Note2 DM Logic - Foundations of Logic 1 Foundations of...

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Unformatted text preview: Foundations of Logic 1 Foundations of Logic Mathematical Logic is a tool for working with complicated compound statements. It includes: • A language for expressing them. • A concise notation for writing them. • A methodology for objectively reasoning about their truth or falsity. • It is the foundation for expressing formal proofs in all branches of mathematics. 2 Foundations of Logic: Overview • Propositional logic (1.1-1.2): – Basic definitions. (1.1) – Equivalence rules & derivations. (1.2) • Predicate logic (1.3-1.4) – Predicates. – Quantified predicate expressions. – Equivalences & derivations. 3 Propositional Logic (1.1) Propositional Logic is the logic of compound statements built from simpler statements using so-called Boolean connectives. Some applications in computer science: • Design of digital electronic circuits. • Expressing conditions in programs. • Queries to databases & search engines. 4 Definition of a Proposition A proposition (p, q, r, …) is simply a statement (i.e., a declarative sentence) with a definite meaning, having a truth value that’s either true (T) or false (F) (never both, neither, or somewhere in between). (However, you might not know the actual truth value, and it might be situation-dependent.) [Later we will study probability theory, in which we assign degrees of certainty to propositions. But for now: think True/False only!] 5 Examples of Propositions • “It is raining.” (In a given situation.) • “Toronto is the capital of Canada.” • “1 + 2 = 3” But, the following are NOT propositions: • “Who’s there?” (interrogative, question) • “La la la la la.” (meaningless interjection) • “Just do it!” (imperative, command) • “Yeah, I sorta dunno, whatever...” (vague) • “1 + 2” (expression with a non-true/false value) 6 Operators / Connectives An operator or connective combines one or more operand expressions into a larger expression. (E.g., “+” in numeric exprs.) Unary operators take 1 operand (e.g., −3); binary operators take 2 operands (eg 3 4). Propositional or Boolean operators operate on propositions or truth values instead of on numbers. 7 Some Popular Boolean Operators Formal Name Nickname Arity Negation operator Conjunction operator NOT AND Disjunction operator OR Exclusive-OR operator XOR Implication operator IMPLIES Biconditional operator IFF Unary Binary Binary Binary Binary Binary Symbol ¬ ↔ 8 The Negation Operator The unary negation operator “¬” (NOT) transforms a prop. into its logical negation. E.g. If p = “I have brown hair.” then ¬p = “I do not have brown hair.” Truth table for NOT: p p T F T :≡ True; F :≡ False “:≡” means “is defined as” F T Operand column Result column 9 The Conjunction Operator The binary conjunction operator “” (AND) combines two propositions to form their ND logical conjunction. E.g. If p=“I will have rice for lunch.” and q=“I will have noodles for dinner.”, then pq=“I will have rice for lunch and I will have noodles for dinner.” Remember: “” points up like an “A”, and it means “ND” 10 Conjunction Truth Table Operand columns • Note that a p q p q conjunction F F F p 1 p 2 … pn F T F of n propositions T F F will have 2n rows in its truth table. T T T • Also: ¬ and operations together are sufficient to express any Boolean truth table! 11 The Disjunction Operator The binary disjunction operator “” (OR) combines two propositions to form their logical disjunction. p=“My car has a bad engine.” q=“ My car has a bad carburetor.” pq=“Either my car has a bad engine, or the downwardmy car has a bad carburetor.” After pointing “axe” of “” Meaning is like “and/or” in English. splits the wood, you can take 1 piece OR the other, or both. 12 Disjunction Truth Table • Note that pq means p q p q that p is true, or q is F F F true, or both are true! Note F T T difference • So, this operation is T F T from AND also called inclusive or, T T T because it includes the possibility that both p and q are true. • “¬” and “” together are also universal. 13 Nested Propositional Expressions • Use parentheses to group sub-expressions: “I just saw my old friend, and either he’s grown or I’ve shrunk.” = f (g s) – (f g) s would mean something different – f g s would be ambiguous • By convention, “¬” takes precedence over both “” and “”. – ¬s f means (¬s) f , not ¬ (s f) 14 A Simple Exercise Let p=“It rained last night”, q=“The sprinklers came on last night,” r=“The lawn was wet this morning.” Translate each of the following into English: ¬p = “It didn’t rain last night.” “The lawn was wet this morning, and r ¬p = it didn’t rain last night.” ¬rpq= “Either the lawn wasn’t wet this morning, or it rained last night, or the sprinklers came on last night.” 15 The Exclusive Or Operator The binary exclusive-or operator “” (XOR) combines two propositions to form their logical “exclusive or” (exjunction?). p = “I will earn an A in this course,” q = “I will drop this course,” p q = “I will either earn an A for this course, or I will drop it (but not both!)” 16 Exclusive-Or Truth Table • Note that pq means p q pq that p is true, or q is F F F true, but not both! F T T • This operation is T F T called exclusive or, T T F because it excludes the possibility that both p and q are true. • “¬” and “” together are not universal. Note difference from OR. 17 The Implication Operator antecedent consequent The implication p q states that p implies q. I.e., If p is true, then q is true; but if p is not true, then q could be either true or false. E.g., let p = “You study hard.” q = “You will get a good grade.” p q = “If you study hard, then you will get a good grade.” (else, it could go either way) 18 Implication Truth Table • p q is false only when p is true but q is not true. • p q does not say that p causes q! • p q does not require that p or q are ever true! • E.g. “(1=0) pigs can fly” is TRUE! The only False case! 19 Examples of Implications • “If this lecture ends, then the sun will rise tomorrow.” True or False? • “If Tuesday is a day of the week, then I am a penguin.” True or False? • “If 1+1=6, then Bush is president.” True or False? • “If the moon is made of green cheese, then I am richer than Bill Gates.” True or False? 20 Why does this seem wrong? • Consider a sentence like, – “If I wear a red shirt tomorrow, then the U.S. will attack Iraq the same day.” • In logic, we consider the sentence True so long as either I don’t wear a red shirt, or the US attacks. • But in normal English conversation, if I were to make this claim, you would think I was lying. – Why this discrepancy between logic & language? 21 Resolving the Discrepancy • In English, a sentence “if p then q” usually really implicitly means something like, – “In all possible situations, if p then q.” • That is, “For p to be true and q false is impossible.” • Or, “I guarantee that no matter what, if p, then q.” • This can be expressed in predicate logic as: – “For all situations s, if p is true in situation s, then q is also true in situation s” – Formally, we could write: s, P(s) → Q(s) • This sentence is logically False in our example, because for me to wear a red shirt and the U.S. not to attack Iraq is a possible (even if not actual) situation. – Natural language and logic then agree with each other. 22 English Phrases Meaning p q • • • • • • • • “p implies q” “if p, then q” “if p, q” “when p, q” “whenever p, q” “q if p” “q when p” “q whenever p” • • • • • “p only if q” “p is sufficient for q” “q is necessary for p” “q follows from p” “q is implied by p” We will see some equivalent logic expressions later. 23 Converse, Inverse, Contrapositive Some terminology, for an implication p q: • Its converse is: q p. • Its inverse is: ¬p ¬q. • Its contrapositive: ¬q ¬ p. • One of these three has the same meaning (same truth table) as p q. Can you figure out which? 24 How do we know for sure? Proving the equivalence of p q and its contrapositive using truth tables: p F F T T q F T F T q T F T F p T T F F p q q p T T T T F F T T 25 The biconditional operator The biconditional p q states that p is true if and only if (IFF) q is true. p = “You can take the flight.” q = “You buy a ticket.” p q = “You can take the flight if and only if you buy a ticket.” 26 Biconditional Truth Table • p q means that p and q have the same truth value. • Note this truth table is the exact opposite of ’s! – p q means ¬(p q) p F F T T q pq F T T F F F T T • p q does not imply p and q are true, or cause each other. 27 Boolean Operations Summary • We have seen 1 unary operator (out of the 4 possible) and 5 binary operators (out of the 16 possible). Their truth tables are below. p F F T T q F T F T p p q p q pq p q pq T F F F T T T F T T T F F F T T F F F T T F T T 28 Some Alternative Notations N a m e : n o ta n do rx o ri m p l i e s i f f P r o p o s i t i o n a ll o g i c : B o o l e a n a l g e b r a : pp q+ C / C + + / J a v a ( w o r d w i s e ) :! & & | | ! = = = C / C + + / J a v a ( b i t w i s e ) : ~&|^ L o g i c g a t e s : 29 Bits and Bit Operations • • • John Tukey A bit is a binary (base 2) digit: 0 or 1. (1915-2000) Bits may be used to represent truth values. By convention: 0 represents “false”; 1 represents “true”. • Boolean algebra is like ordinary algebra except that variables stand for bits, + means “or”, and multiplication means “and”. – See chapter 10 for more details. 30 Bit Strings • A Bit string of length n is an ordered series or sequence of n0 bits. – More on sequences in 3.2. • By convention, bit strings are written left to right: e.g. the first bit of “1001101010” is 1. • When a bit string represents a base-2 number, by convention the first bit is the most significant bit. Ex. 11012=8+4+1=13. 31 Bitwise Operations • Boolean operations can be extended to operate on bit strings as well as single bits. • E.g.: 01 1011 0110 11 0001 1101 11 1011 1111 Bit-wise OR 01 0001 0100 Bit-wise AND 10 1010 1011 Bit-wise XOR 32 End of 1.1 You have learned about: • Propositions: What they are. • Propositional logic operators’ – – – – Symbolic notations. English equivalents. Logical meaning. Truth tables. • Atomic vs. compound propositions. • Alternative notations. • Bits and bit-strings. • Next section: 1.2 – Propositional equivalences. – How to prove them. 33 Propositional Equivalence (1.2) Two syntactically (i.e., textually) different compound propositions may be the semantically identical (i.e., have the same meaning). We call them equivalent. Learn: • Various equivalence rules or laws. • How to prove equivalences using symbolic derivations. 34 Tautologies and Contradictions A tautology is a compound proposition that is true no matter what the truth values of its atomic propositions are! Ex. p p [What is its truth table?] A contradiction is a compound proposition that is false no matter what! Ex. p p [Truth table?] Other compound props. are contingencies. 35 Logical Equivalence Compound proposition p is logically equivalent to compound proposition q, written pq, IFF the compound proposition pq is a tautology. Compound propositions p and q are logically equivalent to each other IFF p and q contain the same truth values as each other in all rows of their truth tables. 36 Proving Equivalence via Truth Tables Ex. Prove that pq (p q). p F F T T q F T F T p q F T T T p T T F F q p q ( p q) T T F F F T T F T F F T 37 Equivalence Laws • These are similar to the arithmetic identities you may have learned in algebra, but for propositional equivalences instead. • They provide a pattern or template that can be used to match all or part of a much more complicated proposition and to find an equivalence for it. 38 Equivalence Laws - Examples • • • • • • Identity: pT p p F p Domination: pT T pF F Idempotent: p p p p p p Double negation: p p Commutative: pq qp pq qp Associative: (pq)r p(qr) (pq)r p(qr) 39 More Equivalence Laws • Distributive: p(qr) (pq)(pr) p(qr) (pq)(pr) • De Morgan’s: (pq) p q (pq) p q • Trivial tautology/contradiction: p p T p p F Augustus De Morgan (1806-1871) 40 Defining Operators via Equivalences Using equivalences, we can define operators in terms of other operators. • Exclusive or: pq (pq)(pq) pq (pq)(qp) • Implies: pq p q • Biconditional: pq (pq) (qp) pq (pq) 41 Review: Propositional Logic (1.1-1.2) • • • • • Atomic propositions: p, q, r, … Boolean operators: Compound propositions: s : (p q) r Equivalences: pq (p q) Proving equivalences using: – Truth tables. – Symbolic derivations. p q r … 42 Predicate Logic (1.3) • Predicate logic is an extension of propositional logic that permits concisely reasoning about whole classes of entities. • Propositional logic (recall) treats simple propositions (sentences) as atomic entities. • In contrast, predicate logic distinguishes the subject of a sentence from its predicate. – Remember these English grammar terms? 43 Applications of Predicate Logic It is the formal notation for writing perfectly clear, concise, and unambiguous mathematical definitions, axioms, and theorems (more on these in chapter 3) for any branch of mathematics. Predicate logic with function symbols, the “=” operator, and a few proof-building rules is sufficient for defining any conceivable mathematical system, and for proving anything that can be proved within that system! 44 Other Applications • Predicate logic is the foundation of the field of mathematical logic, which culminated in Gödel’s incompleteness theorem, which revealed the ultimate limits of mathematical thought: – Given any finitely describable, consistent proof procedure, there will still be some true statements that can never be proven by that procedure. Kurt Gödel 1906-1978 • I.e., we can’t discover all mathematical truths, unless we sometimes resort to making guesses. 45 Practical Applications • Basis for clearly expressed formal specifications for any complex system. • Basis for automatic theorem provers and many other Artificial Intelligence systems. • Supported by some of the more sophisticated database query engines and container class libraries (these are types of programming tools). 46 Subjects and Predicates • In the sentence “The dog is sleeping”: – The phrase “the dog” denotes the subject the object or entity that the sentence is about. – The phrase “is sleeping” denotes the predicatea property that is true of the subject. • In predicate logic, a predicate is modeled as a function P(·) from objects to propositions. – P(x) = “x is sleeping” (where x is any object). 47 More About Predicates • Convention: Lowercase variables x, y, z... denote objects/entities; uppercase variables P, Q, R… denote propositional functions (predicates). • Keep in mind that the result of applying a predicate P to an object x is the proposition P(x). But the predicate P itself (e.g. P=“is sleeping”) is not a proposition (not a complete sentence). – E.g. if P(x) = “x is a prime number”, P(3) is the proposition “3 is a prime number.” 48 Propositional Functions • Predicate logic generalizes the grammatical notion of a predicate to also include propositional functions of any number of arguments, each of which may take any grammatical role that a noun can take. – E.g. let P(x,y,z) = “x gave y the grade z”, then if x=“Mike”, y=“Mary”, z=“A”, then P(x,y,z) = “Mike gave Mary the grade A.” 49 Universes of Discourse (U.D.s) • The power of distinguishing objects from predicates is that it lets you state things about many objects at once. • E.g., let P(x)=“x+1>x”. We can then say, “For any number x, P(x) is true” instead of (0+1>0) (1+1>1) (2+1>2) ... • The collection of values that a variable x can take is called x’s universe of discourse. 50 Quantifier Expressions • Quantifiers provide a notation that allows us to quantify (count) how many objects in the univ. of disc. satisfy a given predicate. • “” is the FORLL or universal quantifier. x P(x) means for all x in the u.d., P holds. • “” is the XISTS or existential quantifier. x P(x) means there exists an x in the u.d. (that is, 1 or more) such that P(x) is true. 51 The Universal Quantifier • Example: Let the u.d. of x be parking spaces at UCSI. Let P(x) be the predicate “x is full.” Then the universal quantification of P(x), x P(x), is the proposition: – “All parking spaces at UCSI are full.” – i.e., “Every parking space at UCSI is full.” – i.e., “For each parking space at UCSI, that space is full.” 52 The Existential Quantifier • Example: Let the u.d. of x be parking spaces at UCSI. Let P(x) be the predicate “x is full.” Then the existential quantification of P(x), x P(x), is the proposition: – – – “Some parking space at UCSI is full.” “There is a parking space at UCSI that is full.” “At least one parking space at UCSI is full.” 53 Free and Bound Variables • An expression like P(x) is said to have a free variable x (meaning, x is undefined). • A quantifier (either or ) operates on an expression having one or more free variables, and binds one or more of those variables, to produce an expression having one or more bound variables. 54 Example of Binding • P(x,y) has 2 free variables, x and y. x P(x,y) has 1 free variable, and one bound variable. [Which is which?] • “P(x), where x=3” is another way to bind x. • An expression with zero free variables is a bonafide (actual) proposition. • An expression with one or more free variables is still only a predicate: x P(x,y) 55 Nesting of Quantifiers (1.4) Example: Let the u.d. of x & y be people. Let L(x,y)=“x likes y” (a predicate w. 2 f.v.’s) Then y L(x,y) = “There is someone whom x likes.” (A predicate w. 1 free variable, x) Then x (y L(x,y)) = “Everyone has someone whom they like.” (A __________ with ___ free variables.) 56 Review: Propositional Logic (1.1-1.2) • • • • • Atomic propositions: p, q, r, … Boolean operators: Compound propositions: s (p q) r Equivalences: pq (p q) Proving equivalences using: – Truth tables. – Symbolic derivations. p q r … 57 Review: Predicate Logic (1.3) • Objects x, y, z, … • Predicates P, Q, R, … are functions mapping objects x to propositions P(x). • Multi-argument predicates P(x, y). • Quantifiers: [x P(x)] :≡ “For all x’s, P(x).” [x P(x)] :≡ “There is an x such that P(x).” • Universes of discourse, bound & free vars. 58 Quantifier Exercise If R(x,y)=“x relies upon y,” express the following in unambiguous English: Everyone has someone to rely on. x(y R(x,y))= There’s a poor overburdened soul whom y(x R(x,y))= everyone relies upon (including himself)! x(y R(x,y))= There’s some needy person who relies upon everybody (including himself). y(x R(x,y))=Everyone has someone who relies upon them. x(y R(x,y))= Everyone relies upon everybody, (including themselves)! 59 Natural language is ambiguous! • “Everybody likes somebody.” – For everybody, there is somebody they like, x y Likes(x,y) [Probably more likely.] – or, there is somebody (a popular person) whom everyone likes? y x Likes(x,y) • “Somebody likes everybody.” – Same problem: Depends on context, emphasis. 60 Still More Conventions • Sometimes the universe of discourse is restricted within the quantification, e.g., x>0 P(x) is shorthand for “For all x that are greater than zero, P(x).” =x (x>0 P(x)) x>0 P(x) is shorthand for “There is an x greater than zero such that P(x).” =x (x>0 P(x)) 61 Review: Predicate Logic (1.3) • Objects x, y, z, … • Predicates P, Q, R, … are functions mapping objects x to propositions P(x). • Multi-argument predicates P(x, y). • Quantifiers: (x P(x)) =“For all x’s, P(x).” (x P(x))=“There is an x such that P(x).” 62 Deduction Example • Definitions: g :≡ Gandhi; H(x) :≡ “x is human”; M(x) :≡ “x is mortal”. • Premises: H(g) Gandhi is human. x H(x)M(x) All humans are mortal. 63 Deduction Example Continued Some valid conclusions you can draw: H(g)M(g) [Instantiate universal.] If Gandhi is human then he is mortal. H(g) M(g) Gandhi is inhuman or mortal. H(g) (H(g) M(g)) Gandhi is human, and also either inhuman or mortal. (H(g) H(g)) (H(g) M(g)) [Apply distributive law.] F (H(g) M(g)) [Trivial contradiction.] H(g) M(g) [Use identity law.] M(g) Gandhi ...
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  • Summer '16
  • Ms.Preethi Subramaniam
  • Logic, Predicate logic

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