DOCSLIB.ORG

Proofs: Finding Solutions

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Proofs: Finding Solutions Linguistics 280: Language and Formal Reasoning Proofs: finding solutions Robert Levine Autumn Quarter, 2010 Strategies for devising proofs: direct proofs Once you understand how to apply inference rules to formulæ, the next step in modeling reasoning as a series of deductions is to work out just how the rules can get you from a given statement to another statement which is a necessary consequence of the first. Take something very basic: the claim that p ∧ q ⊢ p ∨ q. This is a logically valid statement—it holds every time. When p ∧ q has the value 1, p ∨ q has the value 1, without exception, no matter what sentences of English p and q represent. The truth table for the two formulæ makes this crystal-clear: p q p ∧ q p ∨ q 11 1 1 10 0 1 01 0 1 00 0 0 What the table tells you is that the entailment only goes one way: whenever p ∧ q has the truth value 1, p ∨ q also has the value one, which is what p ∧ q ⊢ p ∨ q means—if we have a proof of p ∧ q, then p ∨ q must be true—but the reverse doesn’t hold: there are two cases in which the disjunction is true, but the conjunction is false. But given that p ∧ q ⊢ p ∨ q does hold, it should be possible—if the Gentzen rules really are the deductive counterpart to the truth tables—to prove this entailment. The trick is figure out what the reasoning steps are. Proofs don’t solve themselves, and staring at the problem for long enough isn’t going to make the answer occur to you. Formal proofs are the purest form of logic puzzle. There is, as you may be aware, an explosion in publishing based on logic puzzles. Amazon.com lists more than 1600 titles in this area; everyone from USA today to Johns Hopkins University Press has published collections, and Soduku—which is a purely deductive number-based logic puzzle genre has probably surpassed crossword puzzles in terms of numbers of ‘players’. But a lot of the logic puzzles that you find in these books involve not just deductive tasks, but misleading or tricky formulations or contexts which lead you to jump to false conclusions, from which it’s impossible to reason to the solution. Formal proofs don’t have this element—everything is laid out in front of you, with no way to deceive you or guide you down a false path. It’s just you against the proof. But without a strategy for solving the proof in front of you, you aren’t going to succeed. And to develop such a strategy you need to think very carefully about what it is that you’re given, and what it is you have to end up with. In this case, you start with a conjunction. What does that tell you? Given a conjunction, we know from the Gentzen rules (themselves based on the truth tables) that in p ∧ q, both p and q must be true. That’s what we’re allowed to deduce from p ∧ q, and when you start a proof such as this one, you must review exactly what it is that you can legitimately infer from the statement you’re asked to assume at the beginning. You then have to go from that knowledge to the conclusion. In the very simple case we’re now looking at, that’s eay. We know that p is true, and we have to get from p to p ∨ q. Are we allowed to do that? Is there an entailment of the form p ⊢ p ∨ q anywhere in our rule system? Look at the rule for ∨ Intro, and you’ll see that that’s exactly what the rule allows: ∨ Intro 1: Γ ⊢ ϕ Γ ⊢ ϕ ∨ ψ where ϕ and ψ range over all possible logical statement. What this means is that we have a green light in our logical setup for the conclusion we want, based on what that setup tells us about conjunctions. All we have to do is set up the proof correctly, as a way of ensuring that it’s really valid. Begin at the beginning, with the startup axiom. We need to get the original premise, p∧q, into play, and the line underneath is a reminder that we’re not just staring at the formula, but are about to make an inference from it: p ∧ q ⊢ p ∧ q | {z } |{z} |{z} Γ ϕ ψ The rule for making inferences from conjunctions is ∧ Elim 1: Γ ⊢ ϕ ∧ ψ Γ ⊢ ϕ According to the rule above, with Γ = {p ∧ q}, ϕ = p, ψ = q, we are allowed to infer p ∧ q ⊢ p: p ∧ q ⊢ p ∧ q | {z } |{z} |{z} Γ ϕ ψ ∧E1 p ∧ q ⊢ p | {z } |{z} Γ ϕ And at this point, the labeling that got us through the inference step from p ∧ q ⊢ p ∧ q to p ∧ q ⊢ p is no longer relevant. As explained in the preceding handout on proofs, one we’ve made an inference from a given assumption set, we need to match the formula we’ve inferred to the rule we want to use. And that may be possible only if we interpret symbols such as Γ, ∆ and others that show in the rule as covering different parts of the formula we wish to apply the next rule to than they did in the previous rule. We therefore need to decide how to treat the next reasoning step so that the result of the previous step can undergo the next rule we wish to apply. We have, so far, p ∧ q ⊢ p ∧ q ∧E1 p ∧ q ⊢ p Based on our earlier informal storyline for the proof, we need to apply the particular ∨ Intro rule given earlier. We take Γ to be {p ∧ q}, ϕ = p, and ψ = q (which happen to be the previous values for these; more about that shortly), and get 2 p ∧ q ⊢ p | {z } |{z} Γ ϕ ∨I1 p ∧ q ⊢ p ∨ q | {z } |{z} |{z} Γ ϕ ψ Suppressing the (now irrelevant) labelling, we have p ∧ q ⊢ p ∨I1 p ∧ q ⊢ p ∨ q By combining the first inference step together with this one, we can display the whole history of the proof, which establishes irrefutably that every step of the proof is licensed by the proof system we’ve adopted: p ∧ q ⊢ p ∧ q ∧E1 p ∧ q ⊢ p ∨I1 p ∧ q ⊢ p ∨ q where the black box notation indicates that the proof is now formally terminated successfully. Trickery: proof by contradiction The strategy followed in the foregoing proof was quite simple, because nothing more indirect was necessary. The rule for ∧ Elimination ‘feeds’ the rule for ∨ Introduction. Such moves are quite common in proofs, but it’s unusual for a proof to consist only of such moves. Sometimes something much more devious is called for—even when the theorem itself seems to be nothing but the essence of common sense. For example, the logical statement ¬p ⊢ ¬(p ∧ q) seems so self-evident as to be trivial: if something is false, the conjunction of it and something else cannot be true. And, unsurprisingly, the truth table show that the statement we’ve been asked to prove is indeed true: p q ¬p p ∧ q ¬(p ∧ q) 110 1 0 100 0 1 011 0 1 001 0 1 When ¬p has 1 as its truth value, so does ¬(p ∧ q), just as we expect. Surely nothing very much is required here by way of proof—one or two lines should do it. But when you first encounter this theorem, it’s far from obvious how to translate the admittedly obvious truth of this statement into a proof. The problem is that you cannot directly relate the negation in ¬p to ¬(p ∧ q). A few moments’ thought should make it clear why: ¬p is definitive—p cannot be true—but ¬(p ∧ q) is not; in principle, the latter can be true even if ¬p itself is not true (as, indeed, the truth table makes evidence). So we cannot proceed directly from the premise ¬p to the conclusion ¬(p ∧ q). How then can the proof go forward? Here’s a general rule of thumb: whenever you see negation on the right side of the turnstile—particularly when that negation applies to a complex formula, as in the present case—it’s going to be difficult to establish 3 the negation directly. But you can almost always achieve such proofs by establishing a contradiction (⊥) and then resolving the contradiction by shifting one of the premises which creates the contradiction over to the right if the turnstile, in negated form. This move is exactly what is mandated by the ¬ Intro rule—we’ve already seen evidence of its effectiveness in proving the theorem p ⊃ q ⊢ ¬q ⊃ ¬p. A very useful tactic in the case of proof by contradiction is to work backwards from the last step to the contradiction which would have preceded it and enabled that last step. It’s clear from the Gentzen rule for ¬ Intro—namely ¬ Intro: Γ, ϕ ⊢⊥ Γ ⊢ ¬ϕ —that if we had a stage in the derivation ¬p,p ∧ q ⊢ ⊥, we obtain the proof we seek directly frome the ¬ Intro rule. In other words, using proof by contraction changes our target: if we can prove ¬p,p ∧ q ⊢ ⊥, then the rest of the proof is trivial. But why should proving this statement be any easier than proving the original statement? Once you’ve worked out a few such proofs, the answer to this question will be clear to you in the sharpest way: with proof by contradiction, you’ve moved the thing you have to prove into the assumption set on the left of the turnstile.
Load more

Recommended publications
  • Tt-Satisfiable
    CMPSCI 601: Recall From Last Time Lecture 6 Boolean Syntax: ¡ ¢¤£¦¥¨§¨£ © §¨£ § Boolean variables: A boolean variable represents an atomic statement that may be either true or false. There may be infinitely many of these available. Boolean expressions: £ atomic: , (“top”), (“bottom”) § ! " # $ , , , , , for Boolean expressions Note that any particular expression is a finite string, and thus may use only finitely many variables. £ £ A literal is an atomic expression or its negation: , , , . As you may know, the choice of operators is somewhat arbitary as long as we have a complete set, one that suf- fices to simulate all boolean functions. On HW#1 we ¢ § § ! argued that is already a complete set. 1 CMPSCI 601: Boolean Logic: Semantics Lecture 6 A boolean expression has a meaning, a truth value of true or false, once we know the truth values of all the individual variables. ¢ £ # ¡ A truth assignment is a function ¢ true § false , where is the set of all variables. An as- signment is appropriate to an expression ¤ if it assigns a value to all variables used in ¤ . ¡ The double-turnstile symbol ¥ (read as “models”) de- notes the relationship between a truth assignment and an ¡ ¥ ¤ expression. The statement “ ” (read as “ models ¤ ¤ ”) simply says “ is true under ”. 2 ¡ ¤ ¥ ¤ If is appropriate to , we define when is true by induction on the structure of ¤ : is true and is false for any , £ A variable is true iff says that it is, ¡ ¡ ¡ ¡ " ! ¥ ¤ ¥ ¥ If ¤ , iff both and , ¡ ¡ ¡ ¡ " ¥ ¤ ¥ ¥ If ¤ , iff either or or both, ¡ ¡ ¡ ¡ " # ¥ ¤ ¥ ¥ If ¤ , unless and , ¡ ¡ ¡ ¡ $ ¥ ¤ ¥ ¥ If ¤ , iff and are both true or both false. 3 Definition 6.1 A boolean expression ¤ is satisfiable iff ¡ ¥ ¤ there exists .
    [Show full text]
  • Chapter 9: Initial Theorems About Axiom System
    Initial Theorems about Axiom 9 System AS1 1. Theorems in Axiom Systems versus Theorems about Axiom Systems ..................................2 2. Proofs about Axiom Systems ................................................................................................3 3. Initial Examples of Proofs in the Metalanguage about AS1 ..................................................4 4. The Deduction Theorem.......................................................................................................7 5. Using Mathematical Induction to do Proofs about Derivations .............................................8 6. Setting up the Proof of the Deduction Theorem.....................................................................9 7. Informal Proof of the Deduction Theorem..........................................................................10 8. The Lemmas Supporting the Deduction Theorem................................................................11 9. Rules R1 and R2 are Required for any DT-MP-Logic........................................................12 10. The Converse of the Deduction Theorem and Modus Ponens .............................................14 11. Some General Theorems About ......................................................................................15 12. Further Theorems About AS1.............................................................................................16 13. Appendix: Summary of Theorems about AS1.....................................................................18 2 Hardegree,
    [Show full text]
  • Chapter 13: Formal Proofs and Quantifiers
    Chapter 13: Formal Proofs and Quantifiers § 13.1 Universal quantifier rules Universal Elimination (∀ Elim) ∀x S(x) ❺ S(c) Here x stands for any variable, c stands for any individual constant, and S(c) stands for the result of replacing all free occurrences of x in S(x) with c. Example 1. ∀x ∃y (Adjoins(x, y) ∧ SameSize(y, x)) 2. ∃y (Adjoins(b, y) ∧ SameSize(y, b)) ∀ Elim: 1 General Conditional Proof (∀ Intro) ✾c P(c) Where c does not occur outside Q(c) the subproof where it is introduced. ❺ ∀x (P(x) → Q(x)) There is an important bit of new notation here— ✾c , the “boxed constant” at the beginning of the assumption line. This says, in effect, “let’s call it c.” To enter the boxed constant in Fitch, start a new subproof and click on the downward pointing triangle ❼. This will open a menu that lets you choose the constant you wish to use as a name for an arbitrary object. Your subproof will typically end with some sentence containing this constant. In giving the justification for the universal generalization, we cite the entire subproof (as we do in the case of → Intro). Notice that although c may not occur outside the subproof where it is introduced, it may occur again inside a subproof within the original subproof. Universal Introduction (∀ Intro) ✾c Where c does not occur outside the subproof where it is P(c) introduced. ❺ ∀x P(x) Remember, any time you set up a subproof for ∀ Intro, you must choose a “boxed constant” on the assumption line of the subproof, even if there is no sentence on the assumption line.
    [Show full text]
  • Logic and Proof Release 3.18.4
    Logic and Proof Release 3.18.4 Jeremy Avigad, Robert Y. Lewis, and Floris van Doorn Sep 10, 2021 CONTENTS 1 Introduction 1 1.1 Mathematical Proof ............................................ 1 1.2 Symbolic Logic .............................................. 2 1.3 Interactive Theorem Proving ....................................... 4 1.4 The Semantic Point of View ....................................... 5 1.5 Goals Summarized ............................................ 6 1.6 About this Textbook ........................................... 6 2 Propositional Logic 7 2.1 A Puzzle ................................................. 7 2.2 A Solution ................................................ 7 2.3 Rules of Inference ............................................ 8 2.4 The Language of Propositional Logic ................................... 15 2.5 Exercises ................................................. 16 3 Natural Deduction for Propositional Logic 17 3.1 Derivations in Natural Deduction ..................................... 17 3.2 Examples ................................................. 19 3.3 Forward and Backward Reasoning .................................... 20 3.4 Reasoning by Cases ............................................ 22 3.5 Some Logical Identities .......................................... 23 3.6 Exercises ................................................. 24 4 Propositional Logic in Lean 25 4.1 Expressions for Propositions and Proofs ................................. 25 4.2 More commands ............................................
    [Show full text]
  • 1 Symbols (2286)
    1 Symbols (2286) USV Symbol Macro(s) Description 0009 \textHT <control> 000A \textLF <control> 000D \textCR <control> 0022 ” \textquotedbl QUOTATION MARK 0023 # \texthash NUMBER SIGN \textnumbersign 0024 $ \textdollar DOLLAR SIGN 0025 % \textpercent PERCENT SIGN 0026 & \textampersand AMPERSAND 0027 ’ \textquotesingle APOSTROPHE 0028 ( \textparenleft LEFT PARENTHESIS 0029 ) \textparenright RIGHT PARENTHESIS 002A * \textasteriskcentered ASTERISK 002B + \textMVPlus PLUS SIGN 002C , \textMVComma COMMA 002D - \textMVMinus HYPHEN-MINUS 002E . \textMVPeriod FULL STOP 002F / \textMVDivision SOLIDUS 0030 0 \textMVZero DIGIT ZERO 0031 1 \textMVOne DIGIT ONE 0032 2 \textMVTwo DIGIT TWO 0033 3 \textMVThree DIGIT THREE 0034 4 \textMVFour DIGIT FOUR 0035 5 \textMVFive DIGIT FIVE 0036 6 \textMVSix DIGIT SIX 0037 7 \textMVSeven DIGIT SEVEN 0038 8 \textMVEight DIGIT EIGHT 0039 9 \textMVNine DIGIT NINE 003C < \textless LESS-THAN SIGN 003D = \textequals EQUALS SIGN 003E > \textgreater GREATER-THAN SIGN 0040 @ \textMVAt COMMERCIAL AT 005C \ \textbackslash REVERSE SOLIDUS 005E ^ \textasciicircum CIRCUMFLEX ACCENT 005F _ \textunderscore LOW LINE 0060 ‘ \textasciigrave GRAVE ACCENT 0067 g \textg LATIN SMALL LETTER G 007B { \textbraceleft LEFT CURLY BRACKET 007C | \textbar VERTICAL LINE 007D } \textbraceright RIGHT CURLY BRACKET 007E ~ \textasciitilde TILDE 00A0 \nobreakspace NO-BREAK SPACE 00A1 ¡ \textexclamdown INVERTED EXCLAMATION MARK 00A2 ¢ \textcent CENT SIGN 00A3 £ \textsterling POUND SIGN 00A4 ¤ \textcurrency CURRENCY SIGN 00A5 ¥ \textyen YEN SIGN 00A6
    [Show full text]
  • Data Stream Algorithms Lecture Notes
    CS49: Data Stream Algorithms Lecture Notes, Fall 2011 Amit Chakrabarti Dartmouth College Latest Update: October 14, 2014 DRAFT Acknowledgements These lecture notes began as rough scribe notes for a Fall 2009 offering of the course “Data Stream Algorithms” at Dartmouth College. The initial scribe notes were prepared mostly by students enrolled in the course in 2009. Subsequently, during a Fall 2011 offering of the course, I edited the notes heavily, bringing them into presentable form, with the aim being to create a resource for students and other teachers of this material. I would like to acknowledge the initial effort by the 2009 students that got these notes started: Radhika Bhasin, Andrew Cherne, Robin Chhetri, Joe Cooley, Jon Denning, Alina Dja- mankulova, Ryan Kingston, Ranganath Kondapally, Adrian Kostrubiak, Konstantin Kutzkow, Aarathi Prasad, Priya Natarajan, and Zhenghui Wang. DRAFT Contents 0 Preliminaries: The Data Stream Model 5 0.1 TheBasicSetup................................... ........... 5 0.2 TheQualityofanAlgorithm’sAnswer . ................. 5 0.3 VariationsoftheBasicSetup . ................ 6 1 Finding Frequent Items Deterministically 7 1.1 TheProblem...................................... .......... 7 1.2 TheMisra-GriesAlgorithm. ............... 7 1.3 AnalysisoftheAlgorithm . .............. 7 2 Estimating the Number of Distinct Elements 9 2.1 TheProblem...................................... .......... 9 2.2 TheAlgorithm .................................... .......... 9 2.3 TheQualityoftheAlgorithm’sEstimate . ..................
    [Show full text]
  • 1. Latex and Symbolic Logic: an Introduction
    Symbolic Logic and LATEX David W. Agler June 21, 2013 1 Introduction This document introduces some features of LATEX, the special symbols you will need in Symbolic Logic (PHIL012), and some reasons for why you should use LATEX over traditional word processing programs. This video also ac- companies several video tutorials on how to use LATEX in the Symbolic Logic (PHIL012) course at Penn State. 2 LATEX is not a word processor LATEX is a typesetting system that was originally designed to produce docu- ments with special symbols. LATEX is not a word processor which coordinates text with styling simultaneously. Instead, the LATEX system separates text from styling as it consists of (i) a text document (or *.tex file) which is text that does not contain any formatting and (ii) a compiler that takes the *.tex file and turns it into a readable and professionally stylized document. In other words, the production of a document using LATEX begins with the specification of what you want to say and the structure of what you want to say and this is then processed by a compiler which styles your document. To get a clearer idea of what some of this means, let's look at a very simple LATEX source document: \documentclass[12pt]{article} \begin{document} \title{Symbolic Logic and \LaTeX\ } \author{David W. Agler} 1 \date{\today} \maketitle \section{Introduction} Hello Again World! \end{document} The above specifies the class (or kind) of document you want to produce (an article, rather than a book or letter), commands for begining the document, the title of the article, the author of the article, the date, a command to make the title, a section that will automatically be numbered, some content that will appear as text (\Hello World"), and finally a command to end the document.
    [Show full text]
  • Non-Adaptive Adaptive Sampling on Turnstile Streams
    Non-Adaptive Adaptive Sampling on Turnstile Streams Sepideh Mahabadi∗ Ilya Razenshteyn† David P. Woodruff‡ Samson Zhou§ April 24, 2020 Abstract Adaptive sampling is a useful algorithmic tool for data summarization problems in the clas- sical centralized setting, where the entire dataset is available to the single processor performing the computation. Adaptive sampling repeatedly selects rows of an underlying matrix A Rn×d, where n d, with probabilities proportional to their distances to the subspace of the previously∈ selected rows.≫ Intuitively, adaptive sampling seems to be limited to trivial multi-pass algorithms in the streaming model of computation due to its inherently sequential nature of assigning sam- pling probabilities to each row only after the previous iteration is completed. Surprisingly, we show this is not the case by giving the first one-pass algorithms for adaptive sampling on turnstile streams and using space poly(d, k, log n), where k is the number of adaptive sampling rounds to be performed. Our adaptive sampling procedure has a number of applications to various data summariza- tion problems that either improve state-of-the-art or have only been previously studied in the more relaxed row-arrival model. We give the first relative-error algorithm for column subset selection on turnstile streams. We show our adaptive sampling algorithm also gives the first relative-error algorithm for subspace approximation on turnstile streams that returns k noisy rows of A. The quality of the output can be improved to a (1+ǫ)-approximation at the tradeoff of a bicriteria algorithm that outputs a larger number of rows.
    [Show full text]
  • Basics of Type Theory and Coq Logic Types ∧, ∨, ⇒, ¬, ∀, ∃ ×, +, →, Q, P Michael Shulman
    Type-theoretic foundations Set theory Type theory Basics of type theory and Coq Logic Types ^; _; ); :; 8; 9 ×; +; !; Q; P Michael Shulman Sets Logic January 31, 2012 ×; +; !; Q; P ^; _; ); :; 8; 9 x 2 A is a proposition x : A is a typing judgment 1 / 77 2 / 77 Type theory is programming Typing judgments Type theory consists of rules for manipulating judgments. The most important judgment is a typing judgment: x1 : A1; x2 : A2;::: xn : An ` b : B For now, think of type theory as a programming language. The turnstile ` binds most loosely, followed by commas. • Closely related to functional programming languages like This should be read as: ML, Haskell, Lisp, Scheme. In the context of variables x of type A , x of type A ,..., • More expressive and powerful. 1 1 2 2 and xn of type An, the expression b has type B. • Can manipulate “mathematical objects”. Examples ` 0: N x : N; y : N ` x + y : N f : R ! R; x : R ` f (x): R 1 (n) 1 f : C (R; R); n: N ` f : C (R; R) 4 / 77 5 / 77 Type constructors Derivations The basic rules tell us how to construct valid typing judgments, We write these rules as follows. i.e. how to write programs with given input and output types. This includes: ` A: Type ` B : Type 1 How to construct new types (judgments Γ ` A: Type). ` A ! B : Type 2 How to construct terms of these types. 3 How to use such terms to construct terms of other types. x : A ` b : B A Example (Function types) ` λx :b : A ! B 1 If A: Type and B : Type, then A ! B : Type.
    [Show full text]
  • Class Code List Report
    Class Code List Report MAJOR INTERMEDIATE MINOR DESCRIPTION 0001 FOU 3676 0007 007 A FRAME VEHICLE MOUNTING 0007 014 A FRAME VEHICLE MOUNT TRUCK 0010 BAC 0555 0021 002 ABSORBER ALUMINUM 0035 005 ACCELERATOR LINEAR 0035 007 ACCELERATOR READING 0035 015 ACCELERATOR NETWORK PATH 0038 007 ACCELEROMETER 0049 004 ACCESSORY CRYOGENIC 0049 007 ACCESSORY LAN SYSTEM 0049 010 ACCESSORY, STRAIN GAUGE 0058 007 ACQUISITION SYSTEM 0091 003 ADAPTER AC 0091 014 ADAPTER CAMERA 0091 018 ADAPTER COMPUTER 0091 021 ADAPTER ENLARGER 0091 023 ADAPTER ETHERNET 0091 038 ADAPTER LINE SET 0091 042 ADAPTER MEDICAL INSTRUMENT 0091 044 ADAPTER MICROCASSETTE 0091 045 ADAPTER MICROPHONE 0091 049 ADAPTER MICROSCOPE 0091 056 ADAPTER MONITOR 0091 058 ADAPTER PLAYBACK 0091 060 ADAPTER PROJECTOR 0091 062 ADAPTER RADIO 0091 063 ADAPTER RECORDING 0091 070 ADAPTER SLIDE 0091 075 ADAPTER STATIC RAM 0091 090 ADAPTER TELEVISION 0091 095 ADAPTER TERMINAL 0091 100 ADAPTER TRANSPARENCY 0105 007 ADDRESSOGRAPH 0119 007 AERATOR 0126 007 AERIFIER 0140 021 AEROGRAPH 0142 007 AESTHESIOMETER 0144 007 AFTERCOOLER 0146 007 AGGREGOMETER 0147 007 AGITATOR 0150 007 AID ELECTRONIC VOCAL 0150 008 AID HEARING 0150 009 AID THERAPUTICAL 0151 004 AIDE DELTOID 0154 007 AIR CONDITIONER 0154 014 AIR CONDITIONING UNIT 0154 021 AIR HANDLER 0161 007 AIR TRACK 0168 007 AIRBRUSH 0179 007 AIRPAK Printed on 11/5/2010 at 1:23:00PM Page 1 of 139 MAJOR INTERMEDIATE MINOR DESCRIPTION 0182 007 AIRPLANE 0189 003 ALARM AUTO 0189 007 ALARM BURGLAR 0189 014 ALARM FIRE 0189 021 ALARM INTRUDER 0189 022 ALARM SYSTEM ERROR
    [Show full text]
  • An Introduction to Type Theory Outline
    An Introduction to Type Theory Dan Christensen UWO, May 8, 2014 Outline: A brief history of types Type theory as set theory Type theory as logic History of Type Theory Initial ideas due to Bertrand Russell in early 1900's, to create a foundation for mathematics that avoids Russell's paradox. Studied by many logicians and later computer scientists, particularly Church, whose λ-calculus (1930's and 40's) is a type theory. In 1972, Per Martin-L¨ofextended type theory to include dependent types. It is this form of type theory that we will focus on. One of the key features is that it unifies set theory and logic! In 2006, Awodey and Warren, and Voevodsky, discovered that type theory has homotopical models. Understanding this interpretation will be the goal of the overall seminar, but not of this talk. 2012{2013: a special year at the IAS, which led to a book. Last week: $7.5M Dept of Defense grant for a group led by Awodey. Type theory as set theory: Notation Type theory studies things called types, which can be thought of as sets. The analog of elements of a set are called terms. We write a : A to indicate that a is a term of type A. And we write A : Type to indicate that A is a type. A judgement is a string of symbols that may or may not be provable from the rules of type theory. A judgement always contains the turnstile symbol ` , separating context from the conclusion. Examples (using things we haven't defined yet): ` 0 : N x : N; y : N ` x + y : N f : R ! R; x : R ` f(x): R Type theory as set theory: more judgements Judgements can also involve asserting that something is a type.
    [Show full text]
  • A Package for Logical Symbols
    A Package for Logical Symbols Rett Bull October 10, 2009 The package logicsym supplies logical symbols for classes like Pomona Col- lege's Math 123, CS 80, and CS 81. Known conflicts: the logicsym package redefines existing symbols \mp (minus- plus) and \Re. 1 Options and Customization 1.1 Compatiblity This version introduces a number of name changes|for consistency. Use the option oldnames to add the previous names. 1.2 Truth Constants There are three options for truth constants: topbottom, tfletters, and tfwords. The commands \False and \True may be used in any mode. The default font for words and letters is italic, more specifically \mathit, but it may be changed by redefining \Truthconstantfont. The default is topbottom. Only one of the three options may be used. 1.3 Conjunction The options for conjunction are wedge and ampersand. The default is wedge. 1.4 Implication The choices for arrows are singlearrows or doublearrows. The default is doublearrows. The \if and only if" connective matches the arrow style but can be replaced with ≡ by specifying the option equivalence. 1.5 Turnstiles There is no option for turnstiles, but their size (relative to other symbols) may be adjusted by redefining \TurnstileScale. The default is 1.0. 1 1.6 Crossing Out In natural deduction inferences, we often want to show that an assumption has been discharged by crossing it out. A slash often works well for single letters, but it is too narrow for longer formulas. We provide a macro \Crossbox to make a wide slash, but unfortunately it is difficult to draw a simple diagonal across a TEX box.
    [Show full text]