Rossella Marrano from Scuola Normale Superiore discusses analogies between theories of truth and utility. Specifically, Marrano explores how concepts from utility theory like cardinal vs ordinal measurement, certainty vs risk, and preferences/choices can provide insights for analyzing truth. While there are similarities, Marrano also notes important disanalogies. The talk proposes bringing methods from utility theory to bear on truth theory, focusing on ordinal rather than cardinal conceptions to address criticisms of artificial precision.
- The document discusses elementary numerical analysis and solving problems related to properties of lower triangular matrices.
- It shows that the product of two unit lower triangular matrices is another unit lower triangular matrix, and that the inverse of a lower triangular matrix is also lower triangular.
- These properties are then used to prove the uniqueness of the LU decomposition of an invertible matrix.
The document provides several links to videos and websites about learning more about the ocean, including how many oceans and continents there are, why the ocean is blue, how the ocean affects climate, ocean currents, and what lives in the ocean. It also includes a link about modern piracy and its problems, as well as a site for children to explore and learn about the ocean.
Relatiemarketing | Autosport Community For The Happy Few | door Toine NagelToine Nagel
Relatiemarketing op niveau. Voor vrije beroepen, financiële instellingen, industriële marketing en marketing van kapitaalgoederen is relatiemarketing een belangrijk instrument om het contact met de klant te onderhouden. Met de Golf Incentive en de Voetbal Sky-Box als 'standaard' lijken de mogelijkheden beperkt... Of toch niet?
'Saint Christophe' - Autosport Community For The Happy Few
Four friends - Mateo, Poorna, Elki, and Ken - were making magic wands. Mateo, Poorna, Elki, and Ken were crafting magic wands together. The document lists the names of four friends who were engaged in the creative activity of making magic wands.
ICRB is a leading consumer research and consulting firm that provides pragmatic and cost effective innovative solutions that help clients conceptualise and develop new products and services. We focus on user centered research and design techniques including ethnography and other in- house developed tools and methodologies to capture user behaviors and interactions. We look to understand how consumers/users think and interact with their ecology in order to provide them with the technologies and service solutions of tomorrow.
Beaucoup de touristes étrangers choisissent le printemps pour leur voyage au Vietnam car c’est le moment où les villages fleuris autour de Hanoi sont recouverts par des couleurs vives des fleurs.
The document discusses comparing degrees of truth in a quantitative way using concepts from utility theory. It proposes bringing key concepts like ordinal vs. cardinal, certainty vs. risk vs. uncertainty, and preferences vs. choice to analyze truth. A representation theorem is presented showing that real-valued truth valuations can arise from certain qualitative comparisons between sentences' degrees of truth under some conditions. This approach provides philosophical insights and suggests new ways to address old questions about fuzzy and many-valued logics.
- The document discusses elementary numerical analysis and solving problems related to properties of lower triangular matrices.
- It shows that the product of two unit lower triangular matrices is another unit lower triangular matrix, and that the inverse of a lower triangular matrix is also lower triangular.
- These properties are then used to prove the uniqueness of the LU decomposition of an invertible matrix.
The document provides several links to videos and websites about learning more about the ocean, including how many oceans and continents there are, why the ocean is blue, how the ocean affects climate, ocean currents, and what lives in the ocean. It also includes a link about modern piracy and its problems, as well as a site for children to explore and learn about the ocean.
Relatiemarketing | Autosport Community For The Happy Few | door Toine NagelToine Nagel
Relatiemarketing op niveau. Voor vrije beroepen, financiële instellingen, industriële marketing en marketing van kapitaalgoederen is relatiemarketing een belangrijk instrument om het contact met de klant te onderhouden. Met de Golf Incentive en de Voetbal Sky-Box als 'standaard' lijken de mogelijkheden beperkt... Of toch niet?
'Saint Christophe' - Autosport Community For The Happy Few
Four friends - Mateo, Poorna, Elki, and Ken - were making magic wands. Mateo, Poorna, Elki, and Ken were crafting magic wands together. The document lists the names of four friends who were engaged in the creative activity of making magic wands.
ICRB is a leading consumer research and consulting firm that provides pragmatic and cost effective innovative solutions that help clients conceptualise and develop new products and services. We focus on user centered research and design techniques including ethnography and other in- house developed tools and methodologies to capture user behaviors and interactions. We look to understand how consumers/users think and interact with their ecology in order to provide them with the technologies and service solutions of tomorrow.
Beaucoup de touristes étrangers choisissent le printemps pour leur voyage au Vietnam car c’est le moment où les villages fleuris autour de Hanoi sont recouverts par des couleurs vives des fleurs.
The document discusses comparing degrees of truth in a quantitative way using concepts from utility theory. It proposes bringing key concepts like ordinal vs. cardinal, certainty vs. risk vs. uncertainty, and preferences vs. choice to analyze truth. A representation theorem is presented showing that real-valued truth valuations can arise from certain qualitative comparisons between sentences' degrees of truth under some conditions. This approach provides philosophical insights and suggests new ways to address old questions about fuzzy and many-valued logics.
1) The document discusses interpreting graded notions of truth through degrees of consequence. Standard many-valued logics are inadequate as they focus on degrees of truth rather than consequence.
2) Shifting to a relational perspective of degrees of consequence avoids issues like determining how much truer a statement is. Degrees should be interpreted ordinally rather than cardinally.
3) Intermediate logical values can be rehabilitated if interpreted as degrees of consequence rather than truth, allowing the development of a logic where consequence relations preserve different degrees. This could model graded notions like probability.
This document discusses representing degrees of truth ordinally rather than cardinally. It proposes bringing concepts from utility theory to analyze truth, focusing on comparative judgments of sentences being "more true" or "less true" rather than assigning numerical truth values. The document outlines axioms for a relation comparing sentences ordinally, and proves that if this relation satisfies the axioms, there exists at least one real-valued valuation function that represents it. This provides an ordinal foundation for real-valued semantics as an alternative to direct numerical assignments of truth values.
The document discusses degrees of truth and probability from a conceptual point of view. It aims to provide a unified framework for justifying the formal overlapping between degrees of truth and belief. It proposes looking at more or less true/probable at a more fundamental level. The goal is to shed light on the quantitative aspects by means of representation theorems linking qualitative comparisons to numerical representations. Specifically, it shows how degrees of truth can be understood as objective probabilities under a many-valued logic framework.
This document discusses degrees of truth as objective probabilities by providing a unified framework for truth and probability. It proposes looking at more or less true/probable at a fundamental qualitative level. The aim is to shed light on the quantitative side through representation theorems, justifying the formal overlapping between degrees of truth and belief from a conceptual viewpoint. Ultimately, it argues that by imposing truth-functional belief and norms on beliefs, objective probabilities can be interpreted as degrees of truth.
The document provides an introduction to formal logic. It discusses how to formulate valid arguments through propositional logic and syllogistic logic. Propositional logic uses truth tables to evaluate combinations of propositions and operators like negation and conjunction. Syllogistic logic examines implications of general statements using domains and categories. The key rules of inference for valid arguments are hypothetical syllogism, modus ponens, and modus tollens.
Chapter 1 Logic and ProofPropositional Logic SemanticsPropo.docxcravennichole326
Chapter 1: Logic and Proof
Propositional Logic Semantics
Propositional variables: p, q, r, s, ... (stand for simple sentences)
T: any proposition that is always true
F: any proposition that is always false
Compound propositions: formed from propositional variables and logical operators (all binary except negation):
Negation ¬
Conjunction ∧
Disjunction ∨
Implication →
Biconditional ↔
Exclusive Or ⊕
Truth Tables: assign all possible T, F to all possible variables, and determines all possible T, F of compound propositions; with n variables there are 2n rows in the table
Negation changes T to F and vice versa
Conjunction is only T if both conjuncts are T
Disjunction is only F is both disjuncts are F
Implication is only F is the antecedent is T and the consequent is F
Biconditional is only true if they have the same tvalue
Exclusive Or is only T if they differ in tvalue
Two (compound) propositions are equivalent (≡) iff they always have the same tvalue (see also below)
English translations:
Conjunction: and, but, although, yet, still, ...
Disjunctions: or, unless
Implication: if, if ... then, only if, when, implies, entails, follows from, is sufficient, is necessary, when, whenever
Biconditional: if and only if, just in case, is necessary and sufficient
A set of propositions is consistent iff there is some assignment of tvalue that makes all T
A set of propositions is inconsistent iff there is no assignment of tvalue that makes all T
A tautology is a compound propositions that is always T
A contradiction is a compound propositions that is always F
A contingency is a compound propositions that is sometimes T, sometimes F
A compound proposition is satisfiable iff some assignment of tvalues make it T
A compound proposition is unsatisfiable iff no assignment of tvalues make it T
Two compound propositions p and q are logically equivalent iff p ↔ q is a tautology
Common equivalences:
DeMorgan’s Laws (Dem)
¬(p ∨ q)≡¬p ∧ ¬q
¬(p ∧ q)≡¬p ∨ ¬q
Identity Laws (Id)
p ∧ T ≡p
p ∨ F ≡p
Domination Laws (Dom)
p ∨ T ≡T
p ∧ F ≡F
Idempotent Laws (Idem)
p ∨ T ≡T
p ∧ p ≡p
Double Negation Law (DN)
¬(¬p) ≡ p
Negation Laws (Neg)
p ∨ ¬p ≡T
p ∧ ¬p ≡F
Commutative Laws (Comm)
p ∨ q ≡q ∨ p
p ∧ q ≡q ∧ p
Associative Laws (Assoc)
(p ∨ q) ∨ r ≡p ∨ (q ∨ r)
(p ∧ q) ∧ r ≡p ∧ (q ∧ r)
Distributive Laws (Dist)
p ∨ (q ∧ r) ≡
(p ∨ q) ∧ (p ∨ r)
p ∧ (q ∨ r) ≡
(p ∧ q) ∨ (p ∧ r)
Absorption Laws (Abs)
p ∨ (p ∧ q) ≡ p
p ∧ (p ∨ q) ≡ p
Conditional Laws (Cond)
p →q≡ ¬p ∨ q
¬(p →q)≡ p ∧ ¬q
Biconditional Law (Bicond)
p ↔ q ≡ (p →q) ∧ (q →p)
Quantifier Negation (QNeg)
¬ ∀x P ( x ) ≡ ∃x ¬ P ( x )
¬ ∃x P ( x ) ≡ ∀x ¬ P ( x )
Predicate and Relational Logic (Quantificational Logic, First Order Logic): Semantics
Variables: x, y, z, ...
Predicates/Relations, Propositional Functions: P(x), M(x), Q(x,y), S(x,y,z), ...
Constants: a, b, c, 0, -1, 4, Socrates, ...
Domain (U): set of things the variables range over
Propositional functions are neither T nor F; however, if all the variables are re ...
This document presents an overview of Rossella Marrano's talk on a qualitative perspective on vagueness and degrees of truth. The talk explores representing vagueness through comparative judgments of truth between sentences, rather than precise numerical assignments. It proposes axioms for a binary relation between sentences based on one being "more true" than the other. Representation results show that real-valued truth functions can arise from satisfying the axioms. The approach aims to address objections that assigning precise numerical truth values replaces vagueness with artificial precision.
D. Mayo: Philosophy of Statistics & the Replication Crisis in Sciencejemille6
D. Mayo discusses various disputes-notably the replication crisis in science-in the context of her just released book: Statistical Inference as Severe Testing: How to Get Beyond the Statistics Wars.
Beyond the Metaverse: XV (eXtended meta/uni/Verse)Steve Mann
Beyond the Metaverse: XV (eXtended meta/uni/Verse)
Abstract: We propose the term and concept XV (eXtended meta/omni/uni/Verse) as an alternative to, and generalization of, the shared/social virtual reality widely known as ``metaverse''. XV is shared/social XR. We, and many others, use XR (eXtended Reality) as a broad umbrella term and concept to encompass all the other realities, where X is an ``anything'' variable, like in mathematics, to denote any reality, X ∈ {physical, virtual, augmented, ...} ℝeality. Therefore XV inherits this generality from XR. We begin with a very simple organized taxonomy of all these realities in terms of two simple building blocks: (1) physical reality (PR) as made of ``atoms'', and (2) virtual reality (VR) as made of ``bits''. Next we introduce XV as combining all these realities with extended society as a three-dimensional space and taxonomy of (1) ``atoms'' (physical reality), (2) ``bits'' (virtuality), and (3) ``genes'' (sociality). Thus those working in the liminal space between Virtual Reality (VR), Augmented Reality (AR), metaverse, and their various extensions, can describe their work and research as existing in the new field of XV. XV includes the metaverse along with extensions of reality itself like shared seeing in the infrared, ultraviolet, and shared seeing of electromagnetic radio waves, sound waves, and electric currents in motors. For example, workers in a mechanical room can look at a pump and see a superimposed time-varying waveform of the actual rotating magnetic field inside its motor, in real time, while sharing this vision across multiple sites.
2212.07960.pdf
Index Terms—Metaverse, Omniverse, eXtendiverse, XR, eXtended
Reality, VR, Virtual Reality, AR, Augmented Reality
Authors:
Steve Mann...
Yu Yuan (President-Elect, IEEE Standards Association)
Thomas Furness (“Grandfather of Virtual Reality”)
Joseph Paradiso (Alexander W Dreyfoos Professor and Associate Academic Head of Program in Media Arts and Sciences, MIT Media Lab)
Thomas Coughlin (IEEE President-Elect 2023 / IEEE President 2024)
Probability In Discrete Structure of Computer SciencePrankit Mishra
This document provides an overview of probability, including its basic definition, history, interpretations, theory, and applications. Probability is defined as a measure between 0 and 1 of the likelihood of an event occurring, where 0 is impossible and 1 is certain. It has been given a mathematical formalization and is used in many fields including statistics, science, and artificial intelligence. Historically, the scientific study of probability began in the 17th century and was further developed by thinkers like Bernoulli, Legendre, and Kolmogorov. Probability can be interpreted objectively based on frequencies or subjectively as degrees of belief. Important probability terms covered include experiments, outcomes, events, joint probability, independent events, mutually exclusive events, and conditional probability.
Discrete math is the branch of mathematics that does not rely on limits. It is well-suited to describe computer science concepts precisely as computers operate discretely in discrete steps. The document provides an overview of topics in discrete math including logic, sets, proofs, counting, and graph theory. These topics provide the tools needed for creating and analyzing sophisticated algorithms.
Everything we see is distributed on some scale. Some people are tall, some short and some are neither tall nor short. Once we find out how many are tall, short or middle heighted we get to know how people are distributed when it comes to height. This distribution can also be of chances. For example, we throw, 100 times, an unbalanced dice and find out how many times 1,2,3,4,5 or 6 appeared on top. This knowledge of distribution plays an important role in empirical work.
Prove asymptotic upper and lower hounds for each of the following sp.pdfwasemanivytreenrco51
Prove asymptotic upper and lower hounds for each of the following specified otherwise, assume
that in each case, T(n) = 1 (or any small constant) for small value You may assume that n = c^k
for some constant c that you choose. Make your bounds as tight as (No need to specify the
origin of your guess.) T(n0 = 8T(n/3) + n^1.83838383... T(n) = T(n - 1) = 1/n T(n) = 16T(n/2)
+ (n log n)^4. T(n) = 2T(n/2) + n/lg n. T(n) = T(n - 1) + T(n - 2) + 1 with base case of T(1) = 1
and T(2) = 2
Solution
A statement are often outlined as a declaratory sentence, or a part of a sentence, that\'s capable of
getting a truth-value, like being true or false. So, as an example, the subsequent area unit
statements:
George W. Bush is that the forty third President of the us.
Paris is that the capital of France.
Everyone born on Monday has purple hair.
Sometimes, a press release will contain one or a lot of alternative statements as elements.
contemplate as an example, the subsequent statement:
Either Ganymede may be a moon of Jupiter or Ganymede may be a moon of Saturn.
While the on top of sentence is itself a press release, as a result of it\'s true, the 2 elements,
\"Ganymede may be a moon of Jupiter\" and \"Ganymede may be a moon of Saturn\", area unit
themselves statements, as a result of the primary is true and therefore the second is fake.
The term proposition is typically used synonymously with statement. However, it\'s typically
accustomed name one thing abstract that 2 totally different statements with an equivalent which
means area unit each aforementioned to \"express\". during this usage, nation sentence, \"It is
raining\", and therefore the French sentence \"Il pleut\", would be thought-about to specific an
equivalent proposition; equally, the 2 English sentences, \"Callisto orbits Jupiter\" and \"Jupiter
is orbitted by Callisto\" would even be thought-about to specific an equivalent proposition.
However, the character or existence of propositions as abstract meanings continues to be a matter
of philosophical dispute, and for the needs of this text, the phrases \"statement\" and
\"proposition\" area unit used interchangeably.
Propositional logic, conjointly referred to as linguistic string logic, is that branch of logic that
studies ways that of mixing or neutering statements or propositions to create a lot of difficult
statements or propositions. change of integrity 2 easier propositions with the word \"and\" is one
common approach of mixing statements. once 2 statements area unit joined along side \"and\",
the advanced statement fashioned by them is true if and as long as each the element statements
area unit true. owing to this, associate argument of the subsequent kind is logically valid:
Paris is that the capital of France and Paris contains a population of over 2 million.
Therefore, Paris contains a population of over 2 million.
Propositional logic for the most part involves learning logical connectives like the words \"and\"
and \"or\" and therefo.
Computational Paths and the Calculation of Fundamental GroupsRuy De Queiroz
This document discusses the impact of intuitionistic type theory in mathematics. It introduces identity types as a way to formalize logical equality via proofs of equality between terms. Identity types allow for a connection between term rewriting and geometric concepts like paths and homotopy. Specifically, computational paths can be used to calculate fundamental groups of topological spaces like the fundamental group of a circle. This links type theory with fields like algebraic topology and homotopy theory.
16 USING LINEAR REGRESSION PREDICTING THE FUTURE16 MEDIA LIBRAR.docxhyacinthshackley2629
16 USING LINEAR REGRESSION PREDICTING THE FUTURE
16: MEDIA LIBRARY
Premium Videos
Core Concepts in Stats Video
· Linear Regression
Lightboard Lecture Video
· Multiple Regression
Time to Practice Video
· Chapter 16: Problem 2
Difficulty Scale
(as hard as they get!)
WHAT YOU WILL LEARN IN THIS CHAPTER
· Understanding how prediction works and how it can be used in the social and behavioral sciences
· Understanding how and why linear regression works when predicting one variable on the basis of another
· Judging the accuracy of predictions
· Understanding how multiple regression works and why it is useful
INTRODUCTION TO LINEAR REGRESSION
You’ve seen it all over the news—concern about obesity and how it affects work and daily life. A set of researchers in Sweden was interested in looking at how well mobility disability and/or obesity predicted job strain and whether social support at work can modify this association. The study included more than 35,000 participants, and differences in job strain mean scores were estimated using linear regression, the exact focus of what we are discussing in this chapter. The results found that level of mobile disability did predict job strain and that social support at work significantly modified the association among job strain, mobile disability, and obesity.
Want to know more? Go to the library or go online …
Norrback, M., De Munter, J., Tynelius, P., Ahlstrom, G., & Rasmussen, F. (2016). The association of mobility disability, weight status and job strain: A cross-sectional study. Scandinavian Journal of Public Health, 44, 311–319.
WHAT IS PREDICTION ALL ABOUT?
Here’s the scoop. Not only can you compute the degree to which two variables are related to one another (by computing a correlation coefficient as we did in Chapter 5), but you can also use these correlations to predict the value of one variable based on the value of another. This is a very special case of how correlations can be used, and it is a very powerful tool for social and behavioral sciences researchers.
The basic idea is to use a set of previously collected data (such as data on variables X and Y), calculate how correlated these variables are with one another, and then use that correlation and the knowledge of X to predict Y. Sound difficult? It’s not really, especially once you see it illustrated.
For example, a researcher collects data on total high school grade point average (GPA) and first-year college GPA for 400 students in their freshman year at the state university. He computes the correlation between the two variables. Then, he uses the techniques you’ll learn about later in this chapter to take a new set of high school GPAs and (knowing the relationship between high school GPA and first-year college GPA from the previous set of students) predict what first-year GPA should be for a new student who is just starting out. Pretty nifty, huh?
Here’s another example. A group of kindergarten teachers is interested in finding out how well ex.
16 USING LINEAR REGRESSION PREDICTING THE FUTURE16 MEDIA LIBRAR.docxnovabroom
16 USING LINEAR REGRESSION PREDICTING THE FUTURE
16: MEDIA LIBRARY
Premium Videos
Core Concepts in Stats Video
· Linear Regression
Lightboard Lecture Video
· Multiple Regression
Time to Practice Video
· Chapter 16: Problem 2
Difficulty Scale
(as hard as they get!)
WHAT YOU WILL LEARN IN THIS CHAPTER
· Understanding how prediction works and how it can be used in the social and behavioral sciences
· Understanding how and why linear regression works when predicting one variable on the basis of another
· Judging the accuracy of predictions
· Understanding how multiple regression works and why it is useful
INTRODUCTION TO LINEAR REGRESSION
You’ve seen it all over the news—concern about obesity and how it affects work and daily life. A set of researchers in Sweden was interested in looking at how well mobility disability and/or obesity predicted job strain and whether social support at work can modify this association. The study included more than 35,000 participants, and differences in job strain mean scores were estimated using linear regression, the exact focus of what we are discussing in this chapter. The results found that level of mobile disability did predict job strain and that social support at work significantly modified the association among job strain, mobile disability, and obesity.
Want to know more? Go to the library or go online …
Norrback, M., De Munter, J., Tynelius, P., Ahlstrom, G., & Rasmussen, F. (2016). The association of mobility disability, weight status and job strain: A cross-sectional study. Scandinavian Journal of Public Health, 44, 311–319.
WHAT IS PREDICTION ALL ABOUT?
Here’s the scoop. Not only can you compute the degree to which two variables are related to one another (by computing a correlation coefficient as we did in Chapter 5), but you can also use these correlations to predict the value of one variable based on the value of another. This is a very special case of how correlations can be used, and it is a very powerful tool for social and behavioral sciences researchers.
The basic idea is to use a set of previously collected data (such as data on variables X and Y), calculate how correlated these variables are with one another, and then use that correlation and the knowledge of X to predict Y. Sound difficult? It’s not really, especially once you see it illustrated.
For example, a researcher collects data on total high school grade point average (GPA) and first-year college GPA for 400 students in their freshman year at the state university. He computes the correlation between the two variables. Then, he uses the techniques you’ll learn about later in this chapter to take a new set of high school GPAs and (knowing the relationship between high school GPA and first-year college GPA from the previous set of students) predict what first-year GPA should be for a new student who is just starting out. Pretty nifty, huh?
Here’s another example. A group of kindergarten teachers is interested in finding out how well ex.
Computational Paths and the Calculation of Fundamental GroupsRuy De Queiroz
Apresentação online na Série "Lógicos em Quarentena", iniciativa conjunta da Soc. Brasileira de Lógica e do Grupo de Interesse em Lógica da Soc. Brasileira de Computação, 20/05/2020
Homotopic Foundations of the Theory of ComputationRuy De Queiroz
The document discusses the history and impact of intuitionistic type theory on the foundations of mathematics and computation. It describes how Martin-Löf's introduction of identity types in type theory allowed a connection between term rewriting and concepts in homotopy theory. It discusses how homotopy type theory has since been used to formalize concepts of "sameness" in mathematics like isomorphisms and homotopies. Researchers have used these tools to formally represent and calculate fundamental groups of topological spaces.
This presentation by Katharine Kemp, Associate Professor at the Faculty of Law & Justice at UNSW Sydney, was made during the discussion “The Intersection between Competition and Data Privacy” held at the 143rd meeting of the OECD Competition Committee on 13 June 2024. More papers and presentations on the topic can be found at oe.cd/ibcdp.
This presentation was uploaded with the author’s consent.
This presentation by Juraj Čorba, Chair of OECD Working Party on Artificial Intelligence Governance (AIGO), was made during the discussion “Artificial Intelligence, Data and Competition” held at the 143rd meeting of the OECD Competition Committee on 12 June 2024. More papers and presentations on the topic can be found at oe.cd/aicomp.
This presentation was uploaded with the author’s consent.
1) The document discusses interpreting graded notions of truth through degrees of consequence. Standard many-valued logics are inadequate as they focus on degrees of truth rather than consequence.
2) Shifting to a relational perspective of degrees of consequence avoids issues like determining how much truer a statement is. Degrees should be interpreted ordinally rather than cardinally.
3) Intermediate logical values can be rehabilitated if interpreted as degrees of consequence rather than truth, allowing the development of a logic where consequence relations preserve different degrees. This could model graded notions like probability.
This document discusses representing degrees of truth ordinally rather than cardinally. It proposes bringing concepts from utility theory to analyze truth, focusing on comparative judgments of sentences being "more true" or "less true" rather than assigning numerical truth values. The document outlines axioms for a relation comparing sentences ordinally, and proves that if this relation satisfies the axioms, there exists at least one real-valued valuation function that represents it. This provides an ordinal foundation for real-valued semantics as an alternative to direct numerical assignments of truth values.
The document discusses degrees of truth and probability from a conceptual point of view. It aims to provide a unified framework for justifying the formal overlapping between degrees of truth and belief. It proposes looking at more or less true/probable at a more fundamental level. The goal is to shed light on the quantitative aspects by means of representation theorems linking qualitative comparisons to numerical representations. Specifically, it shows how degrees of truth can be understood as objective probabilities under a many-valued logic framework.
This document discusses degrees of truth as objective probabilities by providing a unified framework for truth and probability. It proposes looking at more or less true/probable at a fundamental qualitative level. The aim is to shed light on the quantitative side through representation theorems, justifying the formal overlapping between degrees of truth and belief from a conceptual viewpoint. Ultimately, it argues that by imposing truth-functional belief and norms on beliefs, objective probabilities can be interpreted as degrees of truth.
The document provides an introduction to formal logic. It discusses how to formulate valid arguments through propositional logic and syllogistic logic. Propositional logic uses truth tables to evaluate combinations of propositions and operators like negation and conjunction. Syllogistic logic examines implications of general statements using domains and categories. The key rules of inference for valid arguments are hypothetical syllogism, modus ponens, and modus tollens.
Chapter 1 Logic and ProofPropositional Logic SemanticsPropo.docxcravennichole326
Chapter 1: Logic and Proof
Propositional Logic Semantics
Propositional variables: p, q, r, s, ... (stand for simple sentences)
T: any proposition that is always true
F: any proposition that is always false
Compound propositions: formed from propositional variables and logical operators (all binary except negation):
Negation ¬
Conjunction ∧
Disjunction ∨
Implication →
Biconditional ↔
Exclusive Or ⊕
Truth Tables: assign all possible T, F to all possible variables, and determines all possible T, F of compound propositions; with n variables there are 2n rows in the table
Negation changes T to F and vice versa
Conjunction is only T if both conjuncts are T
Disjunction is only F is both disjuncts are F
Implication is only F is the antecedent is T and the consequent is F
Biconditional is only true if they have the same tvalue
Exclusive Or is only T if they differ in tvalue
Two (compound) propositions are equivalent (≡) iff they always have the same tvalue (see also below)
English translations:
Conjunction: and, but, although, yet, still, ...
Disjunctions: or, unless
Implication: if, if ... then, only if, when, implies, entails, follows from, is sufficient, is necessary, when, whenever
Biconditional: if and only if, just in case, is necessary and sufficient
A set of propositions is consistent iff there is some assignment of tvalue that makes all T
A set of propositions is inconsistent iff there is no assignment of tvalue that makes all T
A tautology is a compound propositions that is always T
A contradiction is a compound propositions that is always F
A contingency is a compound propositions that is sometimes T, sometimes F
A compound proposition is satisfiable iff some assignment of tvalues make it T
A compound proposition is unsatisfiable iff no assignment of tvalues make it T
Two compound propositions p and q are logically equivalent iff p ↔ q is a tautology
Common equivalences:
DeMorgan’s Laws (Dem)
¬(p ∨ q)≡¬p ∧ ¬q
¬(p ∧ q)≡¬p ∨ ¬q
Identity Laws (Id)
p ∧ T ≡p
p ∨ F ≡p
Domination Laws (Dom)
p ∨ T ≡T
p ∧ F ≡F
Idempotent Laws (Idem)
p ∨ T ≡T
p ∧ p ≡p
Double Negation Law (DN)
¬(¬p) ≡ p
Negation Laws (Neg)
p ∨ ¬p ≡T
p ∧ ¬p ≡F
Commutative Laws (Comm)
p ∨ q ≡q ∨ p
p ∧ q ≡q ∧ p
Associative Laws (Assoc)
(p ∨ q) ∨ r ≡p ∨ (q ∨ r)
(p ∧ q) ∧ r ≡p ∧ (q ∧ r)
Distributive Laws (Dist)
p ∨ (q ∧ r) ≡
(p ∨ q) ∧ (p ∨ r)
p ∧ (q ∨ r) ≡
(p ∧ q) ∨ (p ∧ r)
Absorption Laws (Abs)
p ∨ (p ∧ q) ≡ p
p ∧ (p ∨ q) ≡ p
Conditional Laws (Cond)
p →q≡ ¬p ∨ q
¬(p →q)≡ p ∧ ¬q
Biconditional Law (Bicond)
p ↔ q ≡ (p →q) ∧ (q →p)
Quantifier Negation (QNeg)
¬ ∀x P ( x ) ≡ ∃x ¬ P ( x )
¬ ∃x P ( x ) ≡ ∀x ¬ P ( x )
Predicate and Relational Logic (Quantificational Logic, First Order Logic): Semantics
Variables: x, y, z, ...
Predicates/Relations, Propositional Functions: P(x), M(x), Q(x,y), S(x,y,z), ...
Constants: a, b, c, 0, -1, 4, Socrates, ...
Domain (U): set of things the variables range over
Propositional functions are neither T nor F; however, if all the variables are re ...
This document presents an overview of Rossella Marrano's talk on a qualitative perspective on vagueness and degrees of truth. The talk explores representing vagueness through comparative judgments of truth between sentences, rather than precise numerical assignments. It proposes axioms for a binary relation between sentences based on one being "more true" than the other. Representation results show that real-valued truth functions can arise from satisfying the axioms. The approach aims to address objections that assigning precise numerical truth values replaces vagueness with artificial precision.
D. Mayo: Philosophy of Statistics & the Replication Crisis in Sciencejemille6
D. Mayo discusses various disputes-notably the replication crisis in science-in the context of her just released book: Statistical Inference as Severe Testing: How to Get Beyond the Statistics Wars.
Beyond the Metaverse: XV (eXtended meta/uni/Verse)Steve Mann
Beyond the Metaverse: XV (eXtended meta/uni/Verse)
Abstract: We propose the term and concept XV (eXtended meta/omni/uni/Verse) as an alternative to, and generalization of, the shared/social virtual reality widely known as ``metaverse''. XV is shared/social XR. We, and many others, use XR (eXtended Reality) as a broad umbrella term and concept to encompass all the other realities, where X is an ``anything'' variable, like in mathematics, to denote any reality, X ∈ {physical, virtual, augmented, ...} ℝeality. Therefore XV inherits this generality from XR. We begin with a very simple organized taxonomy of all these realities in terms of two simple building blocks: (1) physical reality (PR) as made of ``atoms'', and (2) virtual reality (VR) as made of ``bits''. Next we introduce XV as combining all these realities with extended society as a three-dimensional space and taxonomy of (1) ``atoms'' (physical reality), (2) ``bits'' (virtuality), and (3) ``genes'' (sociality). Thus those working in the liminal space between Virtual Reality (VR), Augmented Reality (AR), metaverse, and their various extensions, can describe their work and research as existing in the new field of XV. XV includes the metaverse along with extensions of reality itself like shared seeing in the infrared, ultraviolet, and shared seeing of electromagnetic radio waves, sound waves, and electric currents in motors. For example, workers in a mechanical room can look at a pump and see a superimposed time-varying waveform of the actual rotating magnetic field inside its motor, in real time, while sharing this vision across multiple sites.
2212.07960.pdf
Index Terms—Metaverse, Omniverse, eXtendiverse, XR, eXtended
Reality, VR, Virtual Reality, AR, Augmented Reality
Authors:
Steve Mann...
Yu Yuan (President-Elect, IEEE Standards Association)
Thomas Furness (“Grandfather of Virtual Reality”)
Joseph Paradiso (Alexander W Dreyfoos Professor and Associate Academic Head of Program in Media Arts and Sciences, MIT Media Lab)
Thomas Coughlin (IEEE President-Elect 2023 / IEEE President 2024)
Probability In Discrete Structure of Computer SciencePrankit Mishra
This document provides an overview of probability, including its basic definition, history, interpretations, theory, and applications. Probability is defined as a measure between 0 and 1 of the likelihood of an event occurring, where 0 is impossible and 1 is certain. It has been given a mathematical formalization and is used in many fields including statistics, science, and artificial intelligence. Historically, the scientific study of probability began in the 17th century and was further developed by thinkers like Bernoulli, Legendre, and Kolmogorov. Probability can be interpreted objectively based on frequencies or subjectively as degrees of belief. Important probability terms covered include experiments, outcomes, events, joint probability, independent events, mutually exclusive events, and conditional probability.
Discrete math is the branch of mathematics that does not rely on limits. It is well-suited to describe computer science concepts precisely as computers operate discretely in discrete steps. The document provides an overview of topics in discrete math including logic, sets, proofs, counting, and graph theory. These topics provide the tools needed for creating and analyzing sophisticated algorithms.
Everything we see is distributed on some scale. Some people are tall, some short and some are neither tall nor short. Once we find out how many are tall, short or middle heighted we get to know how people are distributed when it comes to height. This distribution can also be of chances. For example, we throw, 100 times, an unbalanced dice and find out how many times 1,2,3,4,5 or 6 appeared on top. This knowledge of distribution plays an important role in empirical work.
Prove asymptotic upper and lower hounds for each of the following sp.pdfwasemanivytreenrco51
Prove asymptotic upper and lower hounds for each of the following specified otherwise, assume
that in each case, T(n) = 1 (or any small constant) for small value You may assume that n = c^k
for some constant c that you choose. Make your bounds as tight as (No need to specify the
origin of your guess.) T(n0 = 8T(n/3) + n^1.83838383... T(n) = T(n - 1) = 1/n T(n) = 16T(n/2)
+ (n log n)^4. T(n) = 2T(n/2) + n/lg n. T(n) = T(n - 1) + T(n - 2) + 1 with base case of T(1) = 1
and T(2) = 2
Solution
A statement are often outlined as a declaratory sentence, or a part of a sentence, that\'s capable of
getting a truth-value, like being true or false. So, as an example, the subsequent area unit
statements:
George W. Bush is that the forty third President of the us.
Paris is that the capital of France.
Everyone born on Monday has purple hair.
Sometimes, a press release will contain one or a lot of alternative statements as elements.
contemplate as an example, the subsequent statement:
Either Ganymede may be a moon of Jupiter or Ganymede may be a moon of Saturn.
While the on top of sentence is itself a press release, as a result of it\'s true, the 2 elements,
\"Ganymede may be a moon of Jupiter\" and \"Ganymede may be a moon of Saturn\", area unit
themselves statements, as a result of the primary is true and therefore the second is fake.
The term proposition is typically used synonymously with statement. However, it\'s typically
accustomed name one thing abstract that 2 totally different statements with an equivalent which
means area unit each aforementioned to \"express\". during this usage, nation sentence, \"It is
raining\", and therefore the French sentence \"Il pleut\", would be thought-about to specific an
equivalent proposition; equally, the 2 English sentences, \"Callisto orbits Jupiter\" and \"Jupiter
is orbitted by Callisto\" would even be thought-about to specific an equivalent proposition.
However, the character or existence of propositions as abstract meanings continues to be a matter
of philosophical dispute, and for the needs of this text, the phrases \"statement\" and
\"proposition\" area unit used interchangeably.
Propositional logic, conjointly referred to as linguistic string logic, is that branch of logic that
studies ways that of mixing or neutering statements or propositions to create a lot of difficult
statements or propositions. change of integrity 2 easier propositions with the word \"and\" is one
common approach of mixing statements. once 2 statements area unit joined along side \"and\",
the advanced statement fashioned by them is true if and as long as each the element statements
area unit true. owing to this, associate argument of the subsequent kind is logically valid:
Paris is that the capital of France and Paris contains a population of over 2 million.
Therefore, Paris contains a population of over 2 million.
Propositional logic for the most part involves learning logical connectives like the words \"and\"
and \"or\" and therefo.
Computational Paths and the Calculation of Fundamental GroupsRuy De Queiroz
This document discusses the impact of intuitionistic type theory in mathematics. It introduces identity types as a way to formalize logical equality via proofs of equality between terms. Identity types allow for a connection between term rewriting and geometric concepts like paths and homotopy. Specifically, computational paths can be used to calculate fundamental groups of topological spaces like the fundamental group of a circle. This links type theory with fields like algebraic topology and homotopy theory.
16 USING LINEAR REGRESSION PREDICTING THE FUTURE16 MEDIA LIBRAR.docxhyacinthshackley2629
16 USING LINEAR REGRESSION PREDICTING THE FUTURE
16: MEDIA LIBRARY
Premium Videos
Core Concepts in Stats Video
· Linear Regression
Lightboard Lecture Video
· Multiple Regression
Time to Practice Video
· Chapter 16: Problem 2
Difficulty Scale
(as hard as they get!)
WHAT YOU WILL LEARN IN THIS CHAPTER
· Understanding how prediction works and how it can be used in the social and behavioral sciences
· Understanding how and why linear regression works when predicting one variable on the basis of another
· Judging the accuracy of predictions
· Understanding how multiple regression works and why it is useful
INTRODUCTION TO LINEAR REGRESSION
You’ve seen it all over the news—concern about obesity and how it affects work and daily life. A set of researchers in Sweden was interested in looking at how well mobility disability and/or obesity predicted job strain and whether social support at work can modify this association. The study included more than 35,000 participants, and differences in job strain mean scores were estimated using linear regression, the exact focus of what we are discussing in this chapter. The results found that level of mobile disability did predict job strain and that social support at work significantly modified the association among job strain, mobile disability, and obesity.
Want to know more? Go to the library or go online …
Norrback, M., De Munter, J., Tynelius, P., Ahlstrom, G., & Rasmussen, F. (2016). The association of mobility disability, weight status and job strain: A cross-sectional study. Scandinavian Journal of Public Health, 44, 311–319.
WHAT IS PREDICTION ALL ABOUT?
Here’s the scoop. Not only can you compute the degree to which two variables are related to one another (by computing a correlation coefficient as we did in Chapter 5), but you can also use these correlations to predict the value of one variable based on the value of another. This is a very special case of how correlations can be used, and it is a very powerful tool for social and behavioral sciences researchers.
The basic idea is to use a set of previously collected data (such as data on variables X and Y), calculate how correlated these variables are with one another, and then use that correlation and the knowledge of X to predict Y. Sound difficult? It’s not really, especially once you see it illustrated.
For example, a researcher collects data on total high school grade point average (GPA) and first-year college GPA for 400 students in their freshman year at the state university. He computes the correlation between the two variables. Then, he uses the techniques you’ll learn about later in this chapter to take a new set of high school GPAs and (knowing the relationship between high school GPA and first-year college GPA from the previous set of students) predict what first-year GPA should be for a new student who is just starting out. Pretty nifty, huh?
Here’s another example. A group of kindergarten teachers is interested in finding out how well ex.
16 USING LINEAR REGRESSION PREDICTING THE FUTURE16 MEDIA LIBRAR.docxnovabroom
16 USING LINEAR REGRESSION PREDICTING THE FUTURE
16: MEDIA LIBRARY
Premium Videos
Core Concepts in Stats Video
· Linear Regression
Lightboard Lecture Video
· Multiple Regression
Time to Practice Video
· Chapter 16: Problem 2
Difficulty Scale
(as hard as they get!)
WHAT YOU WILL LEARN IN THIS CHAPTER
· Understanding how prediction works and how it can be used in the social and behavioral sciences
· Understanding how and why linear regression works when predicting one variable on the basis of another
· Judging the accuracy of predictions
· Understanding how multiple regression works and why it is useful
INTRODUCTION TO LINEAR REGRESSION
You’ve seen it all over the news—concern about obesity and how it affects work and daily life. A set of researchers in Sweden was interested in looking at how well mobility disability and/or obesity predicted job strain and whether social support at work can modify this association. The study included more than 35,000 participants, and differences in job strain mean scores were estimated using linear regression, the exact focus of what we are discussing in this chapter. The results found that level of mobile disability did predict job strain and that social support at work significantly modified the association among job strain, mobile disability, and obesity.
Want to know more? Go to the library or go online …
Norrback, M., De Munter, J., Tynelius, P., Ahlstrom, G., & Rasmussen, F. (2016). The association of mobility disability, weight status and job strain: A cross-sectional study. Scandinavian Journal of Public Health, 44, 311–319.
WHAT IS PREDICTION ALL ABOUT?
Here’s the scoop. Not only can you compute the degree to which two variables are related to one another (by computing a correlation coefficient as we did in Chapter 5), but you can also use these correlations to predict the value of one variable based on the value of another. This is a very special case of how correlations can be used, and it is a very powerful tool for social and behavioral sciences researchers.
The basic idea is to use a set of previously collected data (such as data on variables X and Y), calculate how correlated these variables are with one another, and then use that correlation and the knowledge of X to predict Y. Sound difficult? It’s not really, especially once you see it illustrated.
For example, a researcher collects data on total high school grade point average (GPA) and first-year college GPA for 400 students in their freshman year at the state university. He computes the correlation between the two variables. Then, he uses the techniques you’ll learn about later in this chapter to take a new set of high school GPAs and (knowing the relationship between high school GPA and first-year college GPA from the previous set of students) predict what first-year GPA should be for a new student who is just starting out. Pretty nifty, huh?
Here’s another example. A group of kindergarten teachers is interested in finding out how well ex.
Computational Paths and the Calculation of Fundamental GroupsRuy De Queiroz
Apresentação online na Série "Lógicos em Quarentena", iniciativa conjunta da Soc. Brasileira de Lógica e do Grupo de Interesse em Lógica da Soc. Brasileira de Computação, 20/05/2020
Homotopic Foundations of the Theory of ComputationRuy De Queiroz
The document discusses the history and impact of intuitionistic type theory on the foundations of mathematics and computation. It describes how Martin-Löf's introduction of identity types in type theory allowed a connection between term rewriting and concepts in homotopy theory. It discusses how homotopy type theory has since been used to formalize concepts of "sameness" in mathematics like isomorphisms and homotopies. Researchers have used these tools to formally represent and calculate fundamental groups of topological spaces.
This presentation by Katharine Kemp, Associate Professor at the Faculty of Law & Justice at UNSW Sydney, was made during the discussion “The Intersection between Competition and Data Privacy” held at the 143rd meeting of the OECD Competition Committee on 13 June 2024. More papers and presentations on the topic can be found at oe.cd/ibcdp.
This presentation was uploaded with the author’s consent.
This presentation by Juraj Čorba, Chair of OECD Working Party on Artificial Intelligence Governance (AIGO), was made during the discussion “Artificial Intelligence, Data and Competition” held at the 143rd meeting of the OECD Competition Committee on 12 June 2024. More papers and presentations on the topic can be found at oe.cd/aicomp.
This presentation was uploaded with the author’s consent.
This presentation by OECD, OECD Secretariat, was made during the discussion “Artificial Intelligence, Data and Competition” held at the 143rd meeting of the OECD Competition Committee on 12 June 2024. More papers and presentations on the topic can be found at oe.cd/aicomp.
This presentation was uploaded with the author’s consent.
This presentation by Thibault Schrepel, Associate Professor of Law at Vrije Universiteit Amsterdam University, was made during the discussion “Artificial Intelligence, Data and Competition” held at the 143rd meeting of the OECD Competition Committee on 12 June 2024. More papers and presentations on the topic can be found at oe.cd/aicomp.
This presentation was uploaded with the author’s consent.
This presentation by Professor Alex Robson, Deputy Chair of Australia’s Productivity Commission, was made during the discussion “Competition and Regulation in Professions and Occupations” held at the 77th meeting of the OECD Working Party No. 2 on Competition and Regulation on 10 June 2024. More papers and presentations on the topic can be found at oe.cd/crps.
This presentation was uploaded with the author’s consent.
This presentation by OECD, OECD Secretariat, was made during the discussion “Pro-competitive Industrial Policy” held at the 143rd meeting of the OECD Competition Committee on 12 June 2024. More papers and presentations on the topic can be found at oe.cd/pcip.
This presentation was uploaded with the author’s consent.
This presentation by Yong Lim, Professor of Economic Law at Seoul National University School of Law, was made during the discussion “Artificial Intelligence, Data and Competition” held at the 143rd meeting of the OECD Competition Committee on 12 June 2024. More papers and presentations on the topic can be found at oe.cd/aicomp.
This presentation was uploaded with the author’s consent.
Suzanne Lagerweij - Influence Without Power - Why Empathy is Your Best Friend...Suzanne Lagerweij
This is a workshop about communication and collaboration. We will experience how we can analyze the reasons for resistance to change (exercise 1) and practice how to improve our conversation style and be more in control and effective in the way we communicate (exercise 2).
This session will use Dave Gray’s Empathy Mapping, Argyris’ Ladder of Inference and The Four Rs from Agile Conversations (Squirrel and Fredrick).
Abstract:
Let’s talk about powerful conversations! We all know how to lead a constructive conversation, right? Then why is it so difficult to have those conversations with people at work, especially those in powerful positions that show resistance to change?
Learning to control and direct conversations takes understanding and practice.
We can combine our innate empathy with our analytical skills to gain a deeper understanding of complex situations at work. Join this session to learn how to prepare for difficult conversations and how to improve our agile conversations in order to be more influential without power. We will use Dave Gray’s Empathy Mapping, Argyris’ Ladder of Inference and The Four Rs from Agile Conversations (Squirrel and Fredrick).
In the session you will experience how preparing and reflecting on your conversation can help you be more influential at work. You will learn how to communicate more effectively with the people needed to achieve positive change. You will leave with a self-revised version of a difficult conversation and a practical model to use when you get back to work.
Come learn more on how to become a real influencer!
This presentation by Professor Giuseppe Colangelo, Jean Monnet Professor of European Innovation Policy, was made during the discussion “The Intersection between Competition and Data Privacy” held at the 143rd meeting of the OECD Competition Committee on 13 June 2024. More papers and presentations on the topic can be found at oe.cd/ibcdp.
This presentation was uploaded with the author’s consent.
This presentation by Tim Capel, Director of the UK Information Commissioner’s Office Legal Service, was made during the discussion “The Intersection between Competition and Data Privacy” held at the 143rd meeting of the OECD Competition Committee on 13 June 2024. More papers and presentations on the topic can be found at oe.cd/ibcdp.
This presentation was uploaded with the author’s consent.
The importance of sustainable and efficient computational practices in artificial intelligence (AI) and deep learning has become increasingly critical. This webinar focuses on the intersection of sustainability and AI, highlighting the significance of energy-efficient deep learning, innovative randomization techniques in neural networks, the potential of reservoir computing, and the cutting-edge realm of neuromorphic computing. This webinar aims to connect theoretical knowledge with practical applications and provide insights into how these innovative approaches can lead to more robust, efficient, and environmentally conscious AI systems.
Webinar Speaker: Prof. Claudio Gallicchio, Assistant Professor, University of Pisa
Claudio Gallicchio is an Assistant Professor at the Department of Computer Science of the University of Pisa, Italy. His research involves merging concepts from Deep Learning, Dynamical Systems, and Randomized Neural Systems, and he has co-authored over 100 scientific publications on the subject. He is the founder of the IEEE CIS Task Force on Reservoir Computing, and the co-founder and chair of the IEEE Task Force on Randomization-based Neural Networks and Learning Systems. He is an associate editor of IEEE Transactions on Neural Networks and Learning Systems (TNNLS).
Why Psychological Safety Matters for Software Teams - ACE 2024 - Ben Linders.pdfBen Linders
Psychological safety in teams is important; team members must feel safe and able to communicate and collaborate effectively to deliver value. It’s also necessary to build long-lasting teams since things will happen and relationships will be strained.
But, how safe is a team? How can we determine if there are any factors that make the team unsafe or have an impact on the team’s culture?
In this mini-workshop, we’ll play games for psychological safety and team culture utilizing a deck of coaching cards, The Psychological Safety Cards. We will learn how to use gamification to gain a better understanding of what’s going on in teams. Individuals share what they have learned from working in teams, what has impacted the team’s safety and culture, and what has led to positive change.
Different game formats will be played in groups in parallel. Examples are an ice-breaker to get people talking about psychological safety, a constellation where people take positions about aspects of psychological safety in their team or organization, and collaborative card games where people work together to create an environment that fosters psychological safety.
This presentation by OECD, OECD Secretariat, was made during the discussion “Competition and Regulation in Professions and Occupations” held at the 77th meeting of the OECD Working Party No. 2 on Competition and Regulation on 10 June 2024. More papers and presentations on the topic can be found at oe.cd/crps.
This presentation was uploaded with the author’s consent.
This presentation by Nathaniel Lane, Associate Professor in Economics at Oxford University, was made during the discussion “Pro-competitive Industrial Policy” held at the 143rd meeting of the OECD Competition Committee on 12 June 2024. More papers and presentations on the topic can be found at oe.cd/pcip.
This presentation was uploaded with the author’s consent.
1. Analogies between truth and utility
Rossella Marrano
Scuola Normale Superiore
Joint work with Hykel Hosni
Pisa, 7 July 2014
Rossella Marrano (SNS) Analogies between truth and utility 07/07/2014 1 / 15
2. Motivation
Degrees of truth Utility
similar criticisms
Ordinal revolution
Rigorous notion of utility
Rossella Marrano (SNS) Analogies between truth and utility 07/07/2014 2 / 15
3. Motivation
Degrees of truth Utility
similar criticisms
Ordinal revolution
Rigorous notion of utility
Our proposal
Bringing key concepts and methods of utility theory to bear on the analysis of
truth
1. cardinal – ordinal (representation theorems)
2. certainty – risk – uncertainty – ambiguity
3. preferences – choices
Rossella Marrano (SNS) Analogies between truth and utility 07/07/2014 2 / 15
4. “Truth is rarely pure and never simple”
To say of what is that it is not, or of what is not that it is, is false,
while to say of what is that it is, and of what is not that it is not, is
true. (Aristotle)
Veritas est adaequatio rei et intellectus (Truth is the equation of
thing and intellect). (Thomas Aquinas)
The opinion which is fated to be ultimately agreed to by all who
investigate, is what we mean by the truth. (Peirce)
Rossella Marrano (SNS) Analogies between truth and utility 07/07/2014 3 / 15
5. Classical semantics
I L = fp1; p2; : : : g is a propositional language
I C = f:;^;_;!g is the set of propositional connectives
I SL = f; ; : : : g is the set of sentences on L
Propositional valuations on L are functions v : L ! f0; 1g.
Rossella Marrano (SNS) Analogies between truth and utility 07/07/2014 4 / 15
6. Classical semantics
I L = fp1; p2; : : : g is a propositional language
I C = f:;^;_;!g is the set of propositional connectives
I SL = f; ; : : : g is the set of sentences on L
Propositional valuations on L are functions v : L ! f0; 1g.
Remark
Valuations extend uniquely to SL.
Suppose ; 2 SL, then there exists a function f_ : f0; 1g f0; 1g ! f0; 1g
such that v( _ ) = f_(v(); v()):
p q : p p ^ q p _ q p ! q
1 1 0 1 1 1
1 0 0 0 1 0
0 1 1 0 1 1
0 0 1 0 0 1
Rossella Marrano (SNS) Analogies between truth and utility 07/07/2014 4 / 15
7. Modelling truth (classically)
I truth as property of sentences
I truth as specific value assigned to sentences (truth-value)
Rossella Marrano (SNS) Analogies between truth and utility 07/07/2014 5 / 15
8. Modelling truth (classically)
I truth as property of sentences
I truth as specific value assigned to sentences (truth-value)
I there are exactly two truth-values: true and false [Bivalence]
I each sentence is given exactly one truth-value [Non-contradiction]
I the truth-value of a compound sentence is determined by the
truth-values of its components [Truth-functionality]
Rossella Marrano (SNS) Analogies between truth and utility 07/07/2014 5 / 15
9. Modelling truth (classically)
I truth as property of sentences
I truth as specific value assigned to sentences (truth-value)
I there are exactly two truth-values: true and false [Bivalence]
I each sentence is given exactly one truth-value [Non-contradiction]
I the truth-value of a compound sentence is determined by the
truth-values of its components [Truth-functionality]
Rossella Marrano (SNS) Analogies between truth and utility 07/07/2014 5 / 15
10. Beyond true and false (1920)
Jan Łukasiewicz (1878-1956)
To me, personally, the principle of bivalence
does not appear to be self-evident. Therefore
I am entitled not to recognize it, and to
accept the view that besides truth and
falsehood there exist other truth-values,
including at least one more, the third
truth-value.
One class of such systems seems to have the
same relation to ordinary logic that geometry in
a space of an arbitrary number of dimensions
has to the geometry of Euclid. [. . . ] In these
systems instead of the two truth-values + and
, we have m distinct ‘truth-values’ tl, t2, . . . ,
tm where m is any positive integer.
Emil Post (1897-1954)
Rossella Marrano (SNS) Analogies between truth and utility 07/07/2014 6 / 15
11. Relaxing bivalence
I v : SL ! f0; 1; ?g
Rossella Marrano (SNS) Analogies between truth and utility 07/07/2014 7 / 15
12. Relaxing bivalence
I v : SL ! f0; 1; ?g
I possible, undetermined [Łukasiewicz (1920)]
I shall be in Warsaw at noon on 21 December of the next year
I undetermined by means of algorithms [Kleene (1938)]
Non terminating processes
I nonsense or meaningless [Bochvar (1938)]
This sentence is false
Rossella Marrano (SNS) Analogies between truth and utility 07/07/2014 7 / 15
13. Relaxing bivalence
I v : SL ! f0; 1; ?g
I possible, undetermined [Łukasiewicz (1920)]
I shall be in Warsaw at noon on 21 December of the next year
I undetermined by means of algorithms [Kleene (1938)]
Non terminating processes
I nonsense or meaningless [Bochvar (1938)]
This sentence is false
I v : SL ! [0; 1]
I degrees of truth
What do “degrees of truth” mean? Interpretation and measurability
Rossella Marrano (SNS) Analogies between truth and utility 07/07/2014 7 / 15
14. Cardinal – Ordinal
Jeremy Bentham (1748-1832)
Agents have “utils” in their heads
The amount of pleasure or pain
felt for a good is measurable
Vilfredo Pareto (1848-1923)
Utility has an ordinal meaning
Agents can only tell between two
goods which one they prefer
Rossella Marrano (SNS) Analogies between truth and utility 07/07/2014 8 / 15
15. Representation theorems
[I]n order to examine general economic equilibrium, this measurement [of
the degrees of utility] is unnecessary. It is sufficient to ascertain if one
pleasure is larger or smaller than another. This is the only fact we need
to build a theory. (Pareto, 1898)
I comparative judgments:
preferences or indifference
I pairwise evaluation
I X2
I numerical analysis: utility
function
I point-wise evaluation
I u: X ! R
Rossella Marrano (SNS) Analogies between truth and utility 07/07/2014 9 / 15
16. Representation theorems
[I]n order to examine general economic equilibrium, this measurement [of
the degrees of utility] is unnecessary. It is sufficient to ascertain if one
pleasure is larger or smaller than another. This is the only fact we need
to build a theory. (Pareto, 1898)
I comparative judgments:
preferences or indifference
I pairwise evaluation
I X2
I numerical analysis: utility
function
I point-wise evaluation
I u: X ! R
General form of the representation
Necessary and sufficient conditions on a preference relation for the
existence of a(n equivalence class of a) real-valued utility function u such that
x y () u(x) u(y):
Plausibility (behavioural foundations) and mathematical convenience
Rossella Marrano (SNS) Analogies between truth and utility 07/07/2014 9 / 15
17. Back to truth
Degrees of truth as real numbers
We shall assume that the truth degrees are linearly ordered, with 1
as maximum and 0 as minimum. Thus truth degrees will be coded
by (some) reals. [. . . ] We shall always take the set [0; 1] with its
natural (standard) linear order. (Petr Hájek, Metamathematics of
Fuzzy Logic, 1998)
Rossella Marrano (SNS) Analogies between truth and utility 07/07/2014 10 / 15
18. Back to truth
Degrees of truth as real numbers
We shall assume that the truth degrees are linearly ordered, with 1
as maximum and 0 as minimum. Thus truth degrees will be coded
by (some) reals. [. . . ] We shall always take the set [0; 1] with its
natural (standard) linear order. (Petr Hájek, Metamathematics of
Fuzzy Logic, 1998)
Artificial precision (Pareto strikes back)
I arbitrariness of the choice
how can we justify the choice of the truth value 0.24 over 0.23?
I implausibility of the interpretation
what does it mean for a sentence to be 1= true?
Rossella Marrano (SNS) Analogies between truth and utility 07/07/2014 10 / 15
19. Back to truth
Degrees of truth as real numbers
We shall assume that the truth degrees are linearly ordered, with 1
as maximum and 0 as minimum. Thus truth degrees will be coded
by (some) reals. [. . . ] We shall always take the set [0; 1] with its
natural (standard) linear order. (Petr Hájek, Metamathematics of
Fuzzy Logic, 1998)
Artificial precision (Pareto strikes back)
I arbitrariness of the choice
how can we justify the choice of the truth value 0.24 over 0.23?
I implausibility of the interpretation
what does it mean for a sentence to be 1= true?
The ordinal bell rings!
Rossella Marrano (SNS) Analogies between truth and utility 07/07/2014 10 / 15
20. Ordinal foundations for many-valued semantics
1. natural appeal of the notion being more or less true than
I closeness to the truth
Compare “a square is round” with “a triangle is round”.
I scientific fallibilism
John, when people thought the Earth was flat, they were wrong.
When people thought the Earth was spherical, they were wrong.
But if you think that thinking the Earth is spherical is just as
wrong as thinking the Earth is flat, then your view is wronger
than both of them put together. (Isaac Asimov, The Relativity
of Wrong, 1989)
I mathematical modelling
All models are wrong, but some models are more wrong than others.
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21. Ordinal foundations for many-valued semantics
1. natural appeal of the notion being more or less true than
I closeness to the truth
Compare “a square is round” with “a triangle is round”.
I scientific fallibilism
John, when people thought the Earth was flat, they were wrong.
When people thought the Earth was spherical, they were wrong.
But if you think that thinking the Earth is spherical is just as
wrong as thinking the Earth is flat, then your view is wronger
than both of them put together. (Isaac Asimov, The Relativity
of Wrong, 1989)
I mathematical modelling
All models are wrong, but some models are more wrong than others.
2. axioms as properties
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22. Representation theorems for many-valued semantics
Many-valued valuations can be proved to arise from truth-comparisons under
specific conditions.
The case of Łukasiewicz infinite-valued logic1
(T.1) SL2 is reflexive and transitive
(T.2) , ?
(T.3) `Ł ! =)
(T.4) 1 2; 1 2 =) 1 1 2 2
(T.5) =) : :
If satisfies axioms (T.1)–(T.5) then there exists a Łukasiewicz valuation
v : SL ! [0; 1] such that for all ; 2 SL:
=) v() v():
1Ongoing work with Hykel Hosni and Vincenzo Marra
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23. Pushing the analogy
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24. Pushing the analogy
1. Certainty – Risk – Uncertainty – Ambiguity
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25. Pushing the analogy
1. Certainty – Risk – Uncertainty – Ambiguity
Expected truth-value of a sentence
[. . . ] the average of its truth in all the worlds the agent has not ruled out,
weighted according to how likely the agent thinks it is that each of those
worlds is the actual one. (Smith, Vagueness and Degrees of Truth, 2008)
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26. Pushing the analogy
1. Certainty – Risk – Uncertainty – Ambiguity
Expected truth-value of a sentence
[. . . ] the average of its truth in all the worlds the agent has not ruled out,
weighted according to how likely the agent thinks it is that each of those
worlds is the actual one. (Smith, Vagueness and Degrees of Truth, 2008)
2. Preferences – Choice
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27. Pushing the analogy
1. Certainty – Risk – Uncertainty – Ambiguity
Expected truth-value of a sentence
[. . . ] the average of its truth in all the worlds the agent has not ruled out,
weighted according to how likely the agent thinks it is that each of those
worlds is the actual one. (Smith, Vagueness and Degrees of Truth, 2008)
2. Preferences – Choice
Truth by choice or ‘revealed truth’
[. . . ] the truth of which we regard as absolute [because we have] freely
conferred this certainty on it by looking upon it as a convention.
Conventions, yes; arbitrary, no.
(Poincaré, Science and Hypothesis, 1905)
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28. Conclusion
Degrees of truth Utility
similar criticisms
Ordinal revolution
Rigorous notion of utility
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29. Conclusion
Degrees of truth Utility
similar criticisms
Ordinal revolution
Rigorous notion of utility
1. cardinal – ordinal
2. certainty – risk – uncertainty – ambiguity
3. preferences – choices
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30. Conclusion
Degrees of truth Utility
similar criticisms
Ordinal revolution
Rigorous notion of utility
1. cardinal – ordinal
2. certainty – risk – uncertainty – ambiguity
3. preferences – choices
I along the way: many disanalogies!
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31. Conclusion
Degrees of truth Utility
similar criticisms
Ordinal revolution
Rigorous notion of utility
1. cardinal – ordinal
2. certainty – risk – uncertainty – ambiguity
3. preferences – choices
I along the way: many disanalogies!
True and good as primitive notions
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32. References
Pierpaolo Battigalli, Simone Cerreia-Vioglio, Fabio Maccheroni, Massimo Marinacci
Mixed Extensions of Decision Problems under Uncertainty
2013
Rosanna Keefe.
Theories of vagueness,
Cambridge University Press, 2000.
Petr Hájek.
Metamathematics of Fuzzy Logic,
Kluwer Academic Publishers, 1998.
Roberto Marchionatti and Enrico Gambino
Pareto and Political Economy as a Science: Methodological Revolution and Analytical
Advances in Economic Theory in the 1890s
Journal of Political Economy, Vol. 105, No. 6 (December 1997), pp. 1322-1348.
J. von Neumann, O. Morgenstern.
The Theory of Games and Economic Behavior (2nd ed).
Princeton: Princeton University Press, 1947
George J. Stigler.
The Development of Utility Theory. I
The Journal of Political Economy, Vol. 58, No. 4. (Aug., 1950), pp. 307-327.
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