DOCSLIB.ORG

Sets, Propositional Logic, Predicates, and Quantifiers

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

COMP 182 Algorithmic Thinking Sets, Propositional Logic, Luay Nakhleh Computer Science Predicates, and Quantifiers Rice University !1 Reading Material ❖ Chapter 1, Sections 1, 4, 5 ❖ Chapter 2, Sections 1, 2 !2 ❖ Mathematics is about statements that are either true or false. ❖ Such statements are called propositions. ❖ We use logic to describe them, and proof techniques to prove whether they are true or false. !3 Propositions ❖ 5>7 ❖ The square root of 2 is irrational. ❖ A graph is bipartite if and only if it doesn’t have a cycle of odd length. ❖ For n>1, the sum of the numbers 1,2,3,…,n is n2. !4 Propositions? ❖ E=mc2 ❖ The sun rises from the East every day. ❖ All species on Earth evolved from a common ancestor. ❖ God does not exist. ❖ Everyone eventually dies. !5 ❖ And some of you might already be wondering: “If I wanted to study mathematics, I would have majored in Math. I came here to study computer science.” !6 ❖ Computer Science is mathematics, but we almost exclusively focus on aspects of mathematics that relate to computation (that can be implemented in software and/or hardware). !7 ❖Logic is the language of computer science and, mathematics is the computer scientist’s most essential toolbox. !8 Examples of “CS-relevant” Math ❖ Algorithm A correctly solves problem P. ❖ Algorithm A has a worst-case running time of O(n3). ❖ Problem P has no solution. ❖ Using comparison between two elements as the basic operation, we cannot sort a list of n elements in less than O(n log n) time. ❖ Problem A is NP-Complete. !9 ❖ “Algorithm A is correct” is a proposition that requires a mathematical proof. ❖ All students in the course thinking that it is true is not a proof. ❖ Showing it is true on 1 million examples is not a proof. !10 ❖ “Problem P has no solution” is a proposition that requires a mathematical proof. ❖ Your inability to come up with a solution to Problem P is not a proof that a solution doesn’t exist. ❖ All your 5,000 Facebook friends not being able to come up with a solution doesn’t make the statement true either. !11 ❖ Despite decades of work by so many brilliant researchers, no one has been able to come up with a polynomial-time algorithm for the Traveling Salesman Problem (TSP). ❖ Still, no computer scientist or mathematician would state “TSP has no polynomial-time solution” because such a statement would require a mathematical proof and such a proof has not been found yet. !12 ❖ It is important to note that decades of work by brilliant researchers not resulting in a polynomial-time algorithm for TSP do strengthen our belief that the conjecture that “TSP has no polynomial-time solution” is true. ❖ This belief could, for example, direct other brilliant researchers to focus on proving the conjecture true (rather than false). ❖ However, no matter how strong our belief is, it is still not a proof. !13 Sets ❖ A set is an unordered collection of items. ❖ We write a∈S to denote that a is an element of set S, or that set S contains element a. ❖ Roster method description of sets: B={0,1}, C={a,b,c,d}, D={#,$,%,&,@} ❖ Set builder or set comprehension description of sets: F={x|x is an odd integer}, G={y| y is an integer that is divisible by 7} !14 Sets ❖ An element of a set cannot appear more than once in the set. ❖ For example, {a,b,b,c} is not a set. ❖ A mathematical structure that allows for an element to appear more than once is called multiset or bag. In this course, we will only work with sets. !15 Special Sets ❖ N = 0, 1, 2, 3,... The set of natural numbers { } ❖ The set of integers Z = ..., 2, 1, 0, 1, 2,... { − − } + ❖ Z = 1, 2,... The set of positive integers { } p ❖ The set of rational numbers Q = p, q Z,q =0 { q | 2 6 } ❖ The set of real numbers R + ❖ The set of positive real numbers R !16 The Empty Set ❖ The empty set is the set that contains no elements. ❖ Denoted by ∅ or {}. ❖ Important: The set {∅} is not empty. Rather, it is a set that contains one element that is ∅. !17 Cardinality of Sets ❖ The cardinality of a finite set S, denoted by |S|, is the number of elements in S. ❖ |{a,b,c}| = 3 ❖ |∅| = 0 ❖ |{∅}| = 1 !18 Cardinality of Sets ❖Not all sets are finite. ❖Infinite sets can be countable or uncountable. ❖More on this later in the semester. !19 Propositional Logic !20 Propositions ❖ A proposition is a declarative sentence that is either true or false, but not both. ❖ We use propositional variables (e.g., p, q, r, s,…) to represent propositions. !21 Propositions ❖ Propositions: ❖ 3∈{1,2,4} ❖ |{0,1}|=2 ❖ 7∉{a,b,c} ❖ Not propositions: ❖ 1+1 ❖ {a,b,c} ❖ |{5,12,19}| !22 Compound Propositions ❖ If p is a proposition, ¬p is its negation. ❖ If p and q are two propositions, then ❖ p∧q (“p and q”) is their conjunction ❖ p∨q (“p or q”) is their disjunction !23 Truth Values ❖ The truth value of a proposition is true, denoted by T, if it is a true proposition, and the truth value is false, denoted by F, if it is a false proposition. ❖ True propositions: ❖ |{a,b}|=2 |∅|<|{1}| 7∉{1,5,9,12} ❖ False propositions: ❖ |{∅}|=0 7∈{1,5,9,12}!24 Truth Table ❖ For a compound proposition, one way to determine the truth value of the proposition is by using a truth table. ❖ The truth table has one row for each combination of T and F for the primitive propositions. !25 4 41 / The1 / The Foundations: Foundations: Logic Logic and Proofsand Proofs TableTable 1 displays 1 displays the thetruthtruth table tablefor thefor negationthe negation of a of proposition a propositionp. Thisp. This table table has ahas row a row TABLETABLE 1 The 1 The for each of the two possible truth values of a propositionp p. Each row shows the truth value of TruthTruth Table Table for for for each of the two possible truth values of a proposition . Each row shows the truth value of the Negation of a p correspondingp corresponding to the to truththe truth value value of p offorp thisfor this row. row. the Negation of a ¬ ¬ Proposition.Proposition. TheThe negation negation of a of proposition a proposition can can also also be considered be considered the theresult result of the of operationthe operation of the of the negationnegation operator operatoron aon proposition. a proposition. The The negation negation operator operator constructs constructs a new a new proposition proposition from from p p p p ¬ ¬ a singlea single existing existing proposition. proposition. We Wewill will now now introduce introduce the logical the logical operators operators that that are used are used to form to form T T F F newnew propositions propositions from from two two or more or more existing existing propositions. propositions. These These logical logical operators operators are also are also called called F F T T connectivesconnectives. DEFINITIONDEFINITION 2 2 Let Letp andp andq beq propositions.be propositions. The Theconjunctionconjunctionof pofandp andq, denotedq, denoted by p by pq, isq the, is proposition the proposition ∧ “p and“p andq.” Theq.” The conjunction conjunctionp pq isq trueis true when when both bothp andp andq areq trueare true and∧ and is false is false otherwise. otherwise. ∧ ∧ TableTable 2 displays 2 displays the the truth truth table table of p of pq. Thisq. This table table has has a row a row for each for each of the of four the four possible possible ∧ combinationscombinations of truth of truth values values of p ofandp ∧andq. Theq. The four four rows rows correspond correspond to the to pairs the pairs of truth of truth values values TT,TT, TF, TF, FT, FT, and and FF, FF,where where the firstthe first truth truth value value in the in pair the pair is the is truth the truth value value of p ofandp and the second the second truthtruth value value is the is truththe truth value value of q of. q. NoteNote that that in logic in logic the wordthe word “but” “but” sometimes sometimes is used is used instead instead of “and” of “and” in a in conjunction. a conjunction. For For example,example, the statementthe statement “The “The sun sun is shining, is shining, but it but is it raining” is raining” is another is another way way of saying of saying “The “The sun sun is shiningis shining and and it is it raining.” is raining.” (In natural (In natural language, language, there there is a subtle is a subtle difference difference in meaning in meaning between between “and”“and” and and “but”; “but”; we will we will not notbe concerned be concerned with with this this nuance nuance here.) here.) EXAMPLEEXAMPLE 5 5FindFind the conjunctionthe conjunction of the of propositions the propositionsp andp andq whereq wherep isp theis proposition the proposition “Rebecca’s “Rebecca’s PC has PC has moremore than than 16 GB 16 GB free free hard hard disk disk space” space” and andq isq theis propositionthe proposition “The “The processor processor in Rebecca’s in Rebecca’s PCPC runs runs faster faster than than 1 GHz.” 1 GHz.” Solution:Solution:TheThe conjunction conjunction of these of these propositions, propositions,p pq, isq the, is proposition the proposition “Rebecca’s “Rebecca’s PC has PC has moremore than than 16 GB 16 GB free free hard hard disk disk space, space, and and the processorthe∧ processor∧ in Rebecca’s in Rebecca’s PC runs PC runs faster faster than than 1 1 GHz.”GHz.” This This conjunction conjunction can can be expressed be expressed more more simply simply as “Rebecca’s as “Rebecca’s PC has PC morehas more than than 16 GB 16 GB freefree hard hard disk disk space, space, and and its processor its processor runs runs faster faster than than 1 GHz.” 1 GHz.” For Forthis this conjunction conjunction to be to true, be true, ▲ bothboth conditions conditions given given must must be true.
Recommended publications
  • Logophoricity in Finnish

    Logophoricity in Finnish

    Open Linguistics 2018; 4: 630–656 Research Article Elsi Kaiser* Effects of perspective-taking on pronominal reference to humans and animals: Logophoricity in Finnish https://doi.org/10.1515/opli-2018-0031 Received December 19, 2017; accepted August 28, 2018 Abstract: This paper investigates the logophoric pronoun system of Finnish, with a focus on reference to animals, to further our understanding of the linguistic representation of non-human animals, how perspective-taking is signaled linguistically, and how this relates to features such as [+/-HUMAN]. In contexts where animals are grammatically [-HUMAN] but conceptualized as the perspectival center (whose thoughts, speech or mental state is being reported), can they be referred to with logophoric pronouns? Colloquial Finnish is claimed to have a logophoric pronoun which has the same form as the human-referring pronoun of standard Finnish, hän (she/he). This allows us to test whether a pronoun that may at first blush seem featurally specified to seek [+HUMAN] referents can be used for [-HUMAN] referents when they are logophoric. I used corpus data to compare the claim that hän is logophoric in both standard and colloquial Finnish vs. the claim that the two registers have different logophoric systems. I argue for a unified system where hän is logophoric in both registers, and moreover can be used for logophoric [-HUMAN] referents in both colloquial and standard Finnish. Thus, on its logophoric use, hän does not require its referent to be [+HUMAN]. Keywords: Finnish, logophoric pronouns, logophoricity, anti-logophoricity, animacy, non-human animals, perspective-taking, corpus 1 Introduction A key aspect of being human is our ability to think and reason about our own mental states as well as those of others, and to recognize that others’ perspectives, knowledge or mental states are distinct from our own, an ability known as Theory of Mind (term due to Premack & Woodruff 1978).
  • Predicate Logic

    Predicate Logic

    Logical Form & Predicate Logic Jean Mark Gawron Linguistics San Diego State University [email protected] http://www.rohan.sdsu.edu/∼gawron Logical Form & Predicate Logic – p. 1/30 Predicates and Arguments Logical Form & Predicate Logic – p. 2/30 Logical Form The Logical Form of English sentences can be represented by formulae of Predicate Logic, What do we mean by the logical form of a sentence? We mean: we can capture the truth conditions of complex English expressions in predicate logic, given some account of the denotations (extensions) of the simple expressions (words). Logical Form & Predicate Logic – p. 3/30 The Leading Idea: Extensions Nouns, verbs, and adjectives have predicates as their translations. What does ’predicate’ mean? ’Predicate’ means what it means in predicate logic. Logical Form & Predicate Logic – p. 4/30 Intransitive verbs Translating into predicate logic (1) a. Johnwalks. b. walk(j) (2) a. [[j]] = John b. “The extension of ’j’ is the individual John.” c. [[walk]] = {x | x walks } d. “The extension of ’walk’ is the set of walkers” (3) a. [[John walks]] = [[walk(j)]] b. [[walk(j)]] = true iff [[j]] ∈ [[walk]] Logical Form & Predicate Logic – p. 5/30 Extensional denotations • Remember two kinds of denotation. For work with logic, denotations are always extensions. • We call a denotation such as [[walk]] (a set) an extension, because it is defined by the set of things the word walk describes or “extends over” • The denotation [[j]] is also extensional because it is defined by the individual the word John describes. • Next we extend extensional denotation to transitive verbs, nouns, and sentences.
  • Discrete Mathematics

    Discrete Mathematics

    Discrete Mathematics1 http://lcs.ios.ac.cn/~znj/DM2017 Naijun Zhan March 19, 2017 1Special thanks to Profs Hanpin Wang (PKU) and Lijun Zhang (ISCAS) for their courtesy of the slides on this course. Contents 1. The Foundations: Logic and Proofs 2. Basic Structures: Sets, Functions, Sequences, Sum- s, and Matrices 3. Algorithms 4. Number Theory and Cryptography 5. Induction and Recursion 6. Counting 7. Discrete Probability 8. Advanced Counting Techniques 9. Relations 10. Graphs 11. Trees 12. Boolean Algebra 13. Modeling Computation 1 Chapter 1 The Foundations: Logic and Proofs Logic in Computer Science During the past fifty years there has been extensive, continuous, and growing interaction between logic and computer science. In many respects, logic provides computer science with both a u- nifying foundational framework and a tool for modeling compu- tational systems. In fact, logic has been called the calculus of computer science. The argument is that logic plays a fundamen- tal role in computer science, similar to that played by calculus in the physical sciences and traditional engineering disciplines. Indeed, logic plays an important role in areas of computer sci- ence as disparate as machine architecture, computer-aided de- sign, programming languages, databases, artificial intelligence, algorithms, and computability and complexity. Moshe Vardi 2 • The origins of logic can be dated back to Aristotle's time. • The birth of mathematical logic: { Leibnitz's idea { Russell paradox { Hilbert's plan { Three schools of modern logic: logicism (Frege, Russell, Whitehead) formalism (Hilbert) intuitionism (Brouwer) • One of the central problem for logicians is that: \why is this proof correct/incorrect?" • Boolean algebra owes to George Boole.
  • Pronouns, Logical Variables, and Logophoricity in Abe Author(S): Hilda Koopman and Dominique Sportiche Source: Linguistic Inquiry, Vol

    Pronouns, Logical Variables, and Logophoricity in Abe Author(S): Hilda Koopman and Dominique Sportiche Source: Linguistic Inquiry, Vol

    MIT Press Pronouns, Logical Variables, and Logophoricity in Abe Author(s): Hilda Koopman and Dominique Sportiche Source: Linguistic Inquiry, Vol. 20, No. 4 (Autumn, 1989), pp. 555-588 Published by: MIT Press Stable URL: http://www.jstor.org/stable/4178645 Accessed: 22-10-2015 18:32 UTC Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at http://www.jstor.org/page/ info/about/policies/terms.jsp JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. MIT Press is collaborating with JSTOR to digitize, preserve and extend access to Linguistic Inquiry. http://www.jstor.org This content downloaded from 128.97.27.20 on Thu, 22 Oct 2015 18:32:27 UTC All use subject to JSTOR Terms and Conditions Hilda Koopman Pronouns, Logical Variables, Dominique Sportiche and Logophoricity in Abe 1. Introduction 1.1. Preliminaries In this article we describe and analyze the propertiesof the pronominalsystem of Abe, a Kwa language spoken in the Ivory Coast, which we view as part of the study of pronominalentities (that is, of possible pronominaltypes) and of pronominalsystems (that is, of the cooccurrence restrictionson pronominaltypes in a particulargrammar). Abe has two series of thirdperson pronouns. One type of pronoun(0-pronoun) has basically the same propertiesas pronouns in languageslike English. The other type of pronoun(n-pronoun) very roughly corresponds to what has been called the referential use of pronounsin English(see Evans (1980)).It is also used as what is called a logophoric pronoun-that is, a particularpronoun that occurs in special embedded contexts (the logophoric contexts) to indicate reference to "the person whose speech, thought or perceptions are reported" (Clements (1975)).
  • Chapter 6 Mirativity and the Bulgarian Evidential System Elena Karagjosova Freie Universität Berlin

    Chapter 6 Mirativity and the Bulgarian Evidential System Elena Karagjosova Freie Universität Berlin

    Chapter 6 Mirativity and the Bulgarian evidential system Elena Karagjosova Freie Universität Berlin This paper provides an account of the Bulgarian admirative construction andits place within the Bulgarian evidential system based on (i) new observations on the morphological, temporal, and evidential properties of the admirative, (ii) a criti- cal reexamination of existing approaches to the Bulgarian evidential system, and (iii) insights from a similar mirative construction in Spanish. I argue in particular that admirative sentences are assertions based on evidence of some sort (reporta- tive, inferential, or direct) which are contrasted against the set of beliefs held by the speaker up to the point of receiving the evidence; the speaker’s past beliefs entail a proposition that clashes with the assertion, triggering belief revision and resulting in a sense of surprise. I suggest an analysis of the admirative in terms of a mirative operator that captures the evidential, temporal, aspectual, and modal properties of the construction in a compositional fashion. The analysis suggests that although mirativity and evidentiality can be seen as separate semantic cate- gories, the Bulgarian admirative represents a cross-linguistically relevant case of a mirative extension of evidential verbal forms. Keywords: mirativity, evidentiality, fake past 1 Introduction The Bulgarian evidential system is an ongoing topic of discussion both withre- spect to its interpretation and its morphological buildup. In this paper, I focus on the currently poorly understood admirative construction. The analysis I present is based on largely unacknowledged observations and data involving the mor- phological structure, the syntactic environment, and the evidential meaning of the admirative. Elena Karagjosova.
  • Toward a Shared Syntax for Shifted Indexicals and Logophoric Pronouns

    Toward a Shared Syntax for Shifted Indexicals and Logophoric Pronouns

    Toward a Shared Syntax for Shifted Indexicals and Logophoric Pronouns Mark Baker Rutgers University April 2018 Abstract: I argue that indexical shift is more like logophoricity and complementizer agreement than most previous semantic accounts would have it. In particular, there is evidence of a syntactic requirement at work, such that the antecedent of a shifted “I” must be a superordinate subject, just as the antecedent of a logophoric pronoun or the goal of complementizer agreement must be. I take this to be evidence that the antecedent enters into a syntactic control relationship with a null operator in all three constructions. Comparative data comes from Magahi and Sakha (for indexical shift), Yoruba (for logophoric pronouns), and Lubukusu (for complementizer agreement). 1. Introduction Having had an office next to Lisa Travis’s for 12 formative years, I learned many things from her that still influence my thinking. One is her example of taking semantic notions, such as aspect and event roles, and finding ways to implement them in syntactic structure, so as to advance the study of less familiar languages and topics.1 In that spirit, I offer here some thoughts about how logophoricity and indexical shift, topics often discussed from a more or less semantic point of view, might have syntactic underpinnings—and indeed, the same syntactic underpinnings. On an impressionistic level, it would not seem too surprising for logophoricity and indexical shift to have a common syntactic infrastructure. Canonical logophoricity as it is found in various West African languages involves using a special pronoun inside the finite CP complement of a verb to refer to the subject of that verb.
  • 1 Lecture 13: Quantifier Laws1 1. Laws of Quantifier Negation There

    1 Lecture 13: Quantifier Laws1 1. Laws of Quantifier Negation There

    Ling 409 lecture notes, Lecture 13 Ling 409 lecture notes, Lecture 13 October 19, 2005 October 19, 2005 Lecture 13: Quantifier Laws1 true or there is at least one individual that makes φ(x) true (or both). Those are also equivalent statements. 1. Laws of Quantifier Negation Laws 2 and 3 suggest a fundamental connection between universal quantification and There are a number of equivalences based on the set-theoretic semantics of predicate logic. conjunction and existential quantification and disjunction: Take for instance a statement of the form (∃x)~ φ(x)2. It asserts that there is at least one individual d which makes ~φ(x) true. That means that d makes ϕ(x) false. That, in turn, (∀x) ψ(x) is true iff ψ(a) and ψ(b) and ψ(c) … (where a, b, c … are the names of the means that (∀x) φ(x) is false, which means that ~(∀x) φ(x) is true. The same reasoning individuals in the domain.) can be applied in the reverse direction and the result is the first quantifier law: ∃ ψ ψ ψ ψ ( x) (x) is true iff (a) or (b) or c) … (where a, b, c … are the names of the (1) Laws of Quantifier Negation individuals in the domain.) Law 1: ~(∀x) φ(x) ⇐⇒ (∃x)~ φ(x) From that point of view, Law 1 resembles a generalized DeMorgan’s Law: (Possible English correspondents: Not everyone passed the test and Someone did not pass (4) ~(ϕ(a) & ϕ(b) & ϕ(c) … ) ⇐⇒ ~ϕ(a) v ~ϕ(b) v ϕ(c) … the test.) (5) Law 4: (∀x) ψ(x) v (∀x)φ(x) ⇒ (∀x)( ψ(x) v φ(x)) ϕ ⇐ ϕ Because of the Law of Double Negation (~~ (x) ⇒ (x)), Law 1 could also be written (6) Law 5: (∃x)( ψ(x) & φ(x)) ⇒ (∃x) ψ(x) & (∃x)φ(x) as follows: These two laws are one-way entailments, NOT equivalences.
  • Deduction (I) Tautologies, Contradictions And

    Deduction (I) Tautologies, Contradictions And

    D (I) T, & L L October , Tautologies, contradictions and contingencies Consider the truth table of the following formula: p (p ∨ p) () If you look at the final column, you will notice that the truth value of the whole formula depends on the way a truth value is assigned to p: the whole formula is true if p is true and false if p is false. Contrast the truth table of (p ∨ p) in () with the truth table of (p ∨ ¬p) below: p ¬p (p ∨ ¬p) () If you look at the final column, you will notice that the truth value of the whole formula does not depend on the way a truth value is assigned to p. The formula is always true because of the meaning of the connectives. Finally, consider the truth table table of (p ∧ ¬p): p ¬p (p ∧ ¬p) () This time the formula is always false no matter what truth value p has. Tautology A statement is called a tautology if the final column in its truth table contains only ’s. Contradiction A statement is called a contradiction if the final column in its truth table contains only ’s. Contingency A statement is called a contingency or contingent if the final column in its truth table contains both ’s and ’s. Let’s consider some examples from the book. Can you figure out which of the following sentences are tautologies, which are contradictions and which contingencies? Hint: the answer is the same for all the formulas with a single row. () a. (p ∨ ¬p), (p → p), (p → (q → p)), ¬(p ∧ ¬p) b.
  • Propositional Logic (PDF)

    Propositional Logic (PDF)

    Mathematics for Computer Science Proving Validity 6.042J/18.062J Instead of truth tables, The Logic of can try to prove valid formulas symbolically using Propositions axioms and deduction rules Albert R Meyer February 14, 2014 propositional logic.1 Albert R Meyer February 14, 2014 propositional logic.2 Proving Validity Algebra for Equivalence The text describes a for example, bunch of algebraic rules to the distributive law prove that propositional P AND (Q OR R) ≡ formulas are equivalent (P AND Q) OR (P AND R) Albert R Meyer February 14, 2014 propositional logic.3 Albert R Meyer February 14, 2014 propositional logic.4 1 Algebra for Equivalence Algebra for Equivalence for example, The set of rules for ≡ in DeMorgan’s law the text are complete: ≡ NOT(P AND Q) ≡ if two formulas are , these rules can prove it. NOT(P) OR NOT(Q) Albert R Meyer February 14, 2014 propositional logic.5 Albert R Meyer February 14, 2014 propositional logic.6 A Proof System A Proof System Another approach is to Lukasiewicz’ proof system is a start with some valid particularly elegant example of this idea. formulas (axioms) and deduce more valid formulas using proof rules Albert R Meyer February 14, 2014 propositional logic.7 Albert R Meyer February 14, 2014 propositional logic.8 2 A Proof System Lukasiewicz’ Proof System Lukasiewicz’ proof system is a Axioms: particularly elegant example of 1) (¬P → P) → P this idea. It covers formulas 2) P → (¬P → Q) whose only logical operators are 3) (P → Q) → ((Q → R) → (P → R)) IMPLIES (→) and NOT. The only rule: modus ponens Albert R Meyer February 14, 2014 propositional logic.9 Albert R Meyer February 14, 2014 propositional logic.10 Lukasiewicz’ Proof System Lukasiewicz’ Proof System Prove formulas by starting with Prove formulas by starting with axioms and repeatedly applying axioms and repeatedly applying the inference rule.
  • Lexical Scoping As Universal Quantification

    Lexical Scoping As Universal Quantification

    University of Pennsylvania ScholarlyCommons Technical Reports (CIS) Department of Computer & Information Science March 1989 Lexical Scoping as Universal Quantification Dale Miller University of Pennsylvania Follow this and additional works at: https://repository.upenn.edu/cis_reports Recommended Citation Dale Miller, "Lexical Scoping as Universal Quantification", . March 1989. University of Pennsylvania Department of Computer and Information Science Technical Report No. MS-CIS-89-23. This paper is posted at ScholarlyCommons. https://repository.upenn.edu/cis_reports/785 For more information, please contact [email protected]. Lexical Scoping as Universal Quantification Abstract A universally quantified goal can be interpreted intensionally, that is, the goal ∀x.G(x) succeeds if for some new constant c, the goal G(c) succeeds. The constant c is, in a sense, given a scope: it is introduced to solve this goal and is "discharged" after the goal succeeds or fails. This interpretation is similar to the interpretation of implicational goals: the goal D ⊃ G should succeed if when D is assumed, the goal G succeeds. The assumption D is discharged after G succeeds or fails. An interpreter for a logic programming language containing both universal quantifiers and implications in goals and the body of clauses is described. In its non-deterministic form, this interpreter is sound and complete for intuitionistic logic. Universal quantification can provide lexical scoping of individual, function, and predicate constants. Several examples are presented to show how such scoping can be used to provide a Prolog-like language with facilities for local definition of programs, local declarations in modules, abstract data types, and encapsulation of state.
  • On Axiomatizations of General Many-Valued Propositional Calculi

    On Axiomatizations of General Many-Valued Propositional Calculi

    On axiomatizations of general many-valued propositional calculi Arto Salomaa Turku Centre for Computer Science Joukahaisenkatu 3{5 B, 20520 Turku, Finland asalomaa@utu.fi Abstract We present a general setup for many-valued propositional logics, and compare truth-table and axiomatic stipulations within this setup. Results are obtained concerning cases, where a finitary axiomatization is (resp. is not) possible. Related problems and examples are discussed. 1 Introduction Many-valued systems of logic are constructed by introducing one or more truth- values between truth and falsity. Truth-functions associated with the logical connectives then operate with more than two truth-values. For instance, the truth-function D associated with the 3-valued disjunction might be defined for the truth-values T;I;F (true, intermediate, false) by D(x; y) = max(x; y); where T > I > F: If the truth-function N associated with negation is defined by N(T ) = F; N(F ) = T;N(I) = I; the law of the excluded middle is not valid, provided \validity" means that the truth-value t results for all assignments of values for the variables. On the other hand, many-valuedness is not so obvious if the axiomatic method is used. As such, an ordinary axiomatization is not many-valued. Truth-tables are essential for the latter. However, in many cases it is possi- ble to axiomatize many-valued logics and, conversely, find many-valued models for axiom systems. In this paper, we will investigate (for propositional log- ics) interconnections between such \truth-value stipulations" and \axiomatic stipulations". A brief outline of the contents of this paper follows.
  • CHAPTER 1 the Main Subject of Mathematical Logic Is Mathematical

    CHAPTER 1 the Main Subject of Mathematical Logic Is Mathematical

    CHAPTER 1 Logic The main subject of Mathematical Logic is mathematical proof. In this chapter we deal with the basics of formalizing such proofs and analysing their structure. The system we pick for the representation of proofs is natu- ral deduction as in Gentzen (1935). Our reasons for this choice are twofold. First, as the name says this is a natural notion of formal proof, which means that the way proofs are represented corresponds very much to the way a careful mathematician writing out all details of an argument would go any- way. Second, formal proofs in natural deduction are closely related (via the Curry-Howard correspondence) to terms in typed lambda calculus. This provides us not only with a compact notation for logical derivations (which otherwise tend to become somewhat unmanageable tree-like structures), but also opens up a route to applying the computational techniques which un- derpin lambda calculus. An essential point for Mathematical Logic is to fix the formal language to be used. We take implication ! and the universal quantifier 8 as basic. Then the logic rules correspond precisely to lambda calculus. The additional connectives (i.e., the existential quantifier 9, disjunction _ and conjunction ^) will be added via axioms. Later we will see that these axioms are deter- mined by particular inductive definitions. In addition to the use of inductive definitions as a unifying concept, another reason for that change of empha- sis will be that it fits more readily with the more computational viewpoint adopted there. This chapter does not simply introduce basic proof theory, but in addi- tion there is an underlying theme: to bring out the constructive content of logic, particularly in regard to the relationship between minimal and classical logic.