Explanation-Based Learning and the Logic of Principia Mathematica
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Basic Concepts of Set Theory, Functions and Relations 1. Basic
Ling 310, adapted from UMass Ling 409, Partee lecture notes March 1, 2006 p. 1 Basic Concepts of Set Theory, Functions and Relations 1. Basic Concepts of Set Theory........................................................................................................................1 1.1. Sets and elements ...................................................................................................................................1 1.2. Specification of sets ...............................................................................................................................2 1.3. Identity and cardinality ..........................................................................................................................3 1.4. Subsets ...................................................................................................................................................4 1.5. Power sets .............................................................................................................................................4 1.6. Operations on sets: union, intersection...................................................................................................4 1.7 More operations on sets: difference, complement...................................................................................5 1.8. Set-theoretic equalities ...........................................................................................................................5 Chapter 2. Relations and Functions ..................................................................................................................6 -
Are Large Cardinal Axioms Restrictive?
Are Large Cardinal Axioms Restrictive? Neil Barton∗ 24 June 2020y Abstract The independence phenomenon in set theory, while perva- sive, can be partially addressed through the use of large cardinal axioms. A commonly assumed idea is that large cardinal axioms are species of maximality principles. In this paper, I argue that whether or not large cardinal axioms count as maximality prin- ciples depends on prior commitments concerning the richness of the subset forming operation. In particular I argue that there is a conception of maximality through absoluteness, on which large cardinal axioms are restrictive. I argue, however, that large cardi- nals are still important axioms of set theory and can play many of their usual foundational roles. Introduction Large cardinal axioms are widely viewed as some of the best candi- dates for new axioms of set theory. They are (apparently) linearly ordered by consistency strength, have substantial mathematical con- sequences for questions independent from ZFC (such as consistency statements and Projective Determinacy1), and appear natural to the ∗Fachbereich Philosophie, University of Konstanz. E-mail: neil.barton@uni- konstanz.de. yI would like to thank David Aspero,´ David Fernandez-Bret´ on,´ Monroe Eskew, Sy-David Friedman, Victoria Gitman, Luca Incurvati, Michael Potter, Chris Scam- bler, Giorgio Venturi, Matteo Viale, Kameryn Williams and audiences in Cambridge, New York, Konstanz, and Sao˜ Paulo for helpful discussion. Two anonymous ref- erees also provided helpful comments, and I am grateful for their input. I am also very grateful for the generous support of the FWF (Austrian Science Fund) through Project P 28420 (The Hyperuniverse Programme) and the VolkswagenStiftung through the project Forcing: Conceptual Change in the Foundations of Mathematics. -
THE 1910 PRINCIPIA's THEORY of FUNCTIONS and CLASSES and the THEORY of DESCRIPTIONS*
EUJAP VOL. 3 No. 2 2007 ORIGinal SCienTifiC papeR UDK: 165 THE 1910 PRINCIPIA’S THEORY OF FUNCTIONS AND CLASSES AND THE THEORY OF DESCRIPTIONS* WILLIAM DEMOPOULOS** The University of Western Ontario ABSTRACT 1. Introduction It is generally acknowledged that the 1910 Prin- The 19101 Principia’s theory of proposi- cipia does not deny the existence of classes, but tional functions and classes is officially claims only that the theory it advances can be developed so that any apparent commitment to a “no-classes theory of classes,” a theory them is eliminable by the method of contextual according to which classes are eliminable. analysis. The application of contextual analysis But it is clear from Principia’s solution to ontological questions is widely viewed as the to the class paradoxes that although the central philosophical innovation of Russell’s theory of descriptions. Principia’s “no-classes theory it advances holds that classes are theory of classes” is a striking example of such eliminable, it does not deny their exis- an application. The present paper develops a re- tence. Whitehead and Russell argue from construction of Principia’s theory of functions the supposition that classes involve or and classes that is based on Russell’s epistemo- logical applications of the method of contextual presuppose propositional functions to the analysis. Such a reconstruction is not eliminativ- conclusion that the paradoxical classes ist—indeed, it explicitly assumes the existence of are excluded by the nature of such func- classes—and possesses certain advantages over tions. This supposition rests on the repre- the no–classes theory advocated by Whitehead and Russell. -
AI Thinks Like a Corporation—And That's Worrying
Topics Current edition More Subscribe Log in or sign up Search Open Future Manage subscription Open Voices AI thinks like a corporation—and that’s worrying Arti!cial intelligence was born of organisational decision-making and state power; it needs human ethics, says Jonnie Penn of the University of Cambridge Open Future Nov 26th 2018 | by BY JONNIE PENN Arti!cial intelligence is everywhere but it is considered in a wholly ahistorical way. To understand the impact AI will have on our lives, it is vital to appreciate the context in which the !eld was established. After all, statistics and state control have evolved hand in hand for hundreds of years. Consider computing. Its origins have been traced not only to analytic philosophy, pure mathematics and Alan Turing, but perhaps surprisingly, to the history of public administration. In “The Government Machine: A Revolutionary History of the Computer” from 2003, Jon Agar of University College London charts the development of the British civil service as it ballooned from 16,000 employees in 1797 to 460,000 by 1999. He noticed an uncanny similarity between the functionality of a human bureaucracy and that of the digital electronic computer. (He confessed that he could not tell whether this observation was trivial or profound.) Get our daily newsletter Upgrade your inbox and get our Daily Dispatch and Editor's Picks. Email address Sign up now Both systems processed large quantities of information using a hierarchy of pre-set but adaptable rules. Yet one predated the other. This suggested a telling link between the organisation of human social structures and the digital tools designed to serve them. -
How Peircean Was the “'Fregean' Revolution” in Logic?
HOW PEIRCEAN WAS THE “‘FREGEAN’ REVOLUTION” IN LOGIC? Irving H. Anellis Peirce Edition, Institute for American Thought Indiana University – Purdue University at Indianapolis Indianapolis, IN, USA [email protected] Abstract. The historiography of logic conceives of a Fregean revolution in which modern mathematical logic (also called symbolic logic) has replaced Aristotelian logic. The preeminent expositors of this conception are Jean van Heijenoort (1912–1986) and Don- ald Angus Gillies. The innovations and characteristics that comprise mathematical logic and distinguish it from Aristotelian logic, according to this conception, created ex nihlo by Gottlob Frege (1848–1925) in his Begriffsschrift of 1879, and with Bertrand Rus- sell (1872–1970) as its chief This position likewise understands the algebraic logic of Augustus De Morgan (1806–1871), George Boole (1815–1864), Charles Sanders Peirce (1838–1914), and Ernst Schröder (1841–1902) as belonging to the Aristotelian tradi- tion. The “Booleans” are understood, from this vantage point, to merely have rewritten Aristotelian syllogistic in algebraic guise. The most detailed listing and elaboration of Frege’s innovations, and the characteristics that distinguish mathematical logic from Aristotelian logic, were set forth by van Heijenoort. I consider each of the elements of van Heijenoort’s list and note the extent to which Peirce had also developed each of these aspects of logic. I also consider the extent to which Peirce and Frege were aware of, and may have influenced, one another’s logical writings. AMS (MOS) 2010 subject classifications: Primary: 03-03, 03A05, 03C05, 03C10, 03G27, 01A55; secondary: 03B05, 03B10, 03E30, 08A20; Key words and phrases: Peirce, abstract algebraic logic; propositional logic; first-order logic; quantifier elimina- tion, equational classes, relational systems §0. -
Foundations of Mathematics
Foundations of Mathematics. Does mathematics need foundations? (Not until 1900.) Mathematical approach: Work towards an axiom system of mathematics with purely mathematical means. ( Hilbert’s Programme ). In its naïve interpretation crushed by Gödel’s Incompleteness Theorem. Extra-mathematical approach: Use external arguments for axioms and rules: pragmatic, philosophical, sociological, (theological ?). Foundations of number theory: test case. Core Logic – 2007/08-1ab – p. 2/36 Sets are everything (1). Different areas of mathematics use different primitive notions: ordered pair, function, natural number, real number, transformation, etc. Set theory is able to incorporate all of these in one framework: Ordered Pair. We define hx, y i := {{ x}, {x, y }} . (Kuratowski pair ) Function. A set f is called a function if there are sets X and Y such that f ⊆ X × Y and ′ ′ ′ ∀x, y, y hx, y i ∈ f&hx, y i ∈ f → y = y . Core Logic – 2007/08-1ab – p. 3/36 Sets are everything (2). Set theory incorporates basic notions of mathematics: Natural Numbers. We call a set X inductive if it contains ∅ and for each x ∈ X, we have x ∪ { x} ∈ X. Assume that there is an inductive set. Then define N to be the intersection of all inductive sets. Rational Numbers. We define P := {0, 1} × N × N\{ 0}, then hi, n, m i ∼ h j, k, ℓ i : ⇐⇒ i = j & n · ℓ = m · k, and Q := P/∼. Core Logic – 2007/08-1ab – p. 4/36 Sets are everything (3). Set theory incorporates basic notions of mathematics: Real Numbers. Define an order on Q by hi, n, m i ≤ h j, k, ℓ i : ⇐⇒ i < j ∨ (i = j & n · ℓ ≤ k · m). -
CAP 5636 - Advanced Artificial Intelligence
CAP 5636 - Advanced Artificial Intelligence Introduction This slide-deck is adapted from the one used by Chelsea Finn at CS221 at Stanford. CAP 5636 Instructor: Lotzi Bölöni http://www.cs.ucf.edu/~lboloni/ Slides, homeworks, links etc: http://www.cs.ucf.edu/~lboloni/Teaching/CAP5636_Fall2021/index.html Class hours: Tue, Th 12:00PM - 1:15PM COVID considerations: UCF expects you to get vaccinated and wear a mask Classes will be in-person, but will be recorded on Zoom. Office hours will be over Zoom. Motivating artificial intelligence It is generally not hard to motivate AI these days. There have been some substantial success stories. A lot of the triumphs have been in games, such as Jeopardy! (IBM Watson, 2011), Go (DeepMind’s AlphaGo, 2016), Dota 2 (OpenAI, 2019), Poker (CMU and Facebook, 2019). On non-game tasks, we also have systems that achieve strong performance on reading comprehension, speech recognition, face recognition, and medical imaging benchmarks. Unlike games, however, where the game is the full problem, good performance on a benchmark does not necessarily translate to good performance on the actual task in the wild. Just because you ace an exam doesn’t necessarily mean you have perfect understanding or know how to apply that knowledge to real problems. So, while promising, not all of these results translate to real-world applications Dangers of AI From the non-scientific community, we also see speculation about the future: that it will bring about sweeping societal change due to automation, resulting in massive job loss, not unlike the industrial revolution, or that AI could even surpass human-level intelligence and seek to take control. -
Russell's Paradox: Let a Be the Set of All Sets Which Do Not Contain
Russell’s paradox: Let A be the set of all sets which do not contain themselves = {S | S 6∈ S} Ex: {1} ∈ {{1}, {1, 2}}, but {1} 6∈ {1} Is A∈A? Suppose A∈A. Then by definition of A, A 6∈ A. Suppose A 6∈ A. Then by definition of A, A∈A. Thus we need axioms in order to create mathematical objects. Principia Mathematica by Alfred North Whitehead and Bertrand Russell From: http://plato.stanford.edu/entries/principia-mathematica/ Logicism is the view that (some or all of) mathematics can be reduced to (formal) logic. It is often explained as a two-part thesis. First, it consists of the claim that all mathematical truths can be translated into logical truths or, in other words, that the vocabulary of mathematics constitutes a proper subset of the vocabulary of logic. Second, it consists of the claim that all mathematical proofs can be recast as logical proofs or, in other words, that the theorems of mathematics constitute a proper subset of the theorems of logic. In Bertrand Russell’s words, it is the logicist’s goal ”to show that all pure mathematics follows from purely logical premises and uses only concepts definable in logical terms.”[1] From: http://www.math.vanderbilt.edu/∼schectex/ccc/choice.html Axiom of Choice. Let C be a collection of nonempty sets. Then we can choose a member from each set in that collection. In other words, there exists a function f defined on C with the property that, for each set S in the collection, f(S) is a member of S. -
The Logicist Manifesto: at Long Last Let Logic-Based Artificial Intelligence Become a Field Unto Itself∗
The Logicist Manifesto: At Long Last Let Logic-Based Artificial Intelligence Become a Field Unto Itself∗ Selmer Bringsjord Rensselaer AI & Reasoning (RAIR) Lab Department of Cognitive Science Department of Computer Science Rensselaer Polytechnic Institute (RPI) Troy NY 12180 USA [email protected] version 9.18.08 Contents 1 Introduction 1 2 Background 1 2.1 Logic-Based AI Encapsulated . .1 2.1.1 LAI is Ambitious . .3 2.1.2 LAI is Based on Logical Systems . .4 2.1.3 LAI is a Top-Down Enterprise . .5 2.2 Ignoring the \Strong" vs. \Weak" Distinction . .5 2.3 A Slice in the Day of a Life of a LAI Agent . .6 2.3.1 Knowledge Representation in Elementary Logic . .8 2.3.2 Deductive Reasoning . .8 2.3.3 A Note on Nonmonotonic Reasoning . 12 2.3.4 Beyond Elementary Logical Systems . 13 2.4 Examples of Logic-Based Cognitive Systems . 15 3 Factors Supporting Logicist AI as an Independent Field 15 3.1 History Supports the Divorce . 15 3.2 The Advent of the Web . 16 3.3 The Remarkable Effectiveness of Logic . 16 3.4 Logic Top to Bottom Now Possible . 17 3.5 Learning and Denial . 19 3.6 Logic is an Antidote to \Cheating" in AI . 19 3.7 Logic Our Only Hope Against the Dark AI Future . 21 4 Objections; Rebuttals 22 4.1 \But you are trapped in a fundamental dilemma: your position is either redundant, or false." . 22 4.2 \But you're neglecting probabilistic AI." . 23 4.3 \But we now know that the mind, contra logicists, is continuous, and hence dynamical, not logical, systems are superior." . -
The Substitutional Paradox in Russell's 1907 Letter to Hawtrey
THE SUBSTITUTIONAL PARADOX IN RUSSELL’S 1907 LETTER TO HAWTREY B L Philosophy / U. of Alberta Edmonton, , Canada .@. This note presents a transcription of Russell’s letter to Hawtrey of January accompanied by some proposed emendations. In that letter Russell describes the paradox that he says “pilled” the “substitutional theory” devel- oped just before he turned to the theory of types. A close paraphrase of the deri- vation of the paradox in a contemporary Lemmon-style natural deduction system shows which axioms the theory must assume to govern its characteristic notion of substituting individuals and propositions for each other in other propositions. Other discussions of this paradox in the literature are mentioned. I conclude with remarks about the significance of the paradox for Russell. n the years to Bertrand Russell worked on what is now called the “substitutional theory” with its primitive notion of substi- Ituting one entity for another in a proposition, as a foundation for logic and source of a solution to the paradoxes. Russell abandoned that approach quite abruptly and returned to a logic based on propositional functions, eventually to appear as Principia Mathematica. Almost all of the material on the substitutional theory has remained unpublished; however, much will appear in print as the subject matter of Volume of the Collected Papers of Bertrand Russell. In recent years there has been some discussion of the substitutional theory, most prominently by Including, one may hope, a transcription of the letter which is the topic of this note. russell: the Journal of Bertrand Russell Studies n.s. (winter –): – The Bertrand Russell Research Centre, McMaster U. -
Frameworks for Intelligent Systems
Frameworks for Intelligent Systems CompSci 765 Meeting 3 Pat Langley Department of Computer Science University of Auckland Outline of the Lecture • Computer science as an empirical discipline • Physical symbol systems • List structures and list processing • Reasoning and intelligence • Intelligence and search • Knowledge and intelligence • Implications for social cognition 2 Computer Science as an Empirical Discipline! In their Turing Award article, Newell and Simon (1976) make some important claims: • Computer science is an empirical discipline, rather than a branch of mathematics. • It is a science of the artificial, in that it constructs artifacts of sufficient complexity that formal analysis is not tractable. • We must study these computational artifacts as if they were natural systems, forming hypotheses and collecting evidence. They propose two hypotheses based on their founding work in list processing and artificial intelligence.! 3 Laws of Qualitative Structure! The authors introduce the idea of laws of qualitative structure, which are crucial for any scientific field’s development: • The cell doctrine in biology • Plate tectonics in geology • The germ theory of disease • The atomic theory of matter They propose two such laws, one related to mental structures and another and the other to mental processes. 4 Physical Symbol Systems! Newell and Simon’s first claim, the physical symbol system hypothesis, states that: • A physical symbol system has the necessary and sufficient means for general intelligent action. They emphasize general cognitive abilities, such as humans exhibit, rather than specialized ones. This is a theoretical claim that is subject to empirical tests, but the evidence to date generally supports it.! 5 More on Physical Symbol Systems! What do Newell and Simon mean by a physical symbol system? • Symbols are physical patterns that are stable unless modified. -
Weyl's `Das Kontinuum' — 100 Years Later
Arnon Avron Weyl's `Das Kontinuum' | 100 years later Orevkov'80 Conference St. Petersburg Days of Logic and Computability V April 2020 Prologue All platonists are alike; each anti-platonist is unhappy in her/his own way... My aim in this talk is first of all to present Weyl's views and system, at the time he wrote \Das Kontinuum" (exactly 100 years ago). Then I'll try to describe mine, which I believe are rather close to Weyl's original ideas (but still different). Weyl's Goals \I shall show that the house of analysis is to a large degree built on sand. I believe that I can replace this shifting foundation with pillars of enduring strength. They will not, however, support everything which today is generally considered to be securely grounded. I give up the rest, since I see no other possibility." \I would like to be understood . by all students who have become acquainted with the currently canonical and al- legedly `rigorous' foundations of analysis." \In spite of Dedekind, Cantor, and Weierstrass, the great task which has been facing us since the Pythagorean dis- covery of the irrationals remains today as unfinished as ever" Weyl and P´olya's Wager in 1918 Within 20 years P´olya and the majority of representative mathematicians will admit that the statements 1 Every bounded set of reals has a precise supremum 2 Every infinite set of numbers contains a denumerable subset contain totally vague concepts, such as \number," \set," and \denumerable," and therefore that their truth or fal- sity has the same status as that of the main propositions of Hegel's natural philosophy.