1 / 40

Classical Logic

Classical Logic. Lecture Notes for SWE 623 by Duminda Wijesekera. Propositional and Predicate Logic. Propositional Logic The study of statements and their connectivity structure. Predicate Logic The study of individuals and their properties. Study syntax and semantics for both.

ellie
Télécharger la présentation

Classical Logic

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Classical Logic Lecture Notes for SWE 623 by Duminda Wijesekera Duminda Wijesekera

  2. Propositional and Predicate Logic • Propositional Logic • The study of statements and their connectivity structure. • Predicate Logic • The study of individuals and their properties. • Study syntax and semantics for both. • Propositional logic more abstract and hence less detailed than predicate logic. Duminda Wijesekera

  3. Propositional Logic: Syntax • A collection of atomic propositional symbols. • Say A = { ai : 0 < i }. A special atom _|_for contradiction • A collection of logical connectives. • (and) ^, (or) v, ( not )  , (implies) => • Inductively define propositions as: If X,Y are propositions, so are :– X^ Y, Xv Y, X => Y,  X. • Examples: • a1^a2, (a =>a2)v(a3^( a4)) are propositions. Duminda Wijesekera

  4. Propositional Logic: Semantics • A model M of a propositional language consists of • a collection of atoms, say B = { bi : 0 < i }, where_|_ is excluded from B, and • a partial mapping M from A = { ai : 0 < i } to B = { bi : 0 < i }. • If M(ai) e B, we say that ai is true in M. We write “ai is true in M” as M |= ai. (Read M satisfies ai). • |= is referred to asthe satisfaction relation. Duminda Wijesekera

  5. Propositional Semantics: Continued • Extend M, and therefore the satisfaction relation to all propositions using the following inductive definition: • M |= X^ Y iff M |= XandM |= Y. • M |= Xv Y iff M |= XorM |= Y. • M |= X => Y ifM |= XthenM |= Y. • M |=  X, if it is not the case thatM |= X. • Notice usage of truth tables Duminda Wijesekera

  6. Propositional Logic: Example • B = { a1, a3} where M given as M(a1) = a1 and M(a2) = a2 has the following properties. • M |= a1 • M |= a1^ a3 • M |= a2 • M |= a2 =>a4 • M does not satisfy the following propositions. • M |= a4 • M |= a1 =>a4 Duminda Wijesekera

  7. Propositional Logic: Proofs • What formulas hold in all models ? • I.e. can we check if a given proposition is true in all models without going through all possible models? • Need proofs to answer this question. • We use Natural Deduction proofs. • Recommended: Read Ch 2 of Logic and Computation by L.C. Paulson. Duminda Wijesekera

  8. Natural Deduction for Prop. Logic • Proofs are trees of formulae made by applying inference rules. • An inference rule is of the form: A1 …… An B • Here A1 ….. An are said to be premises (or antecedents) of the rule, and B is said to be the conclusion (consequent) of the rule. Duminda Wijesekera

  9. Natural Deduction for Prop. Logic • Hence a proofs is a trees whose • Root is the theorem to be proved, • Branches are rules, and • Leaves are the assumptions (axioms) of the proof. • Example • A1 A2 A3 C1 C2 Assumptions B1 B2 Applications of rules D  Theorem being proved • There are introduction and elimination rules for each connective in Natural Deduction proof systems. Duminda Wijesekera

  10. Rules for Conjunction • Introduction A B A ^ B • Elimination A ^ BA ^ B A B Duminda Wijesekera

  11. Rules for Disjunction • Introduction A B A v B A v B • Elimination [A] [B] A v B C C C • [X] denotes discharged assumption X. Duminda Wijesekera

  12. Rules for Implication • Introduction [A] B A => B • Elimination (Modus Ponens) A => B A B Duminda Wijesekera

  13. Rules for Negation •  B interpreted as ( B => _|_). Hence we get the following rules from those of implication. •  Introduction  Elimination [B]  B B _|_ _|_ ________ B • Special Contradiction Rule:  B _|_ __________ B Duminda Wijesekera

  14. Propositional Proofs: Examples • Prove: ( A ^ B ) => (A v B) • Notice: • The outermost connective is =>. Hence, the last step of the proof must be an implication introduction. • That means, we must assume ( A ^ B ) and prove (A v B), and then discharge the assumption by using => introduction rule. • In order to prove (A v B) from ( A ^ B ), we must use v –introduction, and hence must prove either A or B from ( A ^ B ). • This plan forms a skeleton of a proof. Duminda Wijesekera

  15. Prop. Proof: Example Continued • Prove: ( A ^ B ) => (A v B) [A ^ B ] A ^elimination A v B v introduction ( A ^ B ) => (A v B) => introduction • Proofs are analyzed backwards, I.e. start unraveling the logical structure of the conclusion and work backwards to the assumptions. Draw out a plan based on your analysis and write down the formal proof. Duminda Wijesekera

  16. Derived Rules • These are rules derived from other rules. • Example: A ^ B B ^ A • Here is the derivation: A ^ BB ^ A B A^elimination B ^ A ^introduction Duminda Wijesekera

  17. Soundness and Completeness • A rule A1 …… An is said to be sound if for every B model in which all of A1 …… An are true, then so is B. I.e. if M |= A1 , …… , M |= An, then M |= B. • A collection of rules are sound if all rules in the collection is sound. • A collection of rules is complete if M |= A for all models M, then A is provable. I.e. there is a proof of A using the given set of rules. (Denoted |R-- A ) where R is the set of rules. Duminda Wijesekera

  18. Predicate Logic • Language to describe properties of individuals. • Thus, syntax is able to describe individuals, their properties (relationships) and functions. • These are to be thought of as names of individuals, properties (relationships) and functions. • Models are “incarnations” of these individuals, properties (relationships) and functions. • More detailed than propositional logic. Duminda Wijesekera

  19. Predicate Logic: Syntax • A collection of constants– say { ci : i >= 0 }. • Constants are names for individuals. E.g.: 0, 1. • Note: not all individuals in a model have names. • A collection of variables– say { xi : i >= 0 }. • Needed to generically refer to individuals. • Think of them as standing in place of pronouns like it, she. • A collection of function symbols- say { fi : i >= 0 }. • May be of different arities, and may be typed. E.g.: +(x,y) • A collection of predicate symbols- say { pi : i >= 0 }. • May be of different arities. • Encodes properties of individuals. E.g.: prime(x). Duminda Wijesekera

  20. Predicate LogicRecursive Definition of Terms • Every variable is a term. • Every constant is a term. • If fi is an n-ary function symbol and t1, .., tn are terms, then fi(t1, .., tn) is a term. • We use {ti : i <=0 } for the collection of terms. • Examples: • f(x, g(2, y)) is a term, where f, g are function symbols and x, y are variables. • +( x, *(3,y)) is a term in arithmetic usually written as x + (3*y) Duminda Wijesekera

  21. Recursive Definition of Formulas • If pi is an n-ary predicate symbol and t1, .., tn are terms, then pi(t1, .., tn ) is an atomic formula. • If A and B are formulas, then so are: • A^ B, A v B,  A, A => B. • xi A(xi),  xi A(xi), where xi is a variable. • ,  are referred to as the universal and existential quantifier, respectively. • A formula that does not have either quantifier is said to be a quantifier free. Duminda Wijesekera

  22. Free and bound Variables • In x A(x), the variable x is said to be bound; meaning the name x plays no significant role. (compare with he, she, it) • A variable x occurs bound in a formula if xor x is a part of it. More precisely, x occurs bound in: • yA(y) or y A(y) if x and y are the same variable. •  A if x occurs bound in A. • A^ B, A v B, A => B if x occurs bound in either A or B. Duminda Wijesekera

  23. Substitutions • If A is a formula, t is a term and x is a variable, then A[t/x] is the formula obtained by substituting t for x in A. • A[t1/x1, … tn/xn] is the formula resulting in simultaneously substituting x1, …xn by t1, …tn. • Note: Simultaneous substitution Q(x,y)[x/y,y/x] yields Q(y,x) but iterated substitution Q(x,y)[x/y][y/x] yields Q(y,y). Duminda Wijesekera

  24. Substituting Terms for Variables • In A[t/x], the free variables of t stand the danger of becoming bound in A. Hence, need a precise definition. • If x is y then yA(y) [x/y] is yA(y). If not let z be a fresh variable (I.e. not in t, x) then (yA(y) )[t/x] is z (A(z/y) [t/x]). • Similar definition for y A(y). • Examples: • y (y = 1) [y/y] is y (y = 1). Here x is y and t is x. • y (y+1 > x) [2y+x/x] is z ((z+1>x)[2y+x/x] I.e. z (z+1>2y+x). Here t is (2y+x). Duminda Wijesekera

  25. Substituting Terms Continued • ( A )[t/x] is  (A [t/x]) • (A^ B) [t/x] is (A[t/x]^ B[t/x]) • (Av B) [t/x] is (A[t/x]vB[t/x]) • (A=> B) [t/x] is (A[t/x]=> B[t/x]) • Pi(t1, .. tn) [t/x] is Pi(t1[t/x], .. tn[t/x]) for predicate symbol Pi. Duminda Wijesekera

  26. Predicate Logic: Semantics • A model consists of • A set (of individuals), say A = { ai : i >= 0 }. • A set of total functions Fn = { fni : i >= 0 } on A. • I.e. fni(aj) is some ak for every aj. • A set of predicates Pr = { pri : i >= 0 } over A. • Do not have to be total. • Can have many arities. Duminda Wijesekera

  27. Interpreting Syntax • Mapping from Syntax to Semantics: • A mapping mCons : { ci : I >= 0 } to A={ai: i >= 0}. • Need not be ONTO A. I.e. there could be unnamed individuals in the semantic domain. • A mapping mFun : { fi : I >= 0 } to Fn={fni: i >= 0}. • Need not be onto. I.e. there could be unnamed functions in the semantic domain. • A mapping mPred: { pi : I >= 0 } to Pr={pri: i >= 0}. • Need not be onto. I.e. there could be unnamed predicates in the semantic domain. Duminda Wijesekera

  28. Interpreting Formulas: naming • We do not interpret formulas with free variables. • In order to interpret quantified formulas, need to expand the syntax by adding a constant in the syntax for each unnamed individual in the model. • I.e. for each ai for which there is no cj such that Fn(cj ) is ai, add a new constant Cai to the syntax. • Now expand the definition of terms to include these new constants. Let newT = { Nti : i >= 0} be the collection of new terms so defined. Duminda Wijesekera

  29. Interpreting Formulas • Let M be a model. We define M |= F for every quantified formula as follows. • For every n-ary predicate symbol pi , and every sequence of new variable free terms Nt1, … Ntn define M |= pi(Nt1, … Ntn ) if and only if mPred(pi)(Nt1, … Ntn ). • I.e. pi(Nt1, … Ntn ) is true in M if and only if its image under the map mPred holds with parameters Nt1, … Ntn . Duminda Wijesekera

  30. Interpreting Formulas: Continued • For every formula A , M |= y A(y) if and only if M |= A(Nti) for every Nti e newT. • For every formula A , M |= y A(y) if and only if there is some Nti e newT satisfying M |= A(Nti). • M |= A ^ B if M |= A and M |= B . • M |= A v B if M |= A or M |= B. • M |= A => B if when M |= A then M |= B. • M |=  A if it is not the case that M |= A. Duminda Wijesekera

  31. Proof Rules for Predicate Logic • Proof rules of introduction and elimination of ^, v, =>, and . • New rules required for introductionand elimination of  and quantifiers. Duminda Wijesekera

  32. Proof Rules for  •  Introduction A(x) provided x is not free in the x A(x) assumptions of A •  Elimination x A(x) A[t/x] Duminda Wijesekera

  33. Proof Rules for  •  Introduction A[t/x] xA(x) •  Elimination [A] provided x is not free xA(x) B in B nor in the B assumptions of B apart from A Duminda Wijesekera

  34. An Example Proof • Prove ((x A(x)) ^ B)=> (x (A(x)^ B)) provided that x is not free in B. • Plan: • Since outer connective is =>, need to use => introduction at the last step. Hence can use (x A(x)) ^ B as an assumption for the steps above. • Now in order to get x (A(x)^ B) using  introduction, we need to get A[t/x] )^ B. • Can use ^ eliminationto (x A(x)) ^ B and obtain B • Can use x elimination to get A[t/x]. Duminda Wijesekera

  35. Example Proof x A(x) ^ Bx A(x) ^ B x A(x) [A(t/x)] B A(t/x) ^ B x(A(x) ^ B • The other directionof the proof appears in the handout page 32. Duminda Wijesekera

  36. Induction Rule [A(x)] A[0/x] A[x+1/x] A(x) Proviso: x is not free in the assumptions of A[x+1/x] apart from A(x). Duminda Wijesekera

  37. Equality Reasoning • Rules for equality • Reflexivity axiom: t = t. • Symmetry rule: t = u . u = t • Transitivity rule: s = t t = u . s = u • Congruence laws for each function and predicate symbol, or substitution rules. Duminda Wijesekera

  38. Equality Reasoning: Continued • Congruence Law for functions: t1 = u1 …. tn = un f(t1, …., tn) = f(u1, ….,un) • Congruence Law for Predicates: t1 = u1 …. tn = un p(t1, …., tn)  p(u1, ….,un) • Substitution Rule: t = u S[t/x] = S[u/x] Duminda Wijesekera

  39. Equality Reasoning: An Example This example is from Page 37, of the Logic handout. x f(x,x) = x f(g(z), g(z)) = g(z) p(f(g(z), g(z))  p(g(z)) p(f(g(z), g(z))   p(g(z)) Duminda Wijesekera

  40. Logic: Suggested Exercises • Go thorough all proofs and suggested exercises in the handout. • Take the midterm and final exams from last semester and attempt the proofs. • Go through the second homework from last semester. • Reference: Chapter 2 of Logic and Computation by L.C. Paulson. Duminda Wijesekera

More Related