finds them insufficient for defining meaning that both humans and machines can understand.
3) The proposed definition explicitly includes the recipient (R) and predefined responses to determine the meaning generated in the recipient's mind from the text, in order to have an objective way to assess meaning without further interpretation.
1. The document discusses three computational modes of the mind: ordinary thought (conscious classical computation and unconscious quantum computation), and meta-thought (unconscious and non-algorithmic).
2. It proposes a quantum meta-language (QML) and probabilistic identity axiom to describe aspects of human reasoning and the disintegration of the self in conditions like schizophrenia.
3. The document introduces the concept of quantum coherent states of the mind, drawing an analogy to coherent states in quantum field theory which are eigenstates of the annihilation operator.
The document discusses various topics relating to knowledge representation in artificial intelligence, including:
1) Different types of knowledge that need representation including declarative, procedural, commonsense, and scientific knowledge.
2) Ontologies define terminology and objects/relationships in a systematic way to enable knowledge sharing between agents.
3) Semantic networks represent knowledge graphically with nodes for objects/events and arcs for relationships, enabling reasoning through inheritance and matching.
4) Conceptual graphs also represent knowledge graphically as a bipartite graph with concepts and relations, and can represent logical expressions.
T4 Introduction to the modelling and verification of, and reasoning about mul...EASSS 2012
This document provides an overview of a course on modelling, verification, and reasoning in multi-agent systems. The course is divided into 6 lectures covering topics such as linear and branching time, cooperative agents, comparing semantics of alternating-time temporal logic, reasoning with examples, and complexity analysis of verification and reasoning problems. It also lists required background knowledge and recommended reading materials on related logics, automata theory, and specification and verification of multi-agent systems.
Objective Bayes Factors for Informed Hypotheses Presentation SIDIM 2009David Torres
Objective Bayes Factors for Informed Hypotheses: "Completing" The Informed Hypothesis and "Splitting" the Bayes Factors.
Lecture presentation in The Seminario Interuniversitario de Investigación en Ciencias Matemáticas (Interuniversity Seminar on Mathematical Sciences Research, SIDIM) at University of Puerto Rico - Rio Piedras Campus on February 2009. http://sidim2009.uprr.pr/
A General Principle of Learning and its Application for Reconciling Einstein’...Jeffrey Huang
This document proposes a general principle of learning based on discovering intrinsic constraint models. It defines intelligence as the ability to understand the world by discovering intrinsic variables and constraints that have minimum entropy. The key goals of learning are to map observations to intrinsic variables and detect constraints to minimize a model entropy objective function. Discovering intrinsic variables is critical for maximizing prediction accuracy, learning efficiency, and generalization power. The principle provides a theoretical foundation for explaining and developing artificial general intelligence.
1. The document provides definitions and examples of logical connectives like conjunction, disjunction, negation, implication, biconditional, and translations between symbolic logic and English sentences.
2. It discusses translating statements involving conditionals, conjunctions, disjunctions, and negations into symbolic logic using abbreviations for terms.
3. The document also addresses identifying the logical form of statements, such as identifying conditionals, conjunctions, and determining if a statement is a valid sentence of symbolic logic.
The document provides an overview of Truth Maintenance Systems (TMS) in artificial intelligence. It discusses key aspects of TMS including:
1. Enforcing logical relations among beliefs by maintaining and updating relations when assumptions change.
2. Generating explanations for conclusions by using cached inferences to avoid re-deriving inferences.
3. Finding solutions to search problems by representing problems as sets of variables, domains, and constraints.
The document also covers justification-based and assumption-based TMS, and how a TMS interacts with a problem solver to add and retract assumptions, detect contradictions, and perform belief revision.
Machine mediated meaning for semantic interoperability pvn 120109 pdfPutcha Narasimham
1) The document proposes a definition of meaning as "the property of valid expression in Natural Language Text (NLT) along with the context, which is capable of generating “Predefined Responses (PreRes)” from a “Specified Class of Recipients (R)” and the “meaning of NLT for R is PreRes”."
2) It examines existing definitions of meaning from dictionaries and scholars and finds them insufficient for defining meaning that both humans and machines can understand.
3) The proposed definition explicitly includes the recipient (R) and predefined responses to determine the meaning generated in the recipient's mind from the text, in order to have an objective way to assess meaning without further interpretation.
Unification Of Randomized Anomaly In Deception Detection Using Fuzzy Logic Un...IJORCS
In the recent era of computer electronic communication we are currently facing the critical impact of Deception which plays its vital role in the mode of affecting efficient information sharing system. Identifying Deception in any mode of communication is a tedious process without using the proper tool for detecting those vulnerabilities. This paper deals with the efficient tools of Deception detection in which combined application implementation is our main focus rather than with its individuality. We propose a research model which comprises Fuzzy logic, Uncertainty and Randomization. This paper deals with an experiment which implements the scenario of mixture application with its revealed results. We also discuss the combined approach rather than with its individual performance.
The document proposes the Information Bottleneck Method as a way to extract relevant information from a signal X about another signal Y. It formalizes this as finding a compressed code for X that maximizes the information about Y while minimizing the code length. This forms a bottleneck that preserves only the most relevant information. The method provides self-consistent equations to determine the optimal coding rules from X to the code and from the code to Y. It generalizes rate-distortion theory by using the relationship between X and Y to determine relevance rather than requiring an externally specified distortion function.
Improved Performance of Unsupervised Method by Renovated K-MeansIJASCSE
Clustering is a separation of data into groups of similar objects. Every group called cluster consists of objects that are similar to one another and dissimilar to objects of other groups. In this paper, the K-Means algorithm is implemented by three distance functions and to identify the optimal distance function for clustering methods. The proposed K-Means algorithm is compared with K-Means, Static Weighted K-Means (SWK-Means) and Dynamic Weighted K-Means (DWK-Means) algorithm by using Davis Bouldin index, Execution Time and Iteration count methods. Experimental results show that the proposed K-Means algorithm performed better on Iris and Wine dataset when compared with other three clustering methods.
This document presents a new model of categorization called Categorization by Elimination (CBE). CBE uses as few cues or features as necessary to make an accurate category assignment, unlike most existing models which use all available cues. CBE orders cues by validity and uses them sequentially, eliminating potential categories after each cue is considered until only one category remains. The authors show that CBE performs as well as humans and other algorithms on categorization tasks while using fewer cues, making it a parsimonious psychological model of fast and frugal categorization.
Knuth's Definitions of Data and Information; Proposed Definition of KNOWLEDGE Putcha Narasimham
The words data and information are used without sufficient delineation of HOW, WHERE and WHEN to use them. They are at times used interchangeably and the dictionary meanings which seek to distinguish data and information end up with cyclic references. The use of these terms in computer science and information technology also follow the same colloquial trend with some pseudo-scientific attributions (raw facts are data and processed data is information) that do not pass simple tests of validity or rigor.
At times the word knowledge is used to explain the meanings of data and information, compounding the confusion and not having its own meaning. Donald E Knuth’s definitions of data and information are sufficiently precise and rigorous to be called scientific. Unfortunately Knuth's definitions of "data & information" do NOT turn up in the search results of Google, Yahoo, Bing, WolframAlpha. It seems perhaps the SINGLE AUTHENTIC source is vanishing. One definition of "data" which comes very close is from Dictionary of Military and Associated Terms.
They are discussed and used as foundation to define knowledge.
Knuth’s definition of “information” includes the word “meaning” which itself is a very complex and is wrongly defined in most places (according to me). I have a proposed definition for meaning also…too long). It is NOT essential to bring in the concept of "meaning" in the definition of "information". This is a new addition (09OCT13).
The available definitions of knowledge are examined and contrasted with Knuth’s definition of Data and Information. It is argued that “knowledge” refers to the “ability of a person or entity” “to provide data or information” “in response to a query”. This provides basis for knowledge representation, authoring and processing (separately described).
1. Logical Argument Mapping (LAM) is a method for building common ground through cognitive change using logical argument diagrams. It aims to make implicit assumptions and limitations explicit to promote reflection.
2. LAM uses valid argument schemes as a normative standard, challenging users to represent arguments fully and address objections. This process reveals gaps and drives users to continually improve understanding.
3. For cognitive change to occur, relevant information must be visible while reducing cognitive load. LAM aims to integrate with the World Wide Argument Web to allow sharing of arguments.
Separations of probabilistic theories via their information processing capab...Matthew Leifer
Talk given at the workshop "Operational Quantum Physics and the Quantum Classical Contrast" at Perimeter Institute in December 2007. It focuses on the results of http://arxiv.org/abs/0707.0620, http://arxiv.org/abs/0712.2265 and http://arxiv.org/abs/0805.3553
The talk was recorded and is viewable online at http://pirsa.org/07060033/
Predicting performance in Recommender Systems - PosterAlejandro Bellogin
This document discusses predicting the performance of recommender systems. It proposes that data commonly available to recommender systems could enable estimating the success of recommendations. Specifically, it aims to 1) define a performance prediction theory for recommender systems, 2) adapt query performance techniques from information retrieval to recommendations, and 3) evaluate appropriate performance metrics. The research also explores applying these models when combining multiple recommendation strategies or hybrid recommender systems.
This document describes a framework for providing mediated access to the algebraic topology software Kenzo. The framework aims to increase Kenzo's accessibility for non-programmers by developing a friendly front-end. It presents an architecture where a client communicates with Kenzo through XML messages validated by Common Lisp modules. The modules handle construction, computation and processing requests by checking restrictions before passing tasks to Kenzo. The document discusses making the framework extensible, efficient, adaptable to different clients, and including new functionality like additional mathematical systems. It proposes a graphical user interface implemented in Common Lisp as a distinguished client.
Teaching and learning mathematics at university levelharisv9
This document provides a summary and table of contents for a book that presents dialogues between two characters - M, a mathematician, and RME, a researcher in mathematics education. The dialogues discuss issues in learning and teaching mathematics at the university level, grounded in data from previous studies on this topic. Chapters 3-8 contain the dialogues, focusing on students' mathematical reasoning, expression, key concepts, and pedagogy. Chapter 8 discusses the relationship between mathematicians and mathematics education researchers. Introductory and concluding chapters provide background on the studies, methodology, and production of the book.
Comprehensive Guide to Taxonomy of Future KnowledgeMd Santo
This document provides a comprehensive guide to taxonomy of future knowledge. It discusses evolving models of knowledge from data-information-knowledge to a nature knowledge continuum informed by consciousness. Key points include: 1) Knowledge is considered an emergent property within nature and the universe, differentiated by infinite levels of consciousness. 2) Human knowledge is part of nature knowledge and is produced through human knowing tools of senses, brain and DNA. 3) A new framework called Human System Biology-based Knowledge Management is presented for understanding knowledge as a psycho-somatic entity with consciousness.
This document summarizes a symposium on awareness in computation held at the University of Birmingham. It discusses using modular neural networks (MANN) to model self-awareness and self-representation for artificial agents. The objectives are to analyze the modular structure of consciousness, design a cognitive architecture for self-awareness, self-representation and representing others, implement models in agents using artificial neural networks, and observe agent behavior in interaction scenarios. MANN are proposed as suitable for modeling functions related to consciousness, with self-awareness emerging from a sense of belonging and interactions between neural network modules.
This document provides an outline for a course on category theory from a logician's perspective. It introduces the instructor, Valeria de Paiva, and their background in category theory through their PhD thesis on Dialectica categories. The course will cover categories, functors, natural transformations, adjunctions, deductive systems as categories, and a taste of glue semantics. It emphasizes viewing proofs as first-class objects and using category theory for proof semantics rather than set-based models. The goal is to represent proofs explicitly rather than just knowing if a proof exists. The course will take an intuitionistic and constructive perspective on logic.
This document introduces the concept of refined concept maps (RCM) as a way to make traditional concept maps more rigorous and less ambiguous. RCM focuses on using a finite set of well-defined semantic relation names instead of loose linking words. This allows knowledge to be represented in a more formal and logically consistent way while still being accessible to novices. The document provides examples of converting traditional concept maps to RCM and discusses how RCM could serve as a bridge between informal and formal knowledge representation.
This document provides a suggested list of mathematical language terms for grade 6. It includes terms related to problem solving, reasoning and proof, communication, connections, representation, number sense and operations, algebra, geometry, measurement, and statistics and probability. The list contains over 150 mathematical terms organized into these conceptual categories.
Min-based qualitative possibilistic networks are one of the effective tools for a compact representation of
decision problems under uncertainty. The exact approaches for computing decision based on possibilistic
networks are limited by the size of the possibility distributions. Generally, these approaches are based on
possibilistic propagation algorithms. An important step in the computation of the decision is the
transformation of the DAG (Direct Acyclic Graph) into a secondary structure, known as the junction trees
(JT). This transformation is known to be costly and represents a difficult problem. We propose in this paper
a new approximate approach for the computation of decision under uncertainty within possibilistic
networks. The computing of the optimal optimistic decision no longer goes through the junction tree
construction step. Instead, it is performed by calculating the degree of normalization in the moral graph
resulting from the merging of the possibilistic network codifying knowledge of the agent and that codifying
its preferences.
This document summarizes a research paper that presents two algorithms for solving Raven's Progressive Matrices tests visually without propositional representations. The paper introduces the Raven's test and existing computational accounts that use propositions. It then describes two new algorithms called "Affine" and "Fractal" that use visual representations and similarity-preserving transformations to solve the problems. The paper analyzes the performance of the algorithms on all 60 problems from the Standard Progressive Matrices test and finds they perform best on problems requiring visual/spatial skills and less on verbal problems.
The document discusses knowledge organization in cell biology and science education. It presents several working hypotheses about how understanding concepts and organizing them helps learning and transforms novices into experts. The importance of knowledge organization in curriculum development and conceptual change is explained. Different ways of representing knowledge like concept maps and semantic networks are described. The methodology section discusses classifying concepts, assigning semantic relations, comparing novice and expert knowledge structures, and developing a three-layer model for knowledge representation.
This document discusses various theories of truth and paradoxes involving truth. It examines proposals by philosophers like Vann McGee, Hartry Field, and JC Beal regarding how to address paradoxes like the liar paradox within formal theories. It also discusses fuzzy logics and Łukasiewicz logic as possible frameworks for modeling graduality and comparative notions of truth. Adding a truth predicate to formal theories is shown to potentially lead to deviations from the intended ontology or revenge paradoxes.
1. The document discusses the relationship between Hilbert systems (H) and natural deduction systems (N), showing how H maps to N via lambda abstraction.
2. It introduces Martin-Löf type theory (ML-ITT) and explains how propositions as types allows representing proofs as terms. ML-ITT can interpret both H and N through this correspondence.
3. Several works are cited that explore how ML-ITT can be viewed as an interpretation of set theory through universes, and how the Curry-Howard correspondence and lambda calculus are fundamental to ML-ITT.
This document discusses treating the truth predicate (Tr) as a logical connective in truth theories like Friedman-Sheared's theory (FS). It analyzes FS from the perspective of proof theoretic semantics, where Tr's introduction and elimination rules are like those of a connective. However, FS violates the "harmony" requirement for connectives, as it is not conservatively extending and proves the consistency of PA. The document then discusses interpreting paradoxical sentences like McGee's using coinduction and how guarded corecursion relates to the failure of Tr's formal commutability in FS.
- The document discusses whether the truth predicate Tr in Friedman-Sheard's truth theory FS can be considered a logical connective.
- It raises the problem that Tr violates the "HARMONY" between its introduction and elimination rules, as FS is ω-inconsistent based on McGee's theorem.
- The source of the problem is that deflationism allows asserting infinite conjunctions of sentences using Tr, while also insisting that Tr not involve ontological changes, as logical connectives do not.
Unification Of Randomized Anomaly In Deception Detection Using Fuzzy Logic Un...IJORCS
In the recent era of computer electronic communication we are currently facing the critical impact of Deception which plays its vital role in the mode of affecting efficient information sharing system. Identifying Deception in any mode of communication is a tedious process without using the proper tool for detecting those vulnerabilities. This paper deals with the efficient tools of Deception detection in which combined application implementation is our main focus rather than with its individuality. We propose a research model which comprises Fuzzy logic, Uncertainty and Randomization. This paper deals with an experiment which implements the scenario of mixture application with its revealed results. We also discuss the combined approach rather than with its individual performance.
The document proposes the Information Bottleneck Method as a way to extract relevant information from a signal X about another signal Y. It formalizes this as finding a compressed code for X that maximizes the information about Y while minimizing the code length. This forms a bottleneck that preserves only the most relevant information. The method provides self-consistent equations to determine the optimal coding rules from X to the code and from the code to Y. It generalizes rate-distortion theory by using the relationship between X and Y to determine relevance rather than requiring an externally specified distortion function.
Improved Performance of Unsupervised Method by Renovated K-MeansIJASCSE
Clustering is a separation of data into groups of similar objects. Every group called cluster consists of objects that are similar to one another and dissimilar to objects of other groups. In this paper, the K-Means algorithm is implemented by three distance functions and to identify the optimal distance function for clustering methods. The proposed K-Means algorithm is compared with K-Means, Static Weighted K-Means (SWK-Means) and Dynamic Weighted K-Means (DWK-Means) algorithm by using Davis Bouldin index, Execution Time and Iteration count methods. Experimental results show that the proposed K-Means algorithm performed better on Iris and Wine dataset when compared with other three clustering methods.
This document presents a new model of categorization called Categorization by Elimination (CBE). CBE uses as few cues or features as necessary to make an accurate category assignment, unlike most existing models which use all available cues. CBE orders cues by validity and uses them sequentially, eliminating potential categories after each cue is considered until only one category remains. The authors show that CBE performs as well as humans and other algorithms on categorization tasks while using fewer cues, making it a parsimonious psychological model of fast and frugal categorization.
Knuth's Definitions of Data and Information; Proposed Definition of KNOWLEDGE Putcha Narasimham
The words data and information are used without sufficient delineation of HOW, WHERE and WHEN to use them. They are at times used interchangeably and the dictionary meanings which seek to distinguish data and information end up with cyclic references. The use of these terms in computer science and information technology also follow the same colloquial trend with some pseudo-scientific attributions (raw facts are data and processed data is information) that do not pass simple tests of validity or rigor.
At times the word knowledge is used to explain the meanings of data and information, compounding the confusion and not having its own meaning. Donald E Knuth’s definitions of data and information are sufficiently precise and rigorous to be called scientific. Unfortunately Knuth's definitions of "data & information" do NOT turn up in the search results of Google, Yahoo, Bing, WolframAlpha. It seems perhaps the SINGLE AUTHENTIC source is vanishing. One definition of "data" which comes very close is from Dictionary of Military and Associated Terms.
They are discussed and used as foundation to define knowledge.
Knuth’s definition of “information” includes the word “meaning” which itself is a very complex and is wrongly defined in most places (according to me). I have a proposed definition for meaning also…too long). It is NOT essential to bring in the concept of "meaning" in the definition of "information". This is a new addition (09OCT13).
The available definitions of knowledge are examined and contrasted with Knuth’s definition of Data and Information. It is argued that “knowledge” refers to the “ability of a person or entity” “to provide data or information” “in response to a query”. This provides basis for knowledge representation, authoring and processing (separately described).
1. Logical Argument Mapping (LAM) is a method for building common ground through cognitive change using logical argument diagrams. It aims to make implicit assumptions and limitations explicit to promote reflection.
2. LAM uses valid argument schemes as a normative standard, challenging users to represent arguments fully and address objections. This process reveals gaps and drives users to continually improve understanding.
3. For cognitive change to occur, relevant information must be visible while reducing cognitive load. LAM aims to integrate with the World Wide Argument Web to allow sharing of arguments.
Separations of probabilistic theories via their information processing capab...Matthew Leifer
Talk given at the workshop "Operational Quantum Physics and the Quantum Classical Contrast" at Perimeter Institute in December 2007. It focuses on the results of http://arxiv.org/abs/0707.0620, http://arxiv.org/abs/0712.2265 and http://arxiv.org/abs/0805.3553
The talk was recorded and is viewable online at http://pirsa.org/07060033/
Predicting performance in Recommender Systems - PosterAlejandro Bellogin
This document discusses predicting the performance of recommender systems. It proposes that data commonly available to recommender systems could enable estimating the success of recommendations. Specifically, it aims to 1) define a performance prediction theory for recommender systems, 2) adapt query performance techniques from information retrieval to recommendations, and 3) evaluate appropriate performance metrics. The research also explores applying these models when combining multiple recommendation strategies or hybrid recommender systems.
This document describes a framework for providing mediated access to the algebraic topology software Kenzo. The framework aims to increase Kenzo's accessibility for non-programmers by developing a friendly front-end. It presents an architecture where a client communicates with Kenzo through XML messages validated by Common Lisp modules. The modules handle construction, computation and processing requests by checking restrictions before passing tasks to Kenzo. The document discusses making the framework extensible, efficient, adaptable to different clients, and including new functionality like additional mathematical systems. It proposes a graphical user interface implemented in Common Lisp as a distinguished client.
Teaching and learning mathematics at university levelharisv9
This document provides a summary and table of contents for a book that presents dialogues between two characters - M, a mathematician, and RME, a researcher in mathematics education. The dialogues discuss issues in learning and teaching mathematics at the university level, grounded in data from previous studies on this topic. Chapters 3-8 contain the dialogues, focusing on students' mathematical reasoning, expression, key concepts, and pedagogy. Chapter 8 discusses the relationship between mathematicians and mathematics education researchers. Introductory and concluding chapters provide background on the studies, methodology, and production of the book.
Comprehensive Guide to Taxonomy of Future KnowledgeMd Santo
This document provides a comprehensive guide to taxonomy of future knowledge. It discusses evolving models of knowledge from data-information-knowledge to a nature knowledge continuum informed by consciousness. Key points include: 1) Knowledge is considered an emergent property within nature and the universe, differentiated by infinite levels of consciousness. 2) Human knowledge is part of nature knowledge and is produced through human knowing tools of senses, brain and DNA. 3) A new framework called Human System Biology-based Knowledge Management is presented for understanding knowledge as a psycho-somatic entity with consciousness.
This document summarizes a symposium on awareness in computation held at the University of Birmingham. It discusses using modular neural networks (MANN) to model self-awareness and self-representation for artificial agents. The objectives are to analyze the modular structure of consciousness, design a cognitive architecture for self-awareness, self-representation and representing others, implement models in agents using artificial neural networks, and observe agent behavior in interaction scenarios. MANN are proposed as suitable for modeling functions related to consciousness, with self-awareness emerging from a sense of belonging and interactions between neural network modules.
This document provides an outline for a course on category theory from a logician's perspective. It introduces the instructor, Valeria de Paiva, and their background in category theory through their PhD thesis on Dialectica categories. The course will cover categories, functors, natural transformations, adjunctions, deductive systems as categories, and a taste of glue semantics. It emphasizes viewing proofs as first-class objects and using category theory for proof semantics rather than set-based models. The goal is to represent proofs explicitly rather than just knowing if a proof exists. The course will take an intuitionistic and constructive perspective on logic.
This document introduces the concept of refined concept maps (RCM) as a way to make traditional concept maps more rigorous and less ambiguous. RCM focuses on using a finite set of well-defined semantic relation names instead of loose linking words. This allows knowledge to be represented in a more formal and logically consistent way while still being accessible to novices. The document provides examples of converting traditional concept maps to RCM and discusses how RCM could serve as a bridge between informal and formal knowledge representation.
This document provides a suggested list of mathematical language terms for grade 6. It includes terms related to problem solving, reasoning and proof, communication, connections, representation, number sense and operations, algebra, geometry, measurement, and statistics and probability. The list contains over 150 mathematical terms organized into these conceptual categories.
Min-based qualitative possibilistic networks are one of the effective tools for a compact representation of
decision problems under uncertainty. The exact approaches for computing decision based on possibilistic
networks are limited by the size of the possibility distributions. Generally, these approaches are based on
possibilistic propagation algorithms. An important step in the computation of the decision is the
transformation of the DAG (Direct Acyclic Graph) into a secondary structure, known as the junction trees
(JT). This transformation is known to be costly and represents a difficult problem. We propose in this paper
a new approximate approach for the computation of decision under uncertainty within possibilistic
networks. The computing of the optimal optimistic decision no longer goes through the junction tree
construction step. Instead, it is performed by calculating the degree of normalization in the moral graph
resulting from the merging of the possibilistic network codifying knowledge of the agent and that codifying
its preferences.
This document summarizes a research paper that presents two algorithms for solving Raven's Progressive Matrices tests visually without propositional representations. The paper introduces the Raven's test and existing computational accounts that use propositions. It then describes two new algorithms called "Affine" and "Fractal" that use visual representations and similarity-preserving transformations to solve the problems. The paper analyzes the performance of the algorithms on all 60 problems from the Standard Progressive Matrices test and finds they perform best on problems requiring visual/spatial skills and less on verbal problems.
The document discusses knowledge organization in cell biology and science education. It presents several working hypotheses about how understanding concepts and organizing them helps learning and transforms novices into experts. The importance of knowledge organization in curriculum development and conceptual change is explained. Different ways of representing knowledge like concept maps and semantic networks are described. The methodology section discusses classifying concepts, assigning semantic relations, comparing novice and expert knowledge structures, and developing a three-layer model for knowledge representation.
This document discusses various theories of truth and paradoxes involving truth. It examines proposals by philosophers like Vann McGee, Hartry Field, and JC Beal regarding how to address paradoxes like the liar paradox within formal theories. It also discusses fuzzy logics and Łukasiewicz logic as possible frameworks for modeling graduality and comparative notions of truth. Adding a truth predicate to formal theories is shown to potentially lead to deviations from the intended ontology or revenge paradoxes.
1. The document discusses the relationship between Hilbert systems (H) and natural deduction systems (N), showing how H maps to N via lambda abstraction.
2. It introduces Martin-Löf type theory (ML-ITT) and explains how propositions as types allows representing proofs as terms. ML-ITT can interpret both H and N through this correspondence.
3. Several works are cited that explore how ML-ITT can be viewed as an interpretation of set theory through universes, and how the Curry-Howard correspondence and lambda calculus are fundamental to ML-ITT.
This document discusses treating the truth predicate (Tr) as a logical connective in truth theories like Friedman-Sheared's theory (FS). It analyzes FS from the perspective of proof theoretic semantics, where Tr's introduction and elimination rules are like those of a connective. However, FS violates the "harmony" requirement for connectives, as it is not conservatively extending and proves the consistency of PA. The document then discusses interpreting paradoxical sentences like McGee's using coinduction and how guarded corecursion relates to the failure of Tr's formal commutability in FS.
- The document discusses whether the truth predicate Tr in Friedman-Sheard's truth theory FS can be considered a logical connective.
- It raises the problem that Tr violates the "HARMONY" between its introduction and elimination rules, as FS is ω-inconsistent based on McGee's theorem.
- The source of the problem is that deflationism allows asserting infinite conjunctions of sentences using Tr, while also insisting that Tr not involve ontological changes, as logical connectives do not.
This document summarizes research on the logic system PAŁTr2 and its relationship to the liar paradox. It discusses two main approaches to addressing the paradox within PAŁTr2: 1) Modifying the truth predicate to avoid inconsistencies, such as using a revision sequence or fuzzy truth values. 2) Claiming that PAŁTr2 avoids inconsistency when interpreted over non-classical logics like fuzzy logic or paraconsistent logic. The document analyzes these approaches through several topics, including the definitions of ≤ and ≺, the hierarchy of n↑ formulas, and criticisms of interpreting PAŁTr2 over classical logic ω-collections. It references several key papers in the field and outlines the ongoing
This is a slide of My talk at Kyoto Nonclassical Logic Workshop (19, November, 2015). This is based on my paper "A constructive naive set theory and infinity" which was accepted to Notre Dame Journal of Formal Logic.
Chapter wise All Notes of First year Basic Civil Engineering.pptxDenish Jangid
Chapter wise All Notes of First year Basic Civil Engineering
Syllabus
Chapter-1
Introduction to objective, scope and outcome the subject
Chapter 2
Introduction: Scope and Specialization of Civil Engineering, Role of civil Engineer in Society, Impact of infrastructural development on economy of country.
Chapter 3
Surveying: Object Principles & Types of Surveying; Site Plans, Plans & Maps; Scales & Unit of different Measurements.
Linear Measurements: Instruments used. Linear Measurement by Tape, Ranging out Survey Lines and overcoming Obstructions; Measurements on sloping ground; Tape corrections, conventional symbols. Angular Measurements: Instruments used; Introduction to Compass Surveying, Bearings and Longitude & Latitude of a Line, Introduction to total station.
Levelling: Instrument used Object of levelling, Methods of levelling in brief, and Contour maps.
Chapter 4
Buildings: Selection of site for Buildings, Layout of Building Plan, Types of buildings, Plinth area, carpet area, floor space index, Introduction to building byelaws, concept of sun light & ventilation. Components of Buildings & their functions, Basic concept of R.C.C., Introduction to types of foundation
Chapter 5
Transportation: Introduction to Transportation Engineering; Traffic and Road Safety: Types and Characteristics of Various Modes of Transportation; Various Road Traffic Signs, Causes of Accidents and Road Safety Measures.
Chapter 6
Environmental Engineering: Environmental Pollution, Environmental Acts and Regulations, Functional Concepts of Ecology, Basics of Species, Biodiversity, Ecosystem, Hydrological Cycle; Chemical Cycles: Carbon, Nitrogen & Phosphorus; Energy Flow in Ecosystems.
Water Pollution: Water Quality standards, Introduction to Treatment & Disposal of Waste Water. Reuse and Saving of Water, Rain Water Harvesting. Solid Waste Management: Classification of Solid Waste, Collection, Transportation and Disposal of Solid. Recycling of Solid Waste: Energy Recovery, Sanitary Landfill, On-Site Sanitation. Air & Noise Pollution: Primary and Secondary air pollutants, Harmful effects of Air Pollution, Control of Air Pollution. . Noise Pollution Harmful Effects of noise pollution, control of noise pollution, Global warming & Climate Change, Ozone depletion, Greenhouse effect
Text Books:
1. Palancharmy, Basic Civil Engineering, McGraw Hill publishers.
2. Satheesh Gopi, Basic Civil Engineering, Pearson Publishers.
3. Ketki Rangwala Dalal, Essentials of Civil Engineering, Charotar Publishing House.
4. BCP, Surveying volume 1
Strategies for Effective Upskilling is a presentation by Chinwendu Peace in a Your Skill Boost Masterclass organisation by the Excellence Foundation for South Sudan on 08th and 09th June 2024 from 1 PM to 3 PM on each day.
How to Manage Your Lost Opportunities in Odoo 17 CRMCeline George
Odoo 17 CRM allows us to track why we lose sales opportunities with "Lost Reasons." This helps analyze our sales process and identify areas for improvement. Here's how to configure lost reasons in Odoo 17 CRM
This slide is special for master students (MIBS & MIFB) in UUM. Also useful for readers who are interested in the topic of contemporary Islamic banking.
How to Fix the Import Error in the Odoo 17Celine George
An import error occurs when a program fails to import a module or library, disrupting its execution. In languages like Python, this issue arises when the specified module cannot be found or accessed, hindering the program's functionality. Resolving import errors is crucial for maintaining smooth software operation and uninterrupted development processes.
ISO/IEC 27001, ISO/IEC 42001, and GDPR: Best Practices for Implementation and...PECB
Denis is a dynamic and results-driven Chief Information Officer (CIO) with a distinguished career spanning information systems analysis and technical project management. With a proven track record of spearheading the design and delivery of cutting-edge Information Management solutions, he has consistently elevated business operations, streamlined reporting functions, and maximized process efficiency.
Certified as an ISO/IEC 27001: Information Security Management Systems (ISMS) Lead Implementer, Data Protection Officer, and Cyber Risks Analyst, Denis brings a heightened focus on data security, privacy, and cyber resilience to every endeavor.
His expertise extends across a diverse spectrum of reporting, database, and web development applications, underpinned by an exceptional grasp of data storage and virtualization technologies. His proficiency in application testing, database administration, and data cleansing ensures seamless execution of complex projects.
What sets Denis apart is his comprehensive understanding of Business and Systems Analysis technologies, honed through involvement in all phases of the Software Development Lifecycle (SDLC). From meticulous requirements gathering to precise analysis, innovative design, rigorous development, thorough testing, and successful implementation, he has consistently delivered exceptional results.
Throughout his career, he has taken on multifaceted roles, from leading technical project management teams to owning solutions that drive operational excellence. His conscientious and proactive approach is unwavering, whether he is working independently or collaboratively within a team. His ability to connect with colleagues on a personal level underscores his commitment to fostering a harmonious and productive workplace environment.
Date: May 29, 2024
Tags: Information Security, ISO/IEC 27001, ISO/IEC 42001, Artificial Intelligence, GDPR
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This presentation includes basic of PCOS their pathology and treatment and also Ayurveda correlation of PCOS and Ayurvedic line of treatment mentioned in classics.
Walmart Business+ and Spark Good for Nonprofits.pdfTechSoup
"Learn about all the ways Walmart supports nonprofit organizations.
You will hear from Liz Willett, the Head of Nonprofits, and hear about what Walmart is doing to help nonprofits, including Walmart Business and Spark Good. Walmart Business+ is a new offer for nonprofits that offers discounts and also streamlines nonprofits order and expense tracking, saving time and money.
The webinar may also give some examples on how nonprofits can best leverage Walmart Business+.
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Main Java[All of the Base Concepts}.docxadhitya5119
This is part 1 of my Java Learning Journey. This Contains Custom methods, classes, constructors, packages, multithreading , try- catch block, finally block and more.
LAND USE LAND COVER AND NDVI OF MIRZAPUR DISTRICT, UPRAHUL
This Dissertation explores the particular circumstances of Mirzapur, a region located in the
core of India. Mirzapur, with its varied terrains and abundant biodiversity, offers an optimal
environment for investigating the changes in vegetation cover dynamics. Our study utilizes
advanced technologies such as GIS (Geographic Information Systems) and Remote sensing to
analyze the transformations that have taken place over the course of a decade.
The complex relationship between human activities and the environment has been the focus
of extensive research and worry. As the global community grapples with swift urbanization,
population expansion, and economic progress, the effects on natural ecosystems are becoming
more evident. A crucial element of this impact is the alteration of vegetation cover, which plays a
significant role in maintaining the ecological equilibrium of our planet.Land serves as the foundation for all human activities and provides the necessary materials for
these activities. As the most crucial natural resource, its utilization by humans results in different
'Land uses,' which are determined by both human activities and the physical characteristics of the
land.
The utilization of land is impacted by human needs and environmental factors. In countries
like India, rapid population growth and the emphasis on extensive resource exploitation can lead
to significant land degradation, adversely affecting the region's land cover.
Therefore, human intervention has significantly influenced land use patterns over many
centuries, evolving its structure over time and space. In the present era, these changes have
accelerated due to factors such as agriculture and urbanization. Information regarding land use and
cover is essential for various planning and management tasks related to the Earth's surface,
providing crucial environmental data for scientific, resource management, policy purposes, and
diverse human activities.
Accurate understanding of land use and cover is imperative for the development planning
of any area. Consequently, a wide range of professionals, including earth system scientists, land
and water managers, and urban planners, are interested in obtaining data on land use and cover
changes, conversion trends, and other related patterns. The spatial dimensions of land use and
cover support policymakers and scientists in making well-informed decisions, as alterations in
these patterns indicate shifts in economic and social conditions. Monitoring such changes with the
help of Advanced technologies like Remote Sensing and Geographic Information Systems is
crucial for coordinated efforts across different administrative levels. Advanced technologies like
Remote Sensing and Geographic Information Systems
9
Changes in vegetation cover refer to variations in the distribution, composition, and overall
structure of plant communities across different temporal and spatial scales. These changes can
occur natural.
1. Background
Formalizing truth degrees in PALTr
counterexample of the formalized truth degree theory
Conclusion
On the meaning of truth degrees
Shunsuke Yatabe
Research Center for Verification and Semantics,
National Institute of Advanced Industrial Science and Technology,
Japan
February 20, 2010
1 / 20
2. Background
Formalizing truth degrees in PALTr
counterexample of the formalized truth degree theory
Conclusion
Abstract
Primary objective: to introduce an a conflict of semantic
account of truth and axiomatic account of truth
Secondary objective: to analyze the truth conception in
fuzzy logics by formalizing “truth degrees”
Motivation: try to explain how truth degrees relate to truth
conception
Methodology: to formalize truth degree theory in axiomatic
truth theory PAŁTr2 .
Discussion: Since PAŁTr2 is ω-inconsistent, the formalized
truth degree theory fails.
2 / 20
3. Background
On the meaning of fuzzy truth values
Formalizing truth degrees in PALTr
Axiomatic approach
counterexample of the formalized truth degree theory
Transparency of the truth conception
Conclusion
The framework logic
Łukasiewicz infinite-valued predicate logic ∀Ł is defined as follows:
(1) Truth values are real numbers in [0, 1],
(2) ϕ0 → ϕ1 = min{1, 1 − ϕ0 + ϕ1 }, ⊥ = 0,
¬A ≡ A → ⊥, etc.
(3) (∀x)ϕ(x) = inf{ ϕ(a) M : a ∈ |M|}.
∀Ł is a sublogic of classical logic (i.e. ∀Ł ϕ implies CL ϕ).
3 / 20
4. Background
On the meaning of fuzzy truth values
Formalizing truth degrees in PALTr
Axiomatic approach
counterexample of the formalized truth degree theory
Transparency of the truth conception
Conclusion
Motivation: semantic account of truth in fuzzy logics
Often said: [0, 1] are truth values,
due to historical reason (e.g. Łukasiewicz, Zadeh)
We can define truth degrees
we can construct degrees of all sentences in any algebra from
a viewpoint of metatheory:
for any sentence A, B,
A ≤ B if and only if A → B = 1
we call this ordering “truth degrees”, and often think that they
represent “degrees of truthhood” of sentences.
4 / 20
5. Background
On the meaning of fuzzy truth values
Formalizing truth degrees in PALTr
Axiomatic approach
counterexample of the formalized truth degree theory
Transparency of the truth conception
Conclusion
Motivation: fuzzy truth values and algebra (1)
Not all semantic objects are called “truth values”:
example: sets of possible worlds, situations, etc.
Many fuzzy logics are characterized by their algebras, but it is
not trivial to say “such algebraic values are truth values of
such fuzzy logics”,
Analogy: intutionistic logic case
it is not complete for Tarskian semantics with truth values
{0, 1}: two truth values are not appropriate for interpretting
intutionistic logic.
To say “the Heyting algebra is a truth value of intuitionistic
logic” is controversial: no constructivist agrees this.
5 / 20
6. Background
On the meaning of fuzzy truth values
Formalizing truth degrees in PALTr
Axiomatic approach
counterexample of the formalized truth degree theory
Transparency of the truth conception
Conclusion
Motivation: fuzzy truth values and algebra (2)
The problem: [0, 1] are not enough to interpret many fuzzy
logics.
some fuzzy predicate logics (as BL∀) are not complete for
[0, 1],
What are truth values of such logics?
Sticking around “fuzzy truth values” comes with a heavy price:
we can’t give a unified account to explain the meaning of
all fuzzy logics.
The first proposal: to have the unified account has the first
priority.
6 / 20
7. Background
On the meaning of fuzzy truth values
Formalizing truth degrees in PALTr
Axiomatic approach
counterexample of the formalized truth degree theory
Transparency of the truth conception
Conclusion
Motivation the relationship between “truth degrees” and truth
Instead of asking the meaning of truth values [0, 1] ....
Question: what is a relationship between the conception of
truth and the so called “truth degrees”?
if [0,1] are truth values, it is trivial,
if not: the ordering of truth degrees in fuzzy logics are
regarded as an abstraction of the order relation in the
algebraic semantics (e.g. Paoli).
but we never call the chain in Heyting algebra “truth degrees”....
truth degrees should be about truth, but relationship between
algebraic value and truth is not trivial.
Asking the meaning of truth degrees is asking the meaning of
“truth values” of fuzzy logic in a roundabout way.
The next goal:
to find a framework to formalize truth degree theory without
mentioning truth values,
to formalize truth degree theory within it.
7 / 20
8. Background
On the meaning of fuzzy truth values
Formalizing truth degrees in PALTr
Axiomatic approach
counterexample of the formalized truth degree theory
Transparency of the truth conception
Conclusion
T-scheme and disquotation
Alternative framework: axiomatic truth theory
First we introduce its motivation.
Tarskian definition of truth:
Tr( ϕ ) ≡ ϕ
for any formula ϕ.
sentence “Snow is white.” is true iff snow is white.
“disquotation view of truth” (Quine, etc.):
The role of truth predicate seems to “disquote” quoted
sentences ϕ (then we get ϕ).
According to them, truth does not have a significant role in
semantics.
8 / 20
9. Background
On the meaning of fuzzy truth values
Formalizing truth degrees in PALTr
Axiomatic approach
counterexample of the formalized truth degree theory
Transparency of the truth conception
Conclusion
The liar paradox
However, we can define the liar sentence:
“This sentence is false” (if the language has an indexical)
L ≡ ¬Tr( L )
in case truth predicate is contained in the language, or it is
definable in that theory,
we can define λ in arithmetic by diagonalization argument.
L ↔ ¬L L ↔ ¬L
[v : L] L → ¬L [w : ¬L] ¬L → L
[v : L] ¬L [w : ¬L] L
L ∨ ¬L ⊥ ⊥ −
⊥ ∨
LC
P
9 / 20
10. Background
On the meaning of fuzzy truth values
Formalizing truth degrees in PALTr
Axiomatic approach
counterexample of the formalized truth degree theory
Transparency of the truth conception
Conclusion
Two ways out
The solution 1: to sustain classical logic:
basic strategy:
to restrict the domain of truth predicate (to exclude the liar
sentence),
axiomatic truth theory case:
to restrict T-scheme not to prove L, e.g. (McGee [M85])
Tr( ϕ ) → ϕ
Such restrictions prevents that the theory implies the liar
sentence as a theorem.
The solution 2: to sustain totality (and full T-scheme) of truth
predicate
abandon classical logic
10 / 20
11. Background
On the meaning of fuzzy truth values
Formalizing truth degrees in PALTr
Axiomatic approach
counterexample of the formalized truth degree theory
Transparency of the truth conception
Conclusion
Our framework PAŁTr2
The framework to analyze truth conception of ∀Ł: PAŁTr2
[HPS00] over ∀Ł
whose axioms are all axioms of classical PA,
the induction scheme for formulae possibly containing the truth
predicate Tr and
T-schemata for a total truth predicate Tr(x)
ϕ ≡ Tr( ϕ )
where ϕ is the Godel code of ϕ.
¨
The total truth predicate is not contradictory in PAŁTr2 .
The liar sentence, L ≡ ¬Tr( L ), dose not imply a
contradiction in ∀Ł: L = 0.5,
We can have a semantically closed language of arithmetic in
∀Ł.
11 / 20
12. Background
On the meaning of fuzzy truth values
Formalizing truth degrees in PALTr
Axiomatic approach
counterexample of the formalized truth degree theory
Transparency of the truth conception
Conclusion
Transparency of the truth conception in PAŁTr2
Since the total truth predicate exists, their truth conceptions
seem to be transparent,
i.e. no theoretical restriction on the domain of Tr (as ”Tr can’t
be applied to the liar sentence) are made
philosophically, they are successors of disquotational view of
truth.
12 / 20
13. Background
Formalizing truth degrees in PALTr
counterexample of the formalized truth degree theory
Conclusion
Formalization: “truth degrees” in terms of axiomatic truth
theory
We define ≤ as follows (call this “degree theoretic ordering”):
ϕ ≤ ψ ≡ϕ→ψ
We define ≺ as follows (call this “ordering of truthhood”):
define an ordering ≺⊆ ω × ω as follows:
ϕ ≺ ψ ≡ Tr( ϕ ) → Tr( ψ )
≺ is defined by the conditionals of the form “a truthhood of
some formula implies a truthhood of another formula”.
Therefore these conditionals definitely represent “degrees of
truthhood” in the sense of truth theory [Fl08].
Truth degree theory says two orderings are identical: for any
formula ϕ, ψ,
ϕ ≤ ψ ≡ ϕ ≺ ψ
13 / 20
14. Background
Formalizing truth degrees in PALTr
counterexample of the formalized truth degree theory
Conclusion
Formzlization: the formalized truth degree theory in PAŁTr2
Before we answer the question, we formalize the truth degree
theory in PAŁTr2 .
We define ≤, ≺ between Godel codes of formulae of PAŁTr2 .
¨
(∀x, y)(Form(x)&Form(y) → [x ≤ y ≡ (Tr(x→y))],
˙
(∀x, y)(Form(x)&Form(y) → [x ≺ y ≡ (Tr(x) → Tr(y))],
The formalized truth degree theory identity degree
theoretic ordering (≤) with degrees of truthhood (≺) :
(∀x, y)(Form(x)&Form(y) → [x ≤ y ≡ x ≺ y])
The ordering need not to be linearly ordered: e.g. Paoli’s
“really fuzzy” truth degrees.
14 / 20
15. Background
Formalizing truth degrees in PALTr
counterexample of the formalized truth degree theory
Conclusion
A (pathological) counter example
We fix PAŁTr2 as a object theory and metatheory.
PAŁTr2 shows that ≤ and ≺ are not identical:
(∀x, y)(Form(x)&Form(y) → [x ≤ y ≡ x ≺ y]) → ⊥
if Tr(x→y) ≡ (Tr(x) → Tr(y)), then mathematical induction
˙
implies the contradictory sentences [HPS00].
In this sense, the assumption of truth degree theory need not
hold.
15 / 20
16. Background
Formalizing truth degrees in PALTr
counterexample of the formalized truth degree theory
Conclusion
Remarks
For any formula ϕ, ψ, PAŁTr2 proves the following:
ϕ ≤ ψ ≡ ϕ ≺ ψ
However, PAŁTr2 proves, the following formalized
commutativity implies a contradiction:
(∀x, y)(Form(x)&Form(y) → [Tr(x→y) ≡ (Tr(x) → Tr(y))])
˙
This fails when x or y is a non-standard natural number
(ω-inconsistency! [R93]).
16 / 20
17. Background
Formalizing truth degrees in PALTr
counterexample of the formalized truth degree theory
Conclusion
An objection and replies (1)
Objection: HPS paradox merely shows that PAŁTr2 is not
suitable framework to analyze the conception of “Truth
degrees”.
ω-inconsistency is a crucial crux [Fl08],
This is not a failure of truth degree theory, but a failure of
axiomatic theory (hajek).
Reply:
Defensive: Even though PAŁTr2 are pathological, it provides
a precise distinction of two concepts (as constructive
mathematics gives a distinction between propositions which
are equivalent in classical logic).
Offensive: Since T-scheme is the key concept of truth, and
induction is essential to arithmetic, we must think the
consequence of PAŁTr2 seriously.
17 / 20
18. Background
Formalizing truth degrees in PALTr
counterexample of the formalized truth degree theory
Conclusion
An objection and replies (2)
Objection: What is a meaning of non-standard elements of
Form?
Reply:
In PAŁTr2 , we can represent an infinite operation on a formula
by some formula: taking a sup of
A, ¬A → A, ¬A → (¬A → A), · · · .
PAŁTr2 is a theory based on an extension of PA to represent
infinite processes in PAŁTr2 itself.
we can define an arithmetical function which corresponds an
infinite operation on codes,
it is interpreted to the real operation on formulae by using Tr,
this enables to treat an infinite process as an object in PAŁTr2 ,
non-standard numbers and non-standard elements of Form
represent such infinite processes.
18 / 20
19. Background
Formalizing truth degrees in PALTr
counterexample of the formalized truth degree theory
Conclusion
Conclusion
We try to formalize truth degree theory without mentioning
truth values.
We can define degree theoretic ordering ≤, but it is not trivial
that how such degree relate truth.
If we want to formalize “truth degree”, we needed a truth
predicate and truth theoretic machinery.
Truth degree theory can be formalized as supposing that
degree theoretic ordering ≤ is isomorphic to degrees of
truthhood ≺.
However some truth theory provides its counterexample
because of ω-inconsistency.
This means that, sometimes semantic anaysis and axiomatic
analysis have differing opinions.
19 / 20
20. Background
Formalizing truth degrees in PALTr
counterexample of the formalized truth degree theory
Conclusion
Reference
Hartry Field. “Saving Truth From Paradox” Oxford (2008)
´
Petr Hajek, Jeff B. Paris, John C. Shepherdson. “ The Liar Paradox
and Fuzzy Logic” Journal of Symbolic Logic, 65(1) (2000) 339-346.
Hannes Leitgeb. “Theories of truth which have no standard models”
Studia Logica, 68 (2001) 69-87.
Vann McGee. “How truthlike can a predicate be? A negative result”
Journal of Philosophical Logic, 17 (1985): 399-410.
Robin Milner, Mads Tofte. “Co-induction in relational semantics”
Theoretical computer science 87 (1991) 209-220.
Greg Restall “Arithmetic and Truth in Łukasiewicz’s Infinitely Valued
Logic” Logique et Analyse 36 (1993) 25-38.
20 / 20