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Propositional Logic

Propositional Logic. Rosen 6 th ed., § 1.1-1.2. Foundations of Logic: Overview. Propositional logic: Basic definitions. Equivalence rules & derivations. Predicate logic Predicates. Quantified predicate expressions. Equivalences & derivations. Propositional Logic.

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Propositional Logic

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  1. Propositional Logic Rosen 6th ed., § 1.1-1.2

  2. Foundations of Logic: Overview • Propositional logic: • Basic definitions. • Equivalence rules & derivations. • Predicate logic • Predicates. • Quantified predicate expressions. • Equivalences & derivations.

  3. Propositional Logic Propositional Logic is the logic of compound statements built from simpler statements using Booleanconnectives. Applications: • 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). [In probability theory, we assign degrees of certainty to propositions. For now: True/False only!]

  5. Examples of Propositions • “It is raining.” (Given a situation.) • “Beijing is the capital of China.” • “1 + 2 = 3” 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. 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:

  8. The Conjunction Operator The binary conjunction operator “” (AND) combines two propositions to form their logical conjunction. E.g. If p=“I will have salad for lunch.” and q=“I will have steak for dinner.”, then pq=“I will have salad for lunch andI will have steak for dinner.”

  9. Conjunction Truth Table • Note that aconjunctionp1p2  … pnof n propositionswill have 2n rowsin its truth table. • ¬ and  operations together are universal, i.e., sufficient to express any truth table!

  10. The Disjunction Operator The binary disjunction operator “” (OR) combines two propositions to form their logical disjunction. p=“That car has a bad engine.” q=“That car has a bad carburetor.” pq=“Either that car has a bad engine, orthat car has a bad carburetor.”

  11. Disjunction Truth Table • Note that pq meansthat p is true, or q istrue, or both are true! • So this operation isalso called inclusive or,because it includes thepossibility that both p and q are true. • “¬” and “” together are also universal.

  12. 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 = r ¬p = ¬ r  p  q = “It didn’t rain last night.” “The lawn was wet this morning, andit didn’t rain last night.” “Either the lawn wasn’t wet this morning, or it rained last night, or the sprinklers came on last night.”

  13. 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!)”

  14. Exclusive-Or Truth Table • Note that pq meansthat p is true, or q istrue, but not both! • This operation iscalled exclusive or,because it excludes thepossibility that both p and q are true. • “¬” and “” together are not universal.

  15. Natural Language is Ambiguous Note that English “or” is by itself ambiguous regarding the “both” case! “Pat is a singer orPat is a writer.” - “Pat is a man orPat is a woman.” - Need context to disambiguate the meaning! For this class, assume “or” means inclusive.  

  16. The Implication Operator The implicationp  q states that p implies q. It is FALSE only in the case that p is TRUE but q is FALSE. E.g., p=“I am elected.”q=“I will lower taxes.” p  q = “If I am elected, then I will lower taxes” (else it could go either way)

  17. Implication Truth Table • p  q is false only whenp is true but q is not true. • p  q does not implythat pcausesq! • p  q does not implythat p or qare ever true! • E.g. “(1=0)  pigs can fly” is TRUE!

  18. 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 George passed the exam.” True or False? • “If the moon is made of green cheese, then I am richer than Bill Gates.” True or False?

  19. Inverse, Converse, Contrapositive Some terminology: • The inverse of p  q is: ¬p  ¬q • The converse of p  q is: q  p. • The contrapositive of p  q is: ¬q  ¬p. • One of these has the same meaning (same truth table) as p q. Can you figure out which? Contrapositive

  20. How do we know for sure? Proving the equivalence of p  q and its contrapositive using truth tables:

  21. The biconditional operator The biconditionalp q states that p is true if and only if(IFF) q is true. It is TRUE when both p  q and q  p are TRUE. p = “It is raining.” q =“The home team wins.” p q = “If and only if it is raining, the home team wins.”

  22. Biconditional Truth Table • p  q means that p and qhave the same truth value. • Note this truth table is theexact opposite of ’s! p  q means ¬(p  q) • p  q does not implyp and q are true, or cause each other.

  23. Boolean Operations Summary • We have seen 1 unary operator (4 possible)and 5 binary operators (16 possible).

  24. Precedence of Logical Operators Operator Precedence

  25. 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)

  26. Some Alternative Notations

  27. Tautologies and Contradictions A tautology is a compound proposition that is trueno matter what the truth values of its atomic propositions are! Ex.p  p [What is its truth table?] A contradiction is a comp. prop. that is false no matter what! Ex.p  p [Truth table?] Other comp. props. are contingencies.

  28. Propositional Equivalence Two syntactically (i.e., textually) different compound propositions may be 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.

  29. Proving Equivalences Compound propositions p and q are logically equivalent to each other IFFp and q contain the same truth values as each other in all rows of their truth tables. Compound proposition p is logically equivalent to compound proposition q, written pq, IFFthe compound proposition pq is a tautology.

  30. Proving Equivalencevia Truth Tables Ex. Prove that pq  (p  q). F T T T F T T F F T T F T F T T F F F T

  31. 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 much more complicated propositions and to find equivalences for them.

  32. Equivalence Laws - Examples • Identity: pT  p pF  p • Domination: pT  T pF  F • Idempotent: pp  p pp  p • Double negation: p  p • Commutative: pq  qp pq  qp • Associative: (pq)r  p(qr) (pq)r  p(qr)

  33. More Equivalence Laws • Distributive: p(qr)  (pq)(pr)p(qr)  (pq)(pr) • De Morgan’s: (pq)  p  q (pq)  p  q

  34. More Equivalence Laws • Absorption:p(pq)  p p (p  q)  p • Trivial tautology/contradiction:p  p  Tp  p  F

  35. Defining Operators via Equivalences Using equivalences, we can define operators in terms of other operators. • Implication: pq  p  q • Biconditional: pq  (pq) (qp)pq  (pq) • Exclusive or: pq  (pq)(pq)pq  (pq)(qp)

  36. An Example Problem • Check using a symbolic derivation whether (p  q)  (p  r)  p  q  r. (p  q)  (p  r) [Expand definition of ] (p  q)  (pr) [Defn. of ]  (p  q)  ((p  r)  (p  r)) [DeMorgan’s Law]  (p  q)  ((p  r)  (p  r))

  37. Example Continued... (p  q)  ((p  r)  (p  r)) [ commutes]  (q  p)  ((p  r)  (p  r))[ associative]  q  (p  ((p  r)  (p  r))) [distrib.  over ]  q  (((p  (p  r))  (p  (p  r))) [assoc.]  q  (((p  p)  r)  (p  (p  r))) [trivial taut.]  q  ((T  r)  (p  (p  r))) [domination] q  (T  (p  (p  r))) [identity]  q  (p  (p  r)) cont.

  38. End of Long Example q  (p  (p  r)) [DeMorgan’s]  q  (p  (p  r)) [Assoc.]  q  ((p  p)  r) [Idempotent]  q  (p  r) [Assoc.]  (q  p)  r [Commut.] p  q  r Q.E.D. (quod erat demonstrandum)

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