Propositional Logic

Propositional Logic

Propositions • A proposition is a declarative sentence that is either true or false. • Examples of propositions: • The Moon is made of green cheese. • Trenton is the capital of New Jersey. • Toronto is the capital of Canada. • 1 + 0 = 1 • 0 + 0 = 2 • Examples that are not propositions. • Sit down! • What time is it? • x + 1 = 2 • x + y = z

Propositional Logic (or Calculus) • Constructing Propositions • Propositional Variables: p, q, r, s, … • The proposition that is always true is denoted by T and the proposition that is always false is denoted by F. • Compound Propositions: constructed from other propositions using logical connectives • Negation ¬ • Conjunction ∧ • Disjunction ∨ • Implication → • Biconditional↔

Compound Propositions: Negation • The negation of a proposition p is denoted by ¬p and has this truth table: • Example: If p denotes “The earth is round” then ¬p denotes “It is not the case that the earth is round,” or more simply “The earth is not round.”

Conjunction • The conjunction of propositions p and q is denoted by p ∧ q and has this truth table: • Example: If p denotes “I am at home” and q denotes “It is raining” then p ∧q denotes “I am at home and it is raining.”

Disjunction • The disjunction of propositions p and q is denoted by p ∨q and has this truth table: • Example: If p denotes “I am at home” and q denotes “It is raining” then p ∨q denotes “I am at home or it is raining.”

The Connective Or in English • In English “or” has two distinct meanings. • InclusiveOr: For p ∨q to be T, either p or q or both must be T Example: “CS202 or Math120 may be taken as a prerequisite.” Meaning: take either one or both • ExclusiveOr (Xor). In p⊕q, eitherporq but not both must be T Example: “Soup or salad comes with this entrée.” Meaning: do not expect to get both soup and salad

Implication • If p and q are propositions, then p →q is a conditional statement or implication, which is read as “if p, then q ” • pis the hypothesis (antecedent or premise) and q is the conclusion (or consequence). • Example: If p denotes “It is raining” and q denotes “The streets are wet” then p →q denotes “If it is raining then streets are wet.”

Understanding Implication • In p →q there does not need to be any connection between p and q. • The meaning of p →q depends only on the truth values of p and q. • Examples of valid but counterintuitive implications: • “If the moon is made of green cheese, then you have more money than Bill Gates” -- True • “If Juan has a smartphone, then 2 + 3 = 6” -- Falseif Juan does have a smartphone, True if he does NOT

Understanding Implication • View logical conditional as an obligation or contract: • “If I am elected, then I will lower taxes” • “If you get 100% on the final, then you will earn an A” • If the politician is elected and does not lower taxes, then he or she has broken the campaign pledge. Similarly for the professor. • This corresponds to the case where p is T and q is F.

Different Ways of Expressing p →q • ifp, thenq • if p, q • pimpliesq • qifp • qwhenp • qwheneverp • Note: (p only if q)≡ (q if p) ≢ (q only if p) • It is raining → streets are wet: T • It is raining only if streets are wet: T • Streets are wet if it is raining: T • Streets are wet only if it’s raining: F • qfollows from p • qunless ¬p • ponly if q • pis sufficient for q • qis necessary for p

Sufficient versus Necessary • Example 1: • p →q : get 100% on final → earn A in class • p is sufficient for q: getting 100% on final is sufficient for earning A • q is necessary for p: earning A is necessary for getting 100% on final • Counterintuitive in English: • “Necessary” suggests a precondition • Example is sequential • Example 2: • Nobel laureate → is intelligent • Being intelligent is necessary for being Nobel laureate • Being a Nobel laureate is sufficient for being intelligent (better if expressed as: implies)

Converse, Inverse, and Contrapositive • From p →q we can form new conditional statements . • q →p is the converse of p →q • ¬ p → ¬ q is the inverse of p →q • ¬q → ¬ p is the contrapositive of p →q • How are these statements related to the original? • How are they related to each other? • Are any of them equivalent?

Converse, Inverse, and Contrapositive • Example: • it’s raining → streets are wet • converse: streets are wet → it’s raining • inverse: it’s not raining → streets are not wet • contrapositive: streets are not wet → it’s not raining • only3is equivalent to original statement • 1and 2 are not equivalent to original statement: streets could be wet for other reasons • 1 and2 are equivalent to each other

Biconditional • If p and q are propositions, then we can form the biconditionalproposition p ↔q, read as “p if and only if q ” • Example: If p denotes “You can take a flight” and q denotes “You buy a ticket” then p ↔qdenotes “You can take a flight if and only if you buy a ticket” • True only if you do both or neither • Doing only one or the other makes the proposition false

Expressing the Biconditional • Alternative ways to say “p if and only if q”: • pis necessary and sufficient for q • ifpthenq, and conversely • piffq

Compound Propositions • conjunction, disjunction, negation, conditionals, and biconditionals can be combined into arbitrarily complex compound proposition • Any proposition can become a term inside another proposition • Propositions can be nested arbitrarily • Example: • p, q, r, t are simple propositions • p ∨q , r ∧t , r →t are compound propositions using logical connectives • (p ∨q) ∧ tand(p ∨q) →tare compound propositions formed by nesting

Precedence of Logical Operators • With multiple operators we need to know the order • To reduce number of parentheses use precedence rules • Example: • p ∨q → ¬ r is equivalent to(p ∨q)→ ¬ r • If the intended meaning is p ∨(q → ¬ r ) then parentheses must be used

Truth Tables of Compound Propositions • Construction of a truth table: • Rows • One for every possible combination of values of all propositional variables • Columns • One for the compound proposition (usually at far right) • One for each expression in the compound proposition as it is built up by nesting

Example Truth Table • Construct a truth table for

Propositional Logic