Why Learning Logic? Logic. Propositional Logic. Compound Propositions

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1 Logic Objectives Propositions and compound propositions Negation, conjunction, disjunction, and exclusive or Implication and biconditional Logic equivalence and satisfiability Application of propositional logic Predicates and quantifiers Valid arguments Rules of inference Reference: 1.1, 1.2, 1.3, 1.4, 1.5, 1.6 Why Learning Logic? Logic has broad applications to mathematics, philosophy, engineering, law, and other fields. Logic is the basis of all mathematical reasoning and of all automated reasoning. Reasoning is the essential ability to infer what is unknown from what is known. It has practical applications to computer hardware design, system specification, computer programming, artificial intelligence, and other areas of CS. 1 2 Propositional Logic A proposition is a declarative sentence that is either true or false. Ex Are the following propositions? 1) Guelph is a city of Ontario. 2) The Earth rotates around the Moon. 3) Who authored the book On the Origin of Species? 4) Please write a Java program to sort integers. 5) 3 7 = 20. 6) 5 + y = 15. Denote propositional variables (PVs) by letters p, q, r, s,, and truth values by T and F. Compound Propositions A compound proposition (CP) is formed from existing propositions using logical operators. Negation of proposition p is p. Truth table of negation Conjunction of p and q: p q Ex p: John played well in the tennis game. q: John lost the game. Disjunction of p and q: p q Ex Take Tylenol when you have headache or body pain. Ex All entrees served with soup or salad. Exclusive or of p and q: p q 3 4 1

2 Compound Proposition: Implication Conditional p q (implication) Common expressions p implies q if p, then q p is sufficient for q p only if q q whenever p q follows from p p: hypothesis, antecedent or premise q: conclusion or consequence Ex p: You take a vacation ; q: You are energized. Causal relation between p and q is not required. Do not confuse p q with if then statement in programming languages. Other conditional statements related to p q 1) Converse q p 2) Contrapositive q p 3) Inverse p q 5 Compound Proposition: Biconditional Biconditional p q Common expressions if p then q, and vice versa p if and only if q p is necessary and sufficient for q p iff q Ex p: Zoe can visit CN Tower ; q: Zoe buys ticket. p q is true when exactly both p q and q pare true. In natural language, a biconditional may be stated as if it is an implication. Ex If you finish the homework, then you can play. 6 Truth Table of Compound Proposition Truth tables of logic operators p q p p q p q p q p q q p p q T T F T T F T T T T F F F T T F T F F T T F T T T F F F F T F F F T T T The truth value of any compound proposition can be determined using truth tables of logic operators. Ex Determine truth table for p (( q) (p q)) Ex What is the truth table for p q p q? Precedence of logic operators (highest),,,,, (lowest) Application of Propositional Logic (PL) Logic has applications in mathematics, philosophy, computer science, engineering, and many other areas. Expressing English Sentences English sentences may be ambiguous while the meaning of logic sentences are precise. Once information in English is translated into logic expressions, they can be used in automated reasoning by artificial intelligence systems. Ex Engine cannot start if battery is dead or there is no gas or electrical cable is broken. In each row of truth table where CP = T, PV values convey the meaning of English sentence

3 Application of PL: Bit Operation Inside computers, information is encoded by bits. A bit is a symbol with values 0 or 1. Parallel to propositional logic Logic F/T PV Bit Op 0/1 Boolean variable NOT AND OR XOR A bit string is a sequence of zero or more bits. The length of a bit string is its number of bits. Ex Determine if a message bit string from network is intended for the computer of address String where bit locations 7 to 12 form the destination address Application of PL: Boolean Searches Logical operators are commonly used for searching large collections of information. Ex Implicit AND in Google Search terms are connected automatically by AND. Ex Case insensitivity by implicit OR Automatically use disjunction of alternative cases Ex Explicit OR in Google Disjunction of terms is specified by OR or Application of PL: Logic Circuits Logic circuits in computer hardware and many control systems are based on propositional logic. Basic units of logic circuits are inverter, AND gate and OR gate, where each input or output signal is a bit. p p p p q p q q p q Combinatorial circuit can be constructed from gates, which can implement a compound proposition. p q r r p q r p (q r) 11 Logical Equivalence: Intuition A key step in any rigorous argument is to replace one statement with another of the same truth value. Ex Persecution claims that the suspect was motivated to commit the crime and was in the crime scene. Defence argues that the suspect was either not motivated for the crime or not in crime scene, or both. Does the argument of defence negate the persecution statement? 12 3

4 Logical Equivalence A tautology is a CP that is always true. A contradiction is a CP that is always false. A contingency is a CP that is neither of the above. Ex Tautology, contradiction, and contingency CPs p and q are logically equivalent, denoted p q, if p q is a tautology. Ex Are p q and q p logically equivalent? Ex Does p (q r) (p q) (p r) hold? Truth table of a CP of n PVs has 2 n rows. Ex Application of De Morgan law Important Equivalences (1) 1. Identity: p T p p F p 2. Domination: p T T p F F 3. Idempotent: p p p p p p 4. Double negation: ( p) p 5. Commutative: p q q p p q q p 6. Associative: (p q) r p (q r) (p q) r p (q r) 7. Distributive: p (q r) (p q) (p r) p (q r) (p q) (p r) 8. De Morgan: (p q) p q (p q) p q Important Equivalences (2) 9. Absorption: p (p q) p p (p q) p 10. Negation: p p T p p F 11. Contraposition: p q q p 12. Implication elimination: p q p q 13. Biconditional elimination: p q (p q) (q p) Constructing New Equivalences There are infinitely many logical equivalences. How do we obtain one when we need it? 1) Verify by truth tables. 2) Construct from known equivalences. Transitivity: If p q and q r, then p r. A proposition u in a CP r can be replaced by a logically equivalent CP v without changing the truth value of r. Ex Show (p q) p q

5 Satisfiability A CP is satisfiable if there is an assignment of truth values to its PVs that makes it true. The assignment is a solution to the satisfiability problem. A CP is unsatisfiable iff it is a contradiction. A CP is unsatisfiable iff its negation is a tautology. How can satisfiability of a CP be determined? Ex Determine satisfiability for each CP below. 1) (p q) (q r) (r p) 2) (p q r) ( p q r) 3) (p q) (q r) (r p) (p q r) ( p q r) Application of PL: System Specifications Ex Decide if the system specs are consistent. 1) Either device x is installed or device y is installed. 2) Device y is not installed. 3) If device y is installed, then device x is installed. Solution method a) Determine PVs. b) Express each system spec as a CP. c) Express the collection of system specs as a conjunction q. d) Determine satisfiability of q. e) The system specs are consistent iff q is satisfiable Application of PL: Logic Puzzles Logic puzzles are fun activities to practice logic and are often used to demo automated reasoning systems. Ex Dad lets kids John and Mary to play outside, and both get mud on foreheads. When they return, dad says At least one of you has a muddy forehead. He then asks kids to answer the question Do you know whether you have a muddy forehead? by Yes or No. He asked twice. What is the answer of each kid at each time? Assumptions 1) A kid can see the other forehead but not his or her own. 2) Both kids answer each question simultaneously. Why Learning Predicate logic Propositional logic cannot adequately support certain expressions and reasoning patterns. Ex From a) and b), we should be able to conclude c). a) Every Canadian province has a capital city. b) Ontario is a Canadian province. c) Ontario has a capital city. Predicate logic is a more powerful type of logic. It is also called first order logic (FL)

6 Predicates Ex A statement with variable ``x is an even number The statement can be expressed in FL by P(x). P denotes predicate is an even number. x is the variable. Statement P(x) is the value of propositional function P at x. Once a value is assigned to x, statement P(x) becomes a proposition and has a truth value. Ex Denote ``Province x has the capital city y by C(x, y). What is the truth value of C(Ontario, Guelph)? A statement P(x 1,,x n ) is the value of propositional function P at n tuple (x 1,,x n ), and P is a n ary predicate. Application of FL: Program Verification The functionality of a program can be stated by a pair of precondition and postcondition. Predicates can be used to specify preconditions and postconditions. Correctness of program is established by showing that whenever the input satisfies precondition, the output will satisfy postcondition. Ex Program to swap values of variables x and y 1. temp := x; 2. x := y; 3. y := temp; The Universal Quantifier We often express that a predicate is true over a range of objects. This is quantification. We do so with words all, many, some, few, one, none, etc. A domain refers to all possible values of one variable. Universal quantification of P(x) is the statement P(x) for all values of x in the domain, and is denoted x P(x). Ex Let P(x) denote x/2 < x. What is the truth value of x P(x), where the domain of x is all positive reals? Ex Answer the above question for the domain of reals. A value x s.t. P(x) is false is a counterexample of x P(x). If the domain is empty, x P(x) is true. The Existential Quantifier Existential quantification of P(x) is the statement There exists a value in the domain s.t P(x), denoted x P(x). Ex Let P(x) denote x/2 < x. What is the truth value of statement x P(x), where the domain is all reals? What if the domain is all negative reals? If the domain is empty, x P(x) is false. When the domain is finite, the quantification x P(x) is equivalent to a disjunction. To decide the truth value for x P(x), loop through each x. If a value is found where P(x) is true, then x P(x) is true. Otherwise, x P(x) is false. For a finite domain, the quantification x P(x) is equivalent to a conjunction, and a similar procedure can be used to decide its truth value

7 Variable Bounding Quantifiers and have higher precedence than logical operators,,,,. Ex x P(x) Q(x) When a quantifier is applied to a variable x, this occurrence of x is bound. If x is neither bound, nor set to a particular value, then x is free. To turn a propositional function into a proposition, all its variables must be bound or set to particular values. The part of a logic expression to which a quantifier is applied is the scope of the quantifier. Logical Equivalences Involving Quantifiers Statements S and T with predicates and quantifiers are logically equivalent, denoted S T, iff they have the same truth value no matter which predicates are substituted into them and which domain is used for their variables. Ex Determine equivalence. S: x P(x). T: x P(x). De Morgan s laws for quantifiers 1) x P(x) x P(x) 2) x P(x) x P(x) Ex Negation of x (x 2 <x) Nested Quantifiers Ex x y (x * y = y * x), where x and y are reals. Everything in the scope of a quantifier can be viewed as a propositional function. View x y P(x,y) as x Q(x), where Q(x) = y P(x,y). Ex Every computer has a CPU. Think nested quantifiers through nested loops. Ex Determine the truth of x y (2x + y = 10), where x is from {1,2,3} and y is from {1, 2,, 10}. Unless all quantifiers are identical, the order of quantifiers is important. 27 Valid Arguments in PL Ex Is the following a valid argument? 1) If the output of the program is wrong, then it is buggy. 2) The output of the program is wrong. 3) Therefore, the program is buggy. A. An argument in PL is a sequence of propositions with the last being conclusion and the rest being premises. B. An argument is valid if the truth of its premises implies the truth of the conclusion. C. An argument form (AF) in PL is a sequence of CPs. D. An AF is valid, if no matter which particular propositions substitute PVs in its premises, the conclusion is true whenever the premises are all true. 28 7

8 Rule of Inference for PL Why do we study the validity of AFs? With a valid AF, if we substitute its PVs with particular propositions, we always obtain a valid argument. An AF with premises p 1, p 2,, p n and conclusion q is valid when (p 1 p 2 p n ) q is a tautology. How do we verify that an AF is valid? What is the limitation of the method? A rule of inference is a simple AF. The alternative Establish the validity of some rules of inference. Use the rules to construct more sophisticated (valid) AFs. 29 Commonly Used Rules of Inference 1. Modus ponens 2. Modus tollens 3. Hypothetical syllogism 4. Disjunctive syllogism 5. Addition 6. Symplification 7. Conjunction 8. Resolution 30 Build Argument by Rules of Inference Ex Show that premises A through D lead to E. A. It s not sunny this afternoon and it s colder than yesterday. B. We will go swimming only if it is sunny. C. If we do not go swimming, then we will take a canoe trip. D. If we take a canoe trip, then we will be home by sunset. E. We will be home by sunset. General procedure to show validity of an argument a) Determine PVs. b) Translate each premise as a CP. c) Derive new CPs and justify each derivation. d) Terminate when the conclusion is derived. If a truth table is used, how many rows are needed? Incorrect Reasoning Ex Are following arguments valid? A. If 3, then 3. We know that 3. Consequently, 3 3 =. B. If I have the flu, then I have fever. I do have fever. Therefore, I have the flu. Summary A valid AF does not ensure the truth of conclusion, if some premises are false. Fallacies resemble rules of inference, but are based on contingencies rather than tautologies

9 Rules of Inference for Quantified Statements 9. Universal instantiation From x P(x), we conclude that P(c) is true, where c is a particular member of the domain. 10. Existential generalization When P(c) is known to be true for a particular member c, we conclude x P(x). 11. Existential instantiation If we know that x P(x) is true, we conclude that there is a member c in the domain for which P(c) is true. We may not know what c is, only that it exists. 12. Universal generalization From the premise that P(c) is true for an arbitrary member c in the domain, we conclude that x P(x) is true Universal instantiation Common Rules of Inference for Quantified Statements 10. Existential generalization 11. Existential instantiation 12. Universal generalization 13. Universal modus ponens 14. Universal modus tollens 34 Examples for Rules of Inference with Quantifiers Ex Show that premises A and B imply conclusion C. A. A sentence in this article is incorrect and B. The article passed proofreading C. Some sentence that passed proofreading is incorrect. Ex The statement D is know to be true. D. For all positive integers n, if n > 4, then n 2 < 2 n. Show < from D. 35 9