Knowing-How and the Deduction Theorem
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The Deduction Rule and Linear and Near-Linear Proof Simulations
The Deduction Rule and Linear and Near-linear Proof Simulations Maria Luisa Bonet¤ Department of Mathematics University of California, San Diego Samuel R. Buss¤ Department of Mathematics University of California, San Diego Abstract We introduce new proof systems for propositional logic, simple deduction Frege systems, general deduction Frege systems and nested deduction Frege systems, which augment Frege systems with variants of the deduction rule. We give upper bounds on the lengths of proofs in Frege proof systems compared to lengths in these new systems. As applications we give near-linear simulations of the propositional Gentzen sequent calculus and the natural deduction calculus by Frege proofs. The length of a proof is the number of lines (or formulas) in the proof. A general deduction Frege proof system provides at most quadratic speedup over Frege proof systems. A nested deduction Frege proof system provides at most a nearly linear speedup over Frege system where by \nearly linear" is meant the ratio of proof lengths is O(®(n)) where ® is the inverse Ackermann function. A nested deduction Frege system can linearly simulate the propositional sequent calculus, the tree-like general deduction Frege calculus, and the natural deduction calculus. Hence a Frege proof system can simulate all those proof systems with proof lengths bounded by O(n ¢ ®(n)). Also we show ¤Supported in part by NSF Grant DMS-8902480. 1 that a Frege proof of n lines can be transformed into a tree-like Frege proof of O(n log n) lines and of height O(log n). As a corollary of this fact we can prove that natural deduction and sequent calculus tree-like systems simulate Frege systems with proof lengths bounded by O(n log n). -
David Hilbert's Contributions to Logical Theory
David Hilbert’s contributions to logical theory CURTIS FRANKS 1. A mathematician’s cast of mind Charles Sanders Peirce famously declared that “no two things could be more directly opposite than the cast of mind of the logician and that of the mathematician” (Peirce 1976, p. 595), and one who would take his word for it could only ascribe to David Hilbert that mindset opposed to the thought of his contemporaries, Frege, Gentzen, Godel,¨ Heyting, Łukasiewicz, and Skolem. They were the logicians par excellence of a generation that saw Hilbert seated at the helm of German mathematical research. Of Hilbert’s numerous scientific achievements, not one properly belongs to the domain of logic. In fact several of the great logical discoveries of the 20th century revealed deep errors in Hilbert’s intuitions—exemplifying, one might say, Peirce’s bald generalization. Yet to Peirce’s addendum that “[i]t is almost inconceivable that a man should be great in both ways” (Ibid.), Hilbert stands as perhaps history’s principle counter-example. It is to Hilbert that we owe the fundamental ideas and goals (indeed, even the name) of proof theory, the first systematic development and application of the methods (even if the field would be named only half a century later) of model theory, and the statement of the first definitive problem in recursion theory. And he did more. Beyond giving shape to the various sub-disciplines of modern logic, Hilbert brought them each under the umbrella of mainstream mathematical activity, so that for the first time in history teams of researchers shared a common sense of logic’s open problems, key concepts, and central techniques. -
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, -
5 Deduction in First-Order Logic
5 Deduction in First-Order Logic The system FOLC. Let C be a set of constant symbols. FOLC is a system of deduction for # the language LC . Axioms: The following are axioms of FOLC. (1) All tautologies. (2) Identity Axioms: (a) t = t for all terms t; (b) t1 = t2 ! (A(x; t1) ! A(x; t2)) for all terms t1 and t2, all variables x, and all formulas A such that there is no variable y occurring in t1 or t2 such that there is free occurrence of x in A in a subformula of A of the form 8yB. (3) Quantifier Axioms: 8xA ! A(x; t) for all formulas A, variables x, and terms t such that there is no variable y occurring in t such that there is a free occurrence of x in A in a subformula of A of the form 8yB. Rules of Inference: A; (A ! B) Modus Ponens (MP) B (A ! B) Quantifier Rule (QR) (A ! 8xB) provided the variable x does not occur free in A. Discussion of the axioms and rules. (1) We would have gotten an equivalent system of deduction if instead of taking all tautologies as axioms we had taken as axioms all instances (in # LC ) of the three schemas on page 16. All instances of these schemas are tautologies, so the change would have not have increased what we could 52 deduce. In the other direction, we can apply the proof of the Completeness Theorem for SL by thinking of all sententially atomic formulas as sentence # letters. The proof so construed shows that every tautology in LC is deducible using MP and schemas (1){(3). -
The Development of Mathematical Logic from Russell to Tarski: 1900–1935
The Development of Mathematical Logic from Russell to Tarski: 1900–1935 Paolo Mancosu Richard Zach Calixto Badesa The Development of Mathematical Logic from Russell to Tarski: 1900–1935 Paolo Mancosu (University of California, Berkeley) Richard Zach (University of Calgary) Calixto Badesa (Universitat de Barcelona) Final Draft—May 2004 To appear in: Leila Haaparanta, ed., The Development of Modern Logic. New York and Oxford: Oxford University Press, 2004 Contents Contents i Introduction 1 1 Itinerary I: Metatheoretical Properties of Axiomatic Systems 3 1.1 Introduction . 3 1.2 Peano’s school on the logical structure of theories . 4 1.3 Hilbert on axiomatization . 8 1.4 Completeness and categoricity in the work of Veblen and Huntington . 10 1.5 Truth in a structure . 12 2 Itinerary II: Bertrand Russell’s Mathematical Logic 15 2.1 From the Paris congress to the Principles of Mathematics 1900–1903 . 15 2.2 Russell and Poincar´e on predicativity . 19 2.3 On Denoting . 21 2.4 Russell’s ramified type theory . 22 2.5 The logic of Principia ......................... 25 2.6 Further developments . 26 3 Itinerary III: Zermelo’s Axiomatization of Set Theory and Re- lated Foundational Issues 29 3.1 The debate on the axiom of choice . 29 3.2 Zermelo’s axiomatization of set theory . 32 3.3 The discussion on the notion of “definit” . 35 3.4 Metatheoretical studies of Zermelo’s axiomatization . 38 4 Itinerary IV: The Theory of Relatives and Lowenheim’s¨ Theorem 41 4.1 Theory of relatives and model theory . 41 4.2 The logic of relatives . -
An Integration of Resolution and Natural Deduction Theorem Proving
From: AAAI-86 Proceedings. Copyright ©1986, AAAI (www.aaai.org). All rights reserved. An Integration of Resolution and Natural Deduction Theorem Proving Dale Miller and Amy Felty Computer and Information Science University of Pennsylvania Philadelphia, PA 19104 Abstract: We present a high-level approach to the integra- ble of translating between them. In order to achieve this goal, tion of such different theorem proving technologies as resolution we have designed a programming language which permits proof and natural deduction. This system represents natural deduc- structures as values and types. This approach builds on and ex- tion proofs as X-terms and resolution refutations as the types of tends the LCF approach to natural deduction theorem provers such X-terms. These type structures, called ezpansion trees, are by replacing the LCF notion of a uakfation with explicit term essentially formulas in which substitution terms are attached to representation of proofs. The terms which represent proofs are quantifiers. As such, this approach to proofs and their types ex- given types which generalize the formulas-as-type notion found tends the formulas-as-type notion found in proof theory. The in proof theory [Howard, 19691. Resolution refutations are seen LCF notion of tactics and tacticals can also be extended to in- aa specifying the type of a natural deduction proofs. This high corporate proofs as typed X-terms. Such extended tacticals can level view of proofs as typed terms can be easily combined with be used to program different interactive and automatic natural more standard aspects of LCF to yield the integration for which Explicit representation of proofs deduction theorem provers. -
CHAPTER 8 Hilbert Proof Systems, Formal Proofs, Deduction Theorem
CHAPTER 8 Hilbert Proof Systems, Formal Proofs, Deduction Theorem The Hilbert proof systems are systems based on a language with implication and contain a Modus Ponens rule as a rule of inference. They are usually called Hilbert style formalizations. We will call them here Hilbert style proof systems, or Hilbert systems, for short. Modus Ponens is probably the oldest of all known rules of inference as it was already known to the Stoics (3rd century B.C.). It is also considered as the most "natural" to our intuitive thinking and the proof systems containing it as the inference rule play a special role in logic. The Hilbert proof systems put major emphasis on logical axioms, keeping the rules of inference to minimum, often in propositional case, admitting only Modus Ponens, as the sole inference rule. 1 Hilbert System H1 Hilbert proof system H1 is a simple proof system based on a language with implication as the only connective, with two axioms (axiom schemas) which characterize the implication, and with Modus Ponens as a sole rule of inference. We de¯ne H1 as follows. H1 = ( Lf)g; F fA1;A2g MP ) (1) where A1;A2 are axioms of the system, MP is its rule of inference, called Modus Ponens, de¯ned as follows: A1 (A ) (B ) A)); A2 ((A ) (B ) C)) ) ((A ) B) ) (A ) C))); MP A ;(A ) B) (MP ) ; B 1 and A; B; C are any formulas of the propositional language Lf)g. Finding formal proofs in this system requires some ingenuity. Let's construct, as an example, the formal proof of such a simple formula as A ) A. -
Propositional Logic: Part II - Syntax & Proofs
Propositional Logic: Part II - Syntax & Proofs 0-0 Outline ² Syntax of Propositional Formulas ² Motivating Proofs ² Syntactic Entailment ` and Proofs ² Proof Rules for Natural Deduction ² Axioms, theories and theorems ² Consistency & completeness 1 Language of Propositional Calculus Def: A propositional formula is constructed inductively from the symbols for ² propositional variables: p; q; r; : : : or p1; p2; : : : ² connectives: :; ^; _; !; $ ² parentheses: (; ) ² constants: >; ? by the following rules: 1. A propositional variable or constant symbol (>; ?) is a formula. 2. If Á and à are formulas, then so are: (:Á); (Á ^ Ã); (Á _ Ã); (Á ! Ã); (Á $ Ã) Note: Removing >; ? from the above provides the def. of sentential formulas in Rubin. Semantically > = T; 2 ? = F though often T; F are used directly in formulas when engineers abuse notation. 3 WFF Smackdown Propositional formulas are also called propositional sentences or well formed formulas (WFF). When we use precedence of logical connectives and associativity of ^; _; $ to drop (smackdown!) parentheses it is understood that this is shorthand for the fully parenthesized expressions. Note: To further reduce use of (; ) some def's of formula use order of precedence: ! ^ ! :; ^; _; instead of :; ; $ _ $ As we will see, PVS uses the 1st order of precedence. 4 BNF and Parse Trees The Backus Naur form (BNF) for the de¯nition of a propositional formula is: Á ::= pj?j>j(:Á)j(Á ^ Á)j(Á _ Á)j(Á ! Á) Here p denotes any propositional variable and each occurrence of Á to the right of ::= represents any formula constructed thus far. We can apply this inductive de¯nition in reverse to construct a formula's parse tree. -
Types of Proof System
Types of proof system Peter Smith October 13, 2010 1 Axioms, rules, and what logic is all about 1.1 Two kinds of proof system There are at least two styles of proof system for propositional logic other than trees that beginners ought to know about. The real interest here, of course, is not in learning yet more about classical proposi- tional logic per se. For how much fun is that? Rather, what we are doing { as always { is illustrating some Big Ideas using propositional logic as a baby example. The two new styles are: 1. Axiomatic systems The first system of formal logic in anything like the contempo- rary sense { Frege's system in his Begriffsschrift of 1879 { is an axiomatic one. What is meant by `axiomatic' in this context? Think of Euclid's geometry for example. In such an axiomatic theory, we are given a \starter pack" of basic as- sumptions or axioms (we are often given a package of supplementary definitions as well, enabling us to introduce new ideas as abbreviations for constructs out of old ideas). And then the theorems of the theory are those claims that can deduced by allowed moves from the axioms, perhaps invoking definitions where appropriate. In Euclid's case, the axioms and definitions are of course geometrical, and he just helps himself to whatever logical inferences he needs to execute the deductions. But, if we are going to go on to axiomatize logic itself, we are going to need to be explicit not just about the basic logical axioms we take as given starting points for deductions, but also about the rules of inference that we are permitted to use. -
First-Order Logic in a Nutshell Syntax
First-Order Logic in a Nutshell 27 numbers is empty, and hence cannot be a member of itself (otherwise, it would not be empty). Now, call a set x good if x is not a member of itself and let C be the col- lection of all sets which are good. Is C, as a set, good or not? If C is good, then C is not a member of itself, but since C contains all sets which are good, C is a member of C, a contradiction. Otherwise, if C is a member of itself, then C must be good, again a contradiction. In order to avoid this paradox we have to exclude the collec- tion C from being a set, but then, we have to give reasons why certain collections are sets and others are not. The axiomatic way to do this is described by Zermelo as follows: Starting with the historically grown Set Theory, one has to search for the principles required for the foundations of this mathematical discipline. In solving the problem we must, on the one hand, restrict these principles sufficiently to ex- clude all contradictions and, on the other hand, take them sufficiently wide to retain all the features of this theory. The principles, which are called axioms, will tell us how to get new sets from already existing ones. In fact, most of the axioms of Set Theory are constructive to some extent, i.e., they tell us how new sets are constructed from already existing ones and what elements they contain. However, before we state the axioms of Set Theory we would like to introduce informally the formal language in which these axioms will be formulated. -
Boolean Logic
Math 1090 Part one: Boolean Logic Saeed Ghasemi York University 18th June 2018 Saeed Ghasemi (York University) Math 1090 18th June 2018 1 / 97 Alphabets of Boolean Logic 0 00 1 Symbols for variables. We denote them by p; q; r;::: or p ; q ; p0; r12 and so on. 2 Symbols for constants. There are two of them, \>" (read \top" or \true") and ? (read \bottom" or \false"). 3 Brackets: \(" and \)". 4 Boolean connectives: :, ^, _, ! and ≡ We will denote the alphabet of propositional (Boolean) logic by V. Some textbooks use less connectives e.g., only : and ^, the other three connectives can be defined using these two. However out alphabet V would contain all five connectives. Saeed Ghasemi (York University) Math 1090 18th June 2018 2 / 97 Definition Given an alphabet A (i.e. a set of symbols) a string (or expression or word) over A is a finite ordered sequence of symbols in A. 0 We use letters A; B; C; A ; B0 ::: to denote strings. Examples If A = fa; b; c; 0; 1g then aaab01, ba1, 01b11 are strings. Note that ab 6= ba since the order matters. These are strings over V pq !:p0? ((p()_ ≡ r ^ rq)(p> The concatenation of two strings A and B is the new string AB. The empty string which we will denote by λ is the unique string such that λA = Aλ = A for any string A. Note that the notion of equality \=" is not part of our alphabet V. It comes from \metatheory" or \metamathematics". Saeed Ghasemi (York University) Math 1090 18th June 2018 3 / 97 Well-formed formulas Definition A string over A is a well-formed formula (wff in short) if it is obtained by applying the following three steps finitely many times 1 Every variable and constant symbol is a wff (these are called \atomic formulas"). -
CS 357: Advanced Topics in Formal Methods Fall 2019 Lecture 7
CS 357: Advanced Topics in Formal Methods Fall 2019 Lecture 7 Aleksandar Zelji´c (materials by Clark Barrett) Stanford University Logical Definitions SupposeΣ is a signature. AΣ -formula is a well-formed formula whose non-logical symbols are contained inΣ. LetΓ be a set ofΣ-formulas. We write j=M Γ[s] to signify that j=M φ[s] for every φ 2 Γ. IfΓ is a set ofΣ-formulas and φ is aΣ-formula, thenΓ logically implies φ, writtenΓ j= φ, iff for every model M ofΣ and every variable assignment s, if j=M Γ[s] then j=M φ[s]. We write j= φ as an abbreviation for f g j= φ. and φ are logically equivalent (written j= = j φ) iff j= φ and φ j= . AΣ-formula φ is valid, written j= φ iff ; j= φ (i.e. j=M φ[s] for every M and s). True 2. Pv1 j= 8 v1 Pv1 False 3. 8 v1 Pv1 j= 9 v2 Pv2 True 4. 9 x 8 y Qxy j= 8 y 9 x Qxy True 5. 8 x 9 y Qxy j= 9 y 8 x Qxy False 6. j= 9 x (Px ! 8 y Py) True Which models satisfy the following sentences? 1. 8 x 8 y x = y Models with exactly one element. 2. 8 x 8 y Qxy Models( A; Q) where Q = A × A. 3. 8 x 9 y Qxy Models( A; Q) where dom(Q) = A. Examples Suppose that P is a unary predicate and Q a binary predicate.