DOCSLIB.ORG

On the Deduction Rule and the Number of Proof Lines (Extended Abstract)

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
On the Deduction Rule and the Number of Proof Lines (Extended Abstract) On the Deduction Rule and the Number of Proof Lines (Extended Abstract) Maria Luisa Bonet¤ Samuel R. Buss¤ Department of Mathematics Department of Mathematics U.C. Berkeley U.C. San Diego Berkeley, California 94720 La Jolla, California 92093 Abstract inference are also sound. A notable example of this is We introduce new proof systems for propositional the deduction rule which states that if a formula B has logic, simple deduction Frege systems, general deduc- a proof from an additional, extra-logical hypothesis A ` ⊃ tion Frege systems and nested deduction Frege systems, (in symbols, A B ) then there is a proof of A B . which augment Frege systems with variants of the This paper considers various strengthenings of this deduction rule. We give upper bounds on the lengths deduction rule and establishes upper bounds on the of proofs in these systems compared to lengths in proof-speedups obtained with these deduction rules. Frege proof systems. As an application we give a near-linear simulation of the propositional Gentzen By a \speedup" of a proof, we mean the amount sequent calculus by Frege proofs. The length of a that proofs can be shortened with additional inference proof is the number of steps or lines in the proof. rules. In this paper, the length of a proof is the number of lines in the proof; where a line consists A general deduction Frege proof system provides at either a formula or a sequent (depending on the proof most quadratic speedup over Frege proof systems. A system). We write B (and A ;:::;As B) nested deduction Frege proof system provides at most k 1 k a nearly linear speedup over Frege system where by to indicate that the formula B has Frege proof of · k lines (from the hypotheses A1;:::;As). More \nearly linear" is meant the ratio of proof lengths is T O(®(n)) where ® is the inverse Ackermann function. generally, we write \ k " to mean \provable in proof A nested deduction Frege system can linearly simulate system T with · k lines". If S and T are proof the propositional sequent calculus and hence a Frege systems we say that S can linearly (respectively, proof system can simulate the propositional sequent quadraticly) simulate T if, for any T -proof of k calculus with proof lengths bounded by O(n ¢ ®(n)). lines, there is an S -proof of the same (or sometimes As a technical tool, we introduce the serial transitive an equivalent) formula of O(k) lines (respectively, of closure problem: given a directed graph and a list of O(k2) lines). We say that T provides at most linear closure edges in the transitive closure of the graph, (respectively, quadratic speedup) over S if S can the problem is to derive all the closure edges. We give linearly (respectively, quadraticly) simulate T . a nearly linear bound on the number of steps in such a derivation when the graph is tree-like. An alternative, commonly used measure of the length of a propositional proof is the number of symbols in 1 Introduction the proof. This is the approach used, for instance, by Cook-Reckhow [3] and Statman [12]. It should be A Frege proof system is an inference system for noticed that the minimum number of lines in a Frege propositional logic in which the only rule of inference proof of a formula is polynomially related to the min- is modus ponens. Although it su±ces to have modus imum number of symbols in an extended Frege proof. ponens as the single inference rule to obtain a complete On the other hand, prior work on proof lengths in proof system, it is well-known that other modes of ¯rst-order logic has frequently measured the number ¤Supported in part by NSF Grant DMS-8902480 of lines in proofs; this includes [2, 4, 6, 7, 8, 10, 11] and others. We next de¯ne extensions to the Frege proof system We begin by de¯ning the main propositional proof that incorporate the deduction theorem as a rule of systems used in this paper. The logical connectives inference. For this purpose, the systems de¯ned below of all our systems are presumed to be :, _, ^ and have proofs in which the lines are sequents of the ² ⊃; however, our results hold for any complete set of form ¡ A; intuitively, the sequent means that the connectives. formulas in ¡ tautologically imply A: operationally, a sequent ¡ ² A means that A has been proved using De¯nition A Frege proof system (denoted F )is the formulas in ¡ as assumptions. characterized by: A general deduction Frege system (denoted dF ) incor- porates a strong version of the deduction rule. Each (1) A ¯nite set of axiom schemata. For example, a line in a general deduction Frege proof is a sequent of ⊃ ⊃ possible axiom schema is (A (B A)); A and the form ¡ ² A where A is a formula and ¡ is a set B represent arbitrary formulas. of formulas. When ¡ is empty, we write just ² A. (2) The only rule of inference is Modus Ponens (MP): The four valid axioms and inference rules in a general deduction Frege proof are: AA⊃B B ² A ¡ A an axiom (3) A proof in this system is a sequence of formulas fAg ² A ¡ Hypothesis A1;:::;An (also called `lines') where each Ai is ¡ ² A ⊃ B ¡ ² A either a substitution instance of an axiom schema 1 2 ¡ [ ² Modus Ponens or is inferred by Modus Ponens (MP) from some ¡1 ¡2 B Aj and Ak with j; k < i. ¡ ² B ¡Deduction Rule ¡ nfAg²A⊃B (4) The proof system must be consistent and com- dF plete. We write A1;:::;An k B to indicate that fA1;:::;Ang²B has a general deduction Frege proof The length of an F -proof is the number of lines in of · k lines. the proof; we write A to indicate that A has k Deduction Frege systems are quite general since they a Frege proof of length · k . We further write allow hypotheses to be \opened" and \closed" (i.e., A ;:::;As B to mean that B is provable from 1 k \assumed" and \discharged") in arbitrary order. A the hypotheses Ai with a Frege proof of · k lines; in more restrictive system is the nested deduction Frege other words, that there is a sequence of · k formulas proof system which requires the hypotheses to be each of which is one of the Ai 's, is an axiom, or is used in a `nested' fashion. The nested deduction inferred by modus ponens from earlier formulas such Frege systems are quite natural since the correspond that B is the ¯nal formula of the proof. Although we to the way mathematicians actually reason while have not speci¯ed the axiom schemata to be used in carrying out proofs. A second reason the nested a Frege proof system, it is easy to see that di®erent deduction Frege system seems quite natural is that we choices of axiom schemata will change the lengths of shall prove below that nested deduction Frege proof proofs only linearly. systems can simulate with linear size proofs both the The simplest form of the deduction theorem states propositional Gentzen sequent calculus and tree-like that if A ` B then ` A ⊃ B . This can be informally general deduction Frege proofs. phrased as a rule in the form The primary feature of the nested deduction Frege A ` B proof system is that hypotheses must be closed in A ⊃ B reverse order of their opening. And after a hypothesis which is called the 1-simple deduction rule; more is closed, any formula proved inside the scope of generally, the simple deduction rule is the hypothesis is now longer available. For example, a nested deduction Frege proof may look like the ` A1;:::An B following: A1 ⊃(¢¢¢(An ⊃B)¢¢¢) hA1i²A1 Hypothesis A1 opened by an example; the fragmentary ndF -proof given . above would be pictorially represented as: 2 hA1;A2i²A2 Hypothesis A2 opened A1 . 6 . 6 . 6 2. hA1;A2i²A3 6 6 A2 hA1i²A2 ⊃A3 Deduction Rule; A2 closed 6 6 6 4 . 6 . 6 h i² 6 A3 A1;A4 A4 Hypothesis A4 opened 6 ⊃ 6 2A2 A3 . 6 . 6 A4 6 6 hA ;A i²A 6 4 . 1 4 5 6 . hA1i²A4 ⊃A5 Deduction Rule; A4 closed 6 6 A5 . 6 . 6 A4 ⊃ A5 6 h i² 4 . A1 A6 . hi²A1 ⊃ A6 Deduction Rule; A1 closed A6 A1 ⊃ A6 A sequent in a nested deduction Frege (ndF ) proof Nested deduction Frege proofs are conceptually simple is of the form ¡ ² A where now ¡ is a sequence of and natural and, in practice, seem to simplify the pro- formulas. An ndF proof is a sequence of sequents cess of discovering proofs. Thus, it is surprising that ¡i ² Ai (i =1;2;:::;n) such that ¡ is taken to 0 a Frege proof system can simulate nested deduction be the empty sequence and, for each i, one of the Frege proofs with near linear size proofs: this fact is following holds: the content of our main theorems below. (a) ¡i =¡i¡1 and Ai is an axiom. 2 Summary of Results (b) ¡i =¡i¡1¤Ai. This opens an assumption, ¤ In this section we outline our primary results giving denotes concatenating a formula to the end of fairly sharp bounds on how much the deduction rule the sequence. can shorten proofs. Most of our results are stated in the form \If proof system X can prove a formula in ¤ ⊃ (c) ¡i¡1 is ¡i B and Ai is B Ai¡1 .
Load more

Recommended publications
  • 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).
    [Show full text]
  • Reflection Principles, Propositional Proof Systems, and Theories
    Reflection principles, propositional proof systems, and theories In memory of Gaisi Takeuti Pavel Pudl´ak ∗ July 30, 2020 Abstract The reflection principle is the statement that if a sentence is provable then it is true. Reflection principles have been studied for first-order theories, but they also play an important role in propositional proof complexity. In this paper we will revisit some results about the reflection principles for propositional proofs systems using a finer scale of reflection principles. We will use the result that proving lower bounds on Resolution proofs is hard in Resolution. This appeared first in the recent article of Atserias and M¨uller [2] as a key lemma and was generalized and simplified in some spin- off papers [11, 13, 12]. We will also survey some results about arithmetical theories and proof systems associated with them. We will show a connection between a conjecture about proof complexity of finite consistency statements and a statement about proof systems associated with a theory. 1 Introduction arXiv:2007.14835v1 [math.LO] 29 Jul 2020 This paper is essentially a survey of some well-known results in proof complexity supple- mented with some observations. In most cases we will also sketch or give an idea of the proofs. Our aim is to focus on some interesting results rather than giving a complete ac- count of known results. We presuppose knowledge of basic concepts and theorems in proof complexity. An excellent source is Kraj´ıˇcek’s last book [19], where you can find the necessary definitions and read more about the results mentioned here.
    [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]
  • 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).
    [Show full text]
  • Knowing-How and the Deduction Theorem
    Knowing-How and the Deduction Theorem Vladimir Krupski ∗1 and Andrei Rodin y2 1Department of Mechanics and Mathematics, Moscow State University 2Institute of Philosophy, Russian Academy of Sciences and Department of Liberal Arts and Sciences, Saint-Petersburg State University July 25, 2017 Abstract: In his seminal address delivered in 1945 to the Royal Society Gilbert Ryle considers a special case of knowing-how, viz., knowing how to reason according to logical rules. He argues that knowing how to use logical rules cannot be reduced to a propositional knowledge. We evaluate this argument in the context of two different types of formal systems capable to represent knowledge and support logical reasoning: Hilbert-style systems, which mainly rely on axioms, and Gentzen-style systems, which mainly rely on rules. We build a canonical syntactic translation between appropriate classes of such systems and demonstrate the crucial role of Deduction Theorem in this construction. This analysis suggests that one’s knowledge of axioms and one’s knowledge of rules ∗[email protected] [email protected] ; the author thanks Jean Paul Van Bendegem, Bart Van Kerkhove, Brendan Larvor, Juha Raikka and Colin Rittberg for their valuable comments and discussions and the Russian Foundation for Basic Research for a financial support (research grant 16-03-00364). 1 under appropriate conditions are also mutually translatable. However our further analysis shows that the epistemic status of logical knowing- how ultimately depends on one’s conception of logical consequence: if one construes the logical consequence after Tarski in model-theoretic terms then the reduction of knowing-how to knowing-that is in a certain sense possible but if one thinks about the logical consequence after Prawitz in proof-theoretic terms then the logical knowledge- how gets an independent status.
    [Show full text]
  • 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.
    [Show full text]
  • Substitution and Propositional Proof Complexity
    Substitution and Propositional Proof Complexity Sam Buss Abstract We discuss substitution rules that allow the substitution of formulas for formula variables. A substitution rule was first introduced by Frege. More recently, substitution is studied in the setting of propositional logic. We state theorems of Urquhart’s giving lower bounds on the number of steps in the substitution Frege system for propositional logic. We give the first superlinear lower bounds on the number of symbols in substitution Frege and multi-substitution Frege proofs. “The length of a proof ought not to be measured by the yard. It is easy to make a proof look short on paper by skipping over many links in the chain of inference and merely indicating large parts of it. Generally people are satisfied if every step in the proof is evidently correct, and this is permissable if one merely wishes to be persuaded that the proposition to be proved is true. But if it is a matter of gaining an insight into the nature of this ‘being evident’, this procedure does not suffice; we must put down all the intermediate steps, that the full light of consciousness may fall upon them.” [G. Frege, Grundgesetze, 1893; translation by M. Furth] 1 Introduction The present article concentrates on the substitution rule allowing the substitution of formulas for formula variables.1 The substitution rule has long pedigree as it was a rule of inference in Frege’s Begriffsschrift [9] and Grundgesetze [10], which contained the first in-depth formalization of logical foundations for mathematics. Since then, substitution has become much less important for the foundations of Sam Buss Department of Mathematics, University of California, San Diego, e-mail: [email protected], Supported in part by Simons Foundation grant 578919.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • Proof Complexity in Algebraic Systems and Bounded Depth Frege Systems with Modular Counting
    PROOF COMPLEXITY IN ALGEBRAIC SYSTEMS AND BOUNDED DEPTH FREGE SYSTEMS WITH MODULAR COUNTING S. Buss, R. Impagliazzo, J. Kraj¶³cek,· P. Pudlak,¶ A. A. Razborov and J. Sgall Abstract. We prove a lower bound of the form N ­(1) on the degree of polynomials in a Nullstellensatz refutation of the Countq polynomials over Zm, where q is a prime not dividing m. In addition, we give an explicit construction of a degree N ­(1) design for the Countq principle over Zm. As a corollary, using Beame et al. (1994) we obtain ­(1) a lower bound of the form 2N for the number of formulas in a constant-depth N Frege proof of the modular counting principle Countq from instances of the counting M principle Countm . We discuss the polynomial calculus proof system and give a method of converting tree-like polynomial calculus derivations into low degree Nullstellensatz derivations. Further we show that a lower bound for proofs in a bounded depth Frege system in the language with the modular counting connective MODp follows from a lower bound on the degree of Nullstellensatz proofs with a constant number of levels of extension axioms, where the extension axioms comprise a formalization of the approximation method of Razborov (1987), Smolensky (1987) (in fact, these two proof systems are basically equivalent). Introduction A propositional proof system is intuitively a system for establishing the validity of propo- sitional tautologies in some ¯xed complete language. The formal de¯nition of propositional proof system is that it is a polynomial time function f which maps strings over an alphabet § onto the set of propositional tautologies (Cook & Reckhow 1979).
    [Show full text]
  • 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.
    [Show full text]
  • 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").
    [Show full text]