Announcements CS243: Discrete Structures First Order Logic, I Homework 1 is due now Rules of Inference I Homework 2 is handed out today I¸sıl Dillig I Homework 2 is due next Tuesday I¸sıl Dillig, CS243: Discrete Structures First Order Logic, Rules of Inference 1/41 I¸sıl Dillig, CS243: Discrete Structures First Order Logic, Rules of Inference 2/41 Review of Last Lecture Translating English into First-Order Logic I Building blocks in FOL: constants, variables, predicates Given predicates student(x), atWM (x), and friends(x, y), how do I Formulas formed using predicates, connectives, and quantifiers we express the following in first-order logic? I Truth value of FOL formulas depend on universe of discourse I ”Every William&Mary student has a friend” and interpretation of predicates and variables I ”At least one W&M student has no friends” I Universal quantification x.P(x) is true if P is true for all ∀ objects in universe of discourse I ”All W&M students are friends with each other” I Existential quantification x.P(x) is true if there exists an ∃ object for which P is true I¸sıl Dillig, CS243: Discrete Structures First Order Logic, Rules of Inference 3/41 I¸sıl Dillig, CS243: Discrete Structures First Order Logic, Rules of Inference 4/41 Satisfiability, Validity in FOL Equivalence I The concepts of satisfiability, validity also important in FOL I Two formulas F and F are equivalent if F F is valid 1 2 1 ↔ 2 I An FOL formula F is satisfiable if there exists some domain I In PL, we could prove equivalence using truth tables, but not and some interpretation such that F evaluates to true possible in FOL I Example: Prove that x.P(x) Q(x) is satisfiable. ∀ → I However, we can still use known equivalences to rewrite one formula as the other I An FOL formula F is valid if, for all domains and all interpretations, F evaluates to true I Example: Prove that ( x. (P(x) Q(x))) and ¬ ∀ → x. (P(x) Q(x)) are equivalent. I Prove that x.P(x) Q(x) is not valid. ∃ ∧ ¬ ∀ → I Example: Prove that x. y.P(x, y) and x. y. P(x, y) are I Formulas that are satisfiable, but not valid are contingent, ¬∃ ∀ ∀ ∃ ¬ equivalent. e.g., x.P(x) Q(x) ∀ → I¸sıl Dillig, CS243: Discrete Structures First Order Logic, Rules of Inference 5/41 I¸sıl Dillig, CS243: Discrete Structures First Order Logic, Rules of Inference 6/41 1 Motivation for Proof Rules Rules of Inference I Proof rules are written as rules of inference: Hypothesis1 I Learned how to express various facts in logic, but this is not Hypothesis2 all that useful on its own ... Conclusion I The reason logic is useful: allows formalizing arguments, constructing validity proofs, and make inferences I An example inference rule: I Rest of lecture: learn about proof rules for logic All men are mortal Socrates is a man I By applying proof rules, can make logical inferences that are ∴ Socrates is mortal correct by construction I Valid inference rule, but too specific I We’ll learn about more general inference rules that will allow constructing formal proofs I¸sıl Dillig, CS243: Discrete Structures First Order Logic, Rules of Inference 7/41 I¸sıl Dillig, CS243: Discrete Structures First Order Logic, Rules of Inference 8/41 Modus Ponens Example Uses of Modus Ponens I Most basic inference rule is modus ponens: I Application of modus ponens in propositional logic: φ1 p q ∧ φ1 φ2 (p q) r → ∧ → φ2 I Application of modus ponens in first-order logic: I This rule is valid because we know φ1 is true, and by definition of implication, if φ is true, then φ must be true 1 2 P(a) P(a) Q(b) I Modus ponens applicable to both propositional logic and → first-order logic I¸sıl Dillig, CS243: Discrete Structures First Order Logic, Rules of Inference 9/41 I¸sıl Dillig, CS243: Discrete Structures First Order Logic, Rules of Inference 10/41 Modus Tollens Example Uses of Modus Tollens I Second imporant inference rule is modus tollens: I Application of modus tollens in propositional logic: φ φ p (q r) 1 → 2 → ∨ φ (q r) ¬ 2 ¬ ∨ φ ¬ 1 I Recall: φ φ and its contrapositive φ φ are I Application of modus tollens in first-order logic: 1 → 2 ¬ 2 → ¬ 1 equivalent to each other Q(a) P(a) Q(a) I Therefore, correctness of this rule follows from modus ponens ¬ → ¬ and equivalence of a formula and its contrapositive. I¸sıl Dillig, CS243: Discrete Structures First Order Logic, Rules of Inference 11/41 I¸sıl Dillig, CS243: Discrete Structures First Order Logic, Rules of Inference 12/41 2 Hypothetical Syllogism (HS) Or Introduction φ φ φ 1 1 → 2 φ φ φ1 φ2 2 → 3 ∨ φ1 φ3 → I Correctness follows from definition of ∨ I Basically says ”implication is transitive” I φ φ is true if either φ or φ is true. 1 ∨ 2 1 2 I Example: I Example application: ”Socrates is a man. Therefore, either P(a) Q(b) Socrates is a man or there are red elephants on the moon.” → Q(b) R(c) → I The book calls this rule addition – feel free to use whichever term is more natural for you I¸sıl Dillig, CS243: Discrete Structures First Order Logic, Rules of Inference 13/41 I¸sıl Dillig, CS243: Discrete Structures First Order Logic, Rules of Inference 14/41 Or Elimination And Introduction φ1 φ2 ∨ φ1 φ2 ¬ φ2 φ 1 φ φ 1 ∧ 2 I Either φ1 or φ2 is true. We know φ2 is false. Therefore, φ1 I This rule just follows from definition of conjunction must be true. I Example application: ”It is Tuesday. It’s the afternoon. I Example application: ”It is either a dog or a cat. It is not a Therefore, it’s Tuesday afternoon”. dog. Therefore, it must be a cat.” I The book calls this rule conjunction; I call it And Intro – use I The book calls this rule disjunctive syllogism; I call it Or whichever you prefer Elimination – use whichever you prefer I¸sıl Dillig, CS243: Discrete Structures First Order Logic, Rules of Inference 15/41 I¸sıl Dillig, CS243: Discrete Structures First Order Logic, Rules of Inference 16/41 And Elimination Resolution I Final inference rule: resolution φ φ φ1 φ2 1 2 ∨ ∧ φ φ φ1 ¬ 1 ∨ 3 φ φ 2 ∨ 3 I This rule also just follows from definition of conjunction I To see why this is correct, observe φ1 is either true or false. I Example application: ”It is Tuesday afternoon. Therefore, it is Tuesday”. I Suppose φ1 is true. Then, φ1 is false. Therefore, by second ¬ hypothesis, φ3 must be true. I The book calls this rule simplification; I call it And Elimination – use whichever you prefer I Suppose φ1 is false. Then, by 1st hypothesis, φ2 must be true. I In any case, either φ or φ must be true; φ φ 2 3 ∴ 2 ∨ 3 I¸sıl Dillig, CS243: Discrete Structures First Order Logic, Rules of Inference 17/41 I¸sıl Dillig, CS243: Discrete Structures First Order Logic, Rules of Inference 18/41 3 Resolution Example Summary Name Rule of Inference φ1 φ1 φ2 Modus ponens → φ2 φ1 φ2 → φ2 I Example 1: Modus tollens ¬ φ1 ¬ P(a) Q(b) φ1 φ2 → ∨ ¬ φ2 φ3 Q(b) R(c) Hypothetical syllogism → ∨ φ1 φ3 → φ1 Or introduction φ1 φ2 ∨ I φ1 φ2 Example 2: ∨ φ2 p q Or elimination ¬ ∨ φ1 p ¬ φ1 φ2 And introduction φ1 φ2 ∧ φ1 φ2 And elimination ∧ φ1 φ1 φ2 ∨ φ1 φ3 Resolution ¬ ∨ φ2 φ3 ∨ I¸sıl Dillig, CS243: Discrete Structures First Order Logic, Rules of Inference 19/41 I¸sıl Dillig, CS243: Discrete Structures First Order Logic, Rules of Inference 20/41 Using the Rules of Inference Encoding in Logic Assume the following hypotheses: I First, encode hypotheses and conclusion as logical formulas. 1. It is not sunny today and it is colder than yesterday. I To do this, identify propositions used in the argument: I s = ”It is sunny today” 2. We will go to the lake only if it is sunny. I c= ”It is colder than yesterday” 3. If we do not go to the lake, then we will go hiking. I l = ”We’ll go to the lake” 4. If we go hiking, then we will be back by sunset. I h = ”We’ll go hiking” Show these lead to the conclusion: ”We will be back by sunset.” I b= ”We’ll be back by sunset” I¸sıl Dillig, CS243: Discrete Structures First Order Logic, Rules of Inference 21/41 I¸sıl Dillig, CS243: Discrete Structures First Order Logic, Rules of Inference 22/41 Encoding in Logic, cont. Formal Proof Using Inference Rules I ”It’s not sunny today and colder than yesterday.” 1. s c Hypothesis I ”We will go to the lake only if it is sunny” ¬ ∧ 2. l s Hypothesis → 3. l h Hypothesis I ”If we do not go to the lake, then we will go hiking.” ¬ → 4. h b Hypothesis → I ”If we go hiking, then we will be back by sunset.” I Conclusion: ”We’ll be back by sunset” I¸sıl Dillig, CS243: Discrete Structures First Order Logic, Rules of Inference 23/41 I¸sıl Dillig, CS243: Discrete Structures First Order Logic, Rules of Inference 24/41 4 Another Example Encoding in Logic Assume the following hypotheses: I First, encode hypotheses and conclusion as logical formulas. 1. It is not raining or Kate has her umbrella I To do this, identify propositions used in the argument: 2.