DOCSLIB.ORG

Polarized Intuitionistic Logic

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Polarized Intuitionistic Logic Combining Classical Logic and Intuitionistic Logic Double Negation, Polarization, Focusing, and Semantics Chuck Liang and Dale Miller Challenges of Combining Intuitionistic Logic with Classical Logic Using a Double Negation Translation I Intuitionistic implication should not collapse into classical implication. Consider A _e (B ⊃ C) −→∼(∼A^ ∼(B ⊃ C)) I How to distinguish the introduction of a translated classical ”connective” from intuitionistic introductions. Introduction of classical disjunction A _e B: ∼A; ∼B; Γ ` ∼A^ ∼B; Γ ` Γ `∼(∼A^ ∼B) No guarantee that sequence won’t be interrupted by other rules. I How to recognize classical ”dualities”. How is ∼(∼A^ ∼B) the ”dual” of (∼A^ ∼B) in an intuitionistic sense, given that ∼∼P 6≡ P. Challenges Continued ... I How to distinguish classical from intuitionistic equivalence. I Classical versus Intuitionistic Cut Elimination. A; Γ ` B ∼A; Γ ` B Γ ` B Admissible in classical logic but not in intuitionistic logic. How do we simulate this cut in an intuitionistic proof system? I What is the meaning of mixed formulas such as A _e (B ⊃ (C _ D))?. Outline and Overview: I Goal: Combine LJ and LC into Polarized Intuitionistic Logic I LC does not include intuitionistic implication I Start with Intuitionistic Logic with a designated atom ?. I ? is just minimal ”false” - this logic predates ICL. I ICL is a stand-alone logic I PIL combine logics I Assign labels, i.e., ”polarities” to formulas. I Define Double-Negation translation. I Use focusing (focalization) to isolate ”classical connectives” I Derive Unified Sequent Calculus I Define Kripke/Algebraic Semantics Syntax and Colors I Formulas freely generated from atoms, ^, _, ⊃, 0 and designated atom ?. I Define :A = A ⊃ ?;(A ⊃ 0 = ∼A) I Formulas are Red or Green as follows: I A ^ B, A _ B, 0, and A ⊃ B where B 6= ? are red. I All atoms are red, except ?, which is green. I :A (A ⊃ ?) is green. 2n I : (R), n > 0, are reddish green (also includes ?) 2n+1 I : (R) are solidly green I Red and Green formulas can be logically equivalent: (A ^ B) ⊃ ? ≡ A ⊃ (B ⊃ ?) I This polarization is not same as duality in linear logic: ?X−◦!Y Recovering Classical ”Dualities” ? I M = :M for red or reddish-green M ? I (:M) = M Syntactic Identity: A?? = A ? I A is convenient way to refer to doubly-negated formulas ? I A is not a connective. ? ? I if A ≡ B, then A is only classically ≡ to B ((A ^ B) ⊃ ?)? = A ^ B, (A ⊃ (B ⊃ ?))? = :(A ⊃ :B) Double Negation as Macro Expansion R red and E green e ? ? ? ? ? I A _ B = (A ^ B ) = :(A ^ B ) e ? ? ? ? ? I A ^ B = (A _ B ) = :(A _ B ) ? ? I 1 = ? = ? ⊃ ?; > = 0 = 0 ⊃ ? To complete the definition of A?, we need missing link: ? I A / B = :(A ⊃ B ) includes special case: (R / 1) = ::R These are not yet new connectives, just labels The following holds: ? ? I 1 = ?; > = 0 e ? ? ? I (A _ B) = A ^ B e ? ? ? I (A ^ B) = A _ B ? ? I (A / B) ≡ A ⊃ B (mod :::P ≡ :P) Caution: do not equate “green” with “classical.” Classical fragment will use _ and _e, ^ and ^e, 0 and ?, 1 and >. The classical fragment will be more LC than LK. Intuitionistic Sequent Calculus LJ A; B; Γ ` D A; Γ ` DB; Γ ` D A ⊃ B; Γ ` AB; Γ ` D ^L _L ⊃L A ^ B; Γ ` D A _ B; Γ ` D A ⊃ B; Γ ` D Γ ` A Γ ` B Γ ` Ai A; Γ ` B ^R _R ⊃R Γ ` A ^ B Γ ` A1 _ A2 Γ ` A ⊃ B Id 0L 1R a; Γ ` a 0; Γ ` D Γ ` 1 ? is considered a special atom _e, ^e and / as Synthetic Connectives in LJ? Distinguish between sequents Γ ` A and Γ ` ?: Correspond to sequents with and without a stoup Introduction of a green formula :A = A ⊃ ?: Id A; Γ ` ? A ⊃ ?; Γ ` A ?; Γ ` ? ⊃R ⊃L Γ ` A ⊃ ? A ⊃ ?; Γ ` ? But LJ is not good enough We don’t want the following: B ⊃ C; Γ ` BC; Γ ` A ⊃ ? A ⊃ ?; Γ ` A ?; Γ ` B B ⊃ C; Γ ` A ⊃ ? A ⊃ ?; Γ ` B Looking for one-to-one mapping between proofs Derive new introduction rules for A _e B = :(A? ^ B?): Γ ` A; B A _e B; Γ ` A? A _e B; Γ ` B? • e ◦ ◦ e e _ R e _ L Γ ` A _ B A _ B; Γ `• Want this to correspond one-to-one with the following fragments: A?; B?; Γ ` ? :(A? ^ B?); Γ ` A? :(A? ^ B?); Γ ` B? A? ^ B?; Γ ` ? :(A? ^ B?); Γ ` A? ^ B? ?; Γ ` ? Γ ` :(A? ^ B?) :(A? ^ B?); Γ ` ? Need focused intuitionistic sequent calculus (LJF) even for unfocused synthetic introduction rules A new dimension of polarization I Atoms are “positive,” except ?, which is “negative” + I _, ^ , 1 and 0 are positive − I ^ , ⊃, are negative I Positives are “synchronous” on the right; Negatives are synchronous on the left I Asynchronous rules are always invertible I Synchronous (and asynchronous) rules can be stringed together into a single phase. e ? + ? ? I A _ B = (A ^ B ) I Caution: Do not confuse positive with red polarities: A ⊃ B is red but negative (red=positive only in LC) Use Delays to Fine-Tune Focusing @+(A) = A ^+ 1; @−(A) = 1 ⊃ A F F ` (left) F r (right) atomic a @−(a) a 0 @−(0) 0 1 @−(1) 1 A ^ B @+(A`) ^− @+(B`) @+(Ar ^− Br ) A _ B @−(A` _ B`) @−(Ar ) _ @−(Br ) A ⊃ B @−(Ar ) ⊃ @+(B`) @+(A` ⊃ Br ) F F ` (left) F r (right) ? ? ? a?, atomic a :(@−(a)) :@−(a) A ^e B :(@−(A?r ) _ @−(B?r )) :@−(A?` _ B?`) A _e B :(@−(A?r ) ^+ @−(B?r )) :@−(A?` ^+ B?`) A / B :(A` ⊃ B?r ) :(@−(Ar ) ⊃ @+(B?`)) Deriving the Sequent Calculus LP Different modes of sequents: ∼ ? ? ∼ I Γ `• A1;:::; An = A1 ;:::; An ; Γ ` ? (Γ `• = Γ ` ?) ∼ I Γ `◦ A = Γ ` A Structural Rules (R red, E green) ? Γ ` E A ; Γ `◦ A • Load=Go Signal=Stop ? Γ `◦ E A ; Γ `• =∼ [Γ;@−(A)] −! [?] [A ⊃ ?; Γ] −! A ? [Γ;@−(A)] −! ? [A ⊃ ?; Γ] −A! [A ⊃ ?; Γ] −! [?] [Γ];@−(A) −! ? [A ⊃ ?; Γ] A−!⊃⊥ [?] [Γ] −! @−(A) ⊃ ? [A ⊃ ?; Γ] −! [?] Γ `◦ A Γ `• B /R Γ `• A / B # [:::; Γ];@+(B?`) −! [?] [:::; Γ] −! @−(Ar ) + ?` R` Rr @ (B ) [:::; Γ] −@−(Ar )! [:::; Γ] −! [?] ⊃L @−(Ar )⊃@+(B?`) [:::; Γ] −! [?] Lf [@−(Ar ) ⊃ @+(B?`); Γ] −! [?] Correspondence between focusing phases and synthetic introduction rules must be relaxed: A / B ≡ (A ⊃ B?) ⊃ ?, which is − followed by + −+, +− are OK, but not + − +. Sequent Calculus LP Structural Rules and Identity ? Γ `• E A; Γ `• Θ A ; Γ `◦ A Signal ? Store ? Load I Γ `◦ E Γ `• A ; Θ A ; Γ `• a; Γ `◦ a Right-Red Introduction Rules Γ `◦ A Γ `◦ B Γ `◦ Ai A; Γ `◦ B ^R _R ⊃R Γ `◦ A ^ B Γ `◦ A1 _ A2 Γ `◦ A ⊃ B Left-Red Introduction Rules A; B; Γ `◦ R A; Γ `◦ RB; Γ `◦ R A ⊃ B; Γ `◦ AB; Γ `◦ R ^L _L ⊃L A ^ B; Γ `◦ R A _ B; Γ `◦ R A ⊃ B; Γ `◦ R Right-Green Introduction Rules Γ ` Γ ` Γ ` A; B • A • B e • e Γ `◦ A Γ `• B e ^ R e _ R /R Γ `• A ^ B Γ `• A _ B Γ `• A / B Rules for Constants Γ `◦ R Γ `• 1R 1L 0L ?R >R Γ `◦ 1 1; Γ `◦ R 0; Γ `◦ R Γ `• ? Γ `• > Extends to First Order Rules for Quantifiers Γ `◦ A[t=x] Γ `◦ A A; Γ `◦ R A[t=x]; Πx:A; Γ `◦ R 9R ΠR 9L ΠL Γ `◦ 9x:A Γ `◦ Πy:A 9y:A; Γ `◦ R Πx:A; Γ `◦ R Γ `• A[t=x] Γ `• A ΣR 8R Here, y is not free in Γ and R. Γ `• Σx:A Γ `• 8y:A Why not remove delays and get focused LPF? Possible, but first ... LC Inside LP ` Γ; N; P; S Γ; P; N `◦ S ^L ` Γ; N _ P; S 7−! Γ; P ^ N `◦ S ` Γ; N; P; Γ ` N; P • e e _ R ` Γ; N _ P; 7−! Γ `• N _ P Γ ` N • Signal ` Γ; P ` ∆; N; Γ `◦ P Γ `◦ N ^R ` Γ∆; P ^ N 7−! Γ `◦ P ^ N LC invariant: no “positive” introductions outside of the stoup ...subsumed by LP invariant: no green introduction in `◦ mode LC is accidentally almost focused, but not LP Independence from Double Negation Translation e e I _ , ^ , /, ? and > are now first-class green connectives and constants. ? I A is now De Morgan negation, defined by “dualities:” _e/^, ^e/_, //⊃, ?/1, >/0, ? I “dual atoms” a/a ; Formulas are in negation normal form. ? ? If A `◦ B is provable, then B `• A is provable. I Reclassification of some formulas: 1 and R ⊃ ? are now red. (R ⊃ ?)? = R / 1 ? I Every green formula is of the form R for some red R. Given A and A?, one is red, the other is green. Kripke Semantics Hybrid Model (Propositional Case): hW; ; C; j=i Requirements and definitions: I is a transitive, reflexive ordering on non-empty set W of “possible worlds.” I j= is a monotonic relation between elements of W and sets of atomic formulas. I C ⊆ W (“classical worlds”) I 4u = fk j k 2 C and u kg (”classical cover” of u) I required: 4k = fkg for all k 2 C. (for propositional models) I if 4u = ; then u is imaginary. Every Kripke Model for IL is immediately a Hybrid Model, with a more structured interpretation of possible worlds. Rules of j= for u; v 2 W; c; k 2 C, green E: I u j= 1 and u 6j= 0 I u j= A _ B iff u j= A or u j= B I u j= A ^ B iff u j= A and u j= B I u j= A ⊃ B iff for all v u, v j= A implies v j= B I u j= E iff for all k 2 4u, k j= E ? I c j= E iff c 6j= E E.g., c j= A / B iff c 6j= A ⊃ B? iff for some v c, v j= A and v 6j= B?.
Load more

Recommended publications
  • 7.1 Rules of Implication I
    Natural Deduction is a method for deriving the conclusion of valid arguments expressed in the symbolism of propositional logic. The method consists of using sets of Rules of Inference (valid argument forms) to derive either a conclusion or a series of intermediate conclusions that link the premises of an argument with the stated conclusion. The First Four Rules of Inference: ◦ Modus Ponens (MP): p q p q ◦ Modus Tollens (MT): p q ~q ~p ◦ Pure Hypothetical Syllogism (HS): p q q r p r ◦ Disjunctive Syllogism (DS): p v q ~p q Common strategies for constructing a proof involving the first four rules: ◦ Always begin by attempting to find the conclusion in the premises. If the conclusion is not present in its entirely in the premises, look at the main operator of the conclusion. This will provide a clue as to how the conclusion should be derived. ◦ If the conclusion contains a letter that appears in the consequent of a conditional statement in the premises, consider obtaining that letter via modus ponens. ◦ If the conclusion contains a negated letter and that letter appears in the antecedent of a conditional statement in the premises, consider obtaining the negated letter via modus tollens. ◦ If the conclusion is a conditional statement, consider obtaining it via pure hypothetical syllogism. ◦ If the conclusion contains a letter that appears in a disjunctive statement in the premises, consider obtaining that letter via disjunctive syllogism. Four Additional Rules of Inference: ◦ Constructive Dilemma (CD): (p q) • (r s) p v r q v s ◦ Simplification (Simp): p • q p ◦ Conjunction (Conj): p q p • q ◦ Addition (Add): p p v q Common Misapplications Common strategies involving the additional rules of inference: ◦ If the conclusion contains a letter that appears in a conjunctive statement in the premises, consider obtaining that letter via simplification.
    [Show full text]
  • Two Sources of Explosion
    Two sources of explosion Eric Kao Computer Science Department Stanford University Stanford, CA 94305 United States of America Abstract. In pursuit of enhancing the deductive power of Direct Logic while avoiding explosiveness, Hewitt has proposed including the law of excluded middle and proof by self-refutation. In this paper, I show that the inclusion of either one of these inference patterns causes paracon- sistent logics such as Hewitt's Direct Logic and Besnard and Hunter's quasi-classical logic to become explosive. 1 Introduction A central goal of a paraconsistent logic is to avoid explosiveness { the inference of any arbitrary sentence β from an inconsistent premise set fp; :pg (ex falso quodlibet). Hewitt [2] Direct Logic and Besnard and Hunter's quasi-classical logic (QC) [1, 5, 4] both seek to preserve the deductive power of classical logic \as much as pos- sible" while still avoiding explosiveness. Their work fits into the ongoing research program of identifying some \reasonable" and \maximal" subsets of classically valid rules and axioms that do not lead to explosiveness. To this end, it is natural to consider which classically sound deductive rules and axioms one can introduce into a paraconsistent logic without causing explo- siveness. Hewitt [3] proposed including the law of excluded middle and the proof by self-refutation rule (a very special case of proof by contradiction) but did not show whether the resulting logic would be explosive. In this paper, I show that for quasi-classical logic and its variant, the addition of either the law of excluded middle or the proof by self-refutation rule in fact leads to explosiveness.
    [Show full text]
  • Contradiction Or Non-Contradiction?  Hegel’S Dialectic Between Brandom and Priest
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Padua@research CONTRADICTION OR NON-CONTRADICTION? HEGEL’S DIALECTIC BETWEEN BRANDOM AND PRIEST by Michela Bordignon Abstract. The aim of the paper is to analyse Brandom’s account of Hegel’s conception of determinate negation and the role this structure plays in the dialectical process with respect to the problem of contradiction. After having shown both the merits and the limits of Brandom’s account, I will refer to Priest’s dialetheistic approach to contradiction as an alternative contemporary perspective from which it is possible to capture essential features of Hegel’s notion of contradiction, and I will test the equation of Hegel’s dialectic with Priest dialetheism. 1. Introduction According to Horstmann, «Hegel thinks of his new logic as being in part incompatible with traditional logic»1. The strongest expression of this new conception of logic is the first thesis of the work Hegel wrote in 1801 in order to earn his teaching habilitation: «contradictio est regula veri, non contradictio falsi»2. Hegel seems to claim that contradictions are true. The Hegelian thesis of the truth of contradiction is highly problematic. This is shown by Popper’s critique based on the principle of ex falso quodlibet: «if a theory contains a contradiction, then it entails everything, and therefore, indeed, nothing […]. A theory which involves a contradiction is therefore entirely useless I thank Graham Wetherall for kindly correcting a previous English translation of this paper and for his suggestions and helpful remarks. Of course, all remaining errors are mine.
    [Show full text]
  • Notes on Proof Theory
    Notes on Proof Theory Master 1 “Informatique”, Univ. Paris 13 Master 2 “Logique Mathématique et Fondements de l’Informatique”, Univ. Paris 7 Damiano Mazza November 2016 1Last edit: March 29, 2021 Contents 1 Propositional Classical Logic 5 1.1 Formulas and truth semantics . 5 1.2 Atomic negation . 8 2 Sequent Calculus 10 2.1 Two-sided formulation . 10 2.2 One-sided formulation . 13 3 First-order Quantification 16 3.1 Formulas and truth semantics . 16 3.2 Sequent calculus . 19 3.3 Ultrafilters . 21 4 Completeness 24 4.1 Exhaustive search . 25 4.2 The completeness proof . 30 5 Undecidability and Incompleteness 33 5.1 Informal computability . 33 5.2 Incompleteness: a road map . 35 5.3 Logical theories . 38 5.4 Arithmetical theories . 40 5.5 The incompleteness theorems . 44 6 Cut Elimination 47 7 Intuitionistic Logic 53 7.1 Sequent calculus . 55 7.2 The relationship between intuitionistic and classical logic . 60 7.3 Minimal logic . 65 8 Natural Deduction 67 8.1 Sequent presentation . 68 8.2 Natural deduction and sequent calculus . 70 8.3 Proof tree presentation . 73 8.3.1 Minimal natural deduction . 73 8.3.2 Intuitionistic natural deduction . 75 1 8.3.3 Classical natural deduction . 75 8.4 Normalization (cut-elimination in natural deduction) . 76 9 The Curry-Howard Correspondence 80 9.1 The simply typed l-calculus . 80 9.2 Product and sum types . 81 10 System F 83 10.1 Intuitionistic second-order propositional logic . 83 10.2 Polymorphic types . 84 10.3 Programming in system F ...................... 85 10.3.1 Free structures .
    [Show full text]
  • Chapter 1 Negation in a Cross-Linguistic Perspective
    Chapter 1 Negation in a cross-linguistic perspective 0. Chapter summary This chapter introduces the empirical scope of our study on the expression and interpretation of negation in natural language. We start with some background notions on negation in logic and language, and continue with a discussion of more linguistic issues concerning negation at the syntax-semantics interface. We zoom in on cross- linguistic variation, both in a synchronic perspective (typology) and in a diachronic perspective (language change). Besides expressions of propositional negation, this book analyzes the form and interpretation of indefinites in the scope of negation. This raises the issue of negative polarity and its relation to negative concord. We present the main facts, criteria, and proposals developed in the literature on this topic. The chapter closes with an overview of the book. We use Optimality Theory to account for the syntax and semantics of negation in a cross-linguistic perspective. This theoretical framework is introduced in Chapter 2. 1 Negation in logic and language The main aim of this book is to provide an account of the patterns of negation we find in natural language. The expression and interpretation of negation in natural language has long fascinated philosophers, logicians, and linguists. Horn’s (1989) Natural history of negation opens with the following statement: “All human systems of communication contain a representation of negation. No animal communication system includes negative utterances, and consequently, none possesses a means for assigning truth value, for lying, for irony, or for coping with false or contradictory statements.” A bit further on the first page, Horn states: “Despite the simplicity of the one-place connective of propositional logic ( ¬p is true if and only if p is not true) and of the laws of inference in which it participate (e.g.
    [Show full text]
  • The Unintended Interpretations of Intuitionistic Logic
    THE UNINTENDED INTERPRETATIONS OF INTUITIONISTIC LOGIC Wim Ruitenburg Department of Mathematics, Statistics and Computer Science Marquette University Milwaukee, WI 53233 Abstract. We present an overview of the unintended interpretations of intuitionis- tic logic that arose after Heyting formalized the “observed regularities” in the use of formal parts of language, in particular, first-order logic and Heyting Arithmetic. We include unintended interpretations of some mild variations on “official” intuitionism, such as intuitionistic type theories with full comprehension and higher order logic without choice principles or not satisfying the right choice sequence properties. We conclude with remarks on the quest for a correct interpretation of intuitionistic logic. §1. The Origins of Intuitionistic Logic Intuitionism was more than twenty years old before A. Heyting produced the first complete axiomatizations for intuitionistic propositional and predicate logic: according to L. E. J. Brouwer, the founder of intuitionism, logic is secondary to mathematics. Some of Brouwer’s papers even suggest that formalization cannot be useful to intuitionism. One may wonder, then, whether intuitionistic logic should itself be regarded as an unintended interpretation of intuitionistic mathematics. I will not discuss Brouwer’s ideas in detail (on this, see [Brouwer 1975], [Hey- ting 1934, 1956]), but some aspects of his philosophy need to be highlighted here. According to Brouwer mathematics is an activity of the human mind, a product of languageless thought. One cannot be certain that language is a perfect reflection of this mental activity. This makes language an uncertain medium (see [van Stigt 1982] for more details on Brouwer’s ideas about language). In “De onbetrouwbaarheid der logische principes” ([Brouwer 1981], pp.
    [Show full text]
  • A Non-Classical Refinement of the Interpolation Property for Classical
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Stirling Online Research Repository – Accepted for publication in Logique & Analyse – A non-classical refinement of the interpolation property for classical propositional logic Peter Milne Abstract We refine the interpolation property of the f^; _; :g-fragment of classical propositional logic, showing that if 2 :φ, 2 and φ then there is an interpolant χ, constructed using at most atomic formulas occurring in both φ and and negation, conjunction and disjunction, such that (i) φ entails χ in Kleene’s strong three-valued logic and (ii) χ entails in Priest’s Logic of Paradox. Keywords: Interpolation theorem for classical propositional logic · Kleene’s strong 3-valued logic · Priest’s Logic of Paradox 1 Introduction Suppose that φ classically entails , that φ is not a classical contradiction and that is not a classical tautology. Then φ and must share non-logical vo- cabulary, for else one could make φ true and false at the same time. That there must be some overlap in non-logical vocabulary between premise and con- clusion is obvious. Possession of the interpolation property takes this line of thought much further: If 2 :φ, 2 and φ then there is a formula χ containing only atomic formulas common to φ and and such that φ χ and χ .1 1Surprisingly few textbooks prove this theorem. (Hodges 2001) and (Hunter 1971) are excep- tions. They prove it in the slightly stronger form if φ and at least one atomic formula is common to φ and then there is a formula χ containing only atomic formulas common to φ and and such that φ χ and χ .
    [Show full text]
  • Intuitionistic Three-Valued Logic and Logic Programming Informatique Théorique Et Applications, Tome 25, No 6 (1991), P
    INFORMATIQUE THÉORIQUE ET APPLICATIONS J. VAUZEILLES A. STRAUSS Intuitionistic three-valued logic and logic programming Informatique théorique et applications, tome 25, no 6 (1991), p. 557-587 <http://www.numdam.org/item?id=ITA_1991__25_6_557_0> © AFCET, 1991, tous droits réservés. L’accès aux archives de la revue « Informatique théorique et applications » im- plique l’accord avec les conditions générales d’utilisation (http://www.numdam. org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Informatique théorique et Applications/Theoretical Informaties and Applications (vol. 25, n° 6, 1991, p. 557 à 587) INTUITIONISTIC THREE-VALUED LOGIC AMD LOGIC PROGRAMM1NG (*) by J. VAUZEILLES (*) and A. STRAUSS (2) Communicated by J. E. PIN Abstract. - In this paper, we study the semantics of logic programs with the help of trivalued hgic, introduced by Girard in 1973. Trivalued sequent calculus enables to extend easily the results of classical SLD-resolution to trivalued logic. Moreover, if one allows négation in the head and in the body of Hom clauses, one obtains a natural semantics for such programs regarding these clauses as axioms of a theory written in the intuitionistic fragment of that logic. Finally^ we define in the same calculus an intuitionistic trivalued version of Clark's completion, which gives us a déclarative semantics for programs with négation in the body of the clauses, the évaluation method being SLDNF-resolution. Resumé. — Dans cet article, nous étudions la sémantique des programmes logiques à l'aide de la logique trivaluée^ introduite par Girard en 1973.
    [Show full text]
  • Chapter 10: Symbolic Trails and Formal Proofs of Validity, Part 2
    Essential Logic Ronald C. Pine CHAPTER 10: SYMBOLIC TRAILS AND FORMAL PROOFS OF VALIDITY, PART 2 Introduction In the previous chapter there were many frustrating signs that something was wrong with our formal proof method that relied on only nine elementary rules of validity. Very simple, intuitive valid arguments could not be shown to be valid. For instance, the following intuitively valid arguments cannot be shown to be valid using only the nine rules. Somalia and Iran are both foreign policy risks. Therefore, Iran is a foreign policy risk. S I / I Either Obama or McCain was President of the United States in 2009.1 McCain was not President in 2010. So, Obama was President of the United States in 2010. (O v C) ~(O C) ~C / O If the computer networking system works, then Johnson and Kaneshiro will both be connected to the home office. Therefore, if the networking system works, Johnson will be connected to the home office. N (J K) / N J Either the Start II treaty is ratified or this landmark treaty will not be worth the paper it is written on. Therefore, if the Start II treaty is not ratified, this landmark treaty will not be worth the paper it is written on. R v ~W / ~R ~W 1 This or statement is obviously exclusive, so note the translation. 427 If the light is on, then the light switch must be on. So, if the light switch in not on, then the light is not on. L S / ~S ~L Thus, the nine elementary rules of validity covered in the previous chapter must be only part of a complete system for constructing formal proofs of validity.
    [Show full text]
  • Intuitionistic Logic
    Intuitionistic Logic Nick Bezhanishvili and Dick de Jongh Institute for Logic, Language and Computation Universiteit van Amsterdam Contents 1 Introduction 2 2 Intuitionism 3 3 Kripke models, Proof systems and Metatheorems 8 3.1 Other proof systems . 8 3.2 Arithmetic and analysis . 10 3.3 Kripke models . 15 3.4 The Disjunction Property, Admissible Rules . 20 3.5 Translations . 22 3.6 The Rieger-Nishimura Lattice and Ladder . 24 3.7 Complexity of IPC . 25 3.8 Mezhirov's game for IPC . 28 4 Heyting algebras 30 4.1 Lattices, distributive lattices and Heyting algebras . 30 4.2 The connection of Heyting algebras with Kripke frames and topologies . 33 4.3 Algebraic completeness of IPC and its extensions . 38 4.4 The connection of Heyting and closure algebras . 40 5 Jankov formulas and intermediate logics 42 5.1 n-universal models . 42 5.2 Formulas characterizing point generated subsets . 44 5.3 The Jankov formulas . 46 5.4 Applications of Jankov formulas . 47 5.5 The logic of the Rieger-Nishimura ladder . 52 1 1 Introduction In this course we give an introduction to intuitionistic logic. We concentrate on the propositional calculus mostly, make some minor excursions to the predicate calculus and to the use of intuitionistic logic in intuitionistic formal systems, in particular Heyting Arithmetic. We have chosen a selection of topics that show various sides of intuitionistic logic. In no way we strive for a complete overview in this short course. Even though we approach the subject for the most part only formally, it is good to have a general introduction to intuitionism.
    [Show full text]
  • The Proper Explanation of Intuitionistic Logic: on Brouwer's Demonstration
    The proper explanation of intuitionistic logic: on Brouwer’s demonstration of the Bar Theorem Göran Sundholm, Mark van Atten To cite this version: Göran Sundholm, Mark van Atten. The proper explanation of intuitionistic logic: on Brouwer’s demonstration of the Bar Theorem. van Atten, Mark Heinzmann, Gerhard Boldini, Pascal Bourdeau, Michel. One Hundred Years of Intuitionism (1907-2007). The Cerisy Conference, Birkhäuser, pp.60- 77, 2008, 978-3-7643-8652-8. halshs-00791550 HAL Id: halshs-00791550 https://halshs.archives-ouvertes.fr/halshs-00791550 Submitted on 24 Jan 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Distributed under a Creative Commons Attribution - NonCommercial| 4.0 International License The proper explanation of intuitionistic logic: on Brouwer’s demonstration of the Bar Theorem Göran Sundholm Philosophical Institute, Leiden University, P.O. Box 2315, 2300 RA Leiden, The Netherlands. [email protected] Mark van Atten SND (CNRS / Paris IV), 1 rue Victor Cousin, 75005 Paris, France. [email protected] Der … geführte Beweis scheint mir aber trotzdem . Basel: Birkhäuser, 2008, 60–77. wegen der in seinem Gedankengange enthaltenen Aussagen Interesse zu besitzen. (Brouwer 1927B, n. 7)1 Brouwer’s demonstration of his Bar Theorem gives rise to provocative ques- tions regarding the proper explanation of the logical connectives within intu- itionistic and constructivist frameworks, respectively, and, more generally, re- garding the role of logic within intuitionism.
    [Show full text]
  • List of Rules of Inference 1 List of Rules of Inference
    List of rules of inference 1 List of rules of inference This is a list of rules of inference, logical laws that relate to mathematical formulae. Introduction Rules of inference are syntactical transform rules which one can use to infer a conclusion from a premise to create an argument. A set of rules can be used to infer any valid conclusion if it is complete, while never inferring an invalid conclusion, if it is sound. A sound and complete set of rules need not include every rule in the following list, as many of the rules are redundant, and can be proven with the other rules. Discharge rules permit inference from a subderivation based on a temporary assumption. Below, the notation indicates such a subderivation from the temporary assumption to . Rules for classical sentential calculus Sentential calculus is also known as propositional calculus. Rules for negations Reductio ad absurdum (or Negation Introduction) Reductio ad absurdum (related to the law of excluded middle) Noncontradiction (or Negation Elimination) Double negation elimination Double negation introduction List of rules of inference 2 Rules for conditionals Deduction theorem (or Conditional Introduction) Modus ponens (or Conditional Elimination) Modus tollens Rules for conjunctions Adjunction (or Conjunction Introduction) Simplification (or Conjunction Elimination) Rules for disjunctions Addition (or Disjunction Introduction) Separation of Cases (or Disjunction Elimination) Disjunctive syllogism List of rules of inference 3 Rules for biconditionals Biconditional introduction Biconditional Elimination Rules of classical predicate calculus In the following rules, is exactly like except for having the term everywhere has the free variable . Universal Introduction (or Universal Generalization) Restriction 1: does not occur in .
    [Show full text]