Propositional Logic

Propositional Logic Rather than jumping right into FOL, we begin with propositional logic A logic involves: • Language (with a syntax) • Semantics • Proof (Inference) System

Example of k-rep in prop calc • R : “It is raining” • B : “Take the bus to class” • W : “Walk to class” • Some things to tell our agent • R  B (“If it is raining, (then) take the bus to class”) • R  W (“If it is not raining, (then) walk to class”) • Ideally, we would like our agent to sense that it is raining & then decide to take the bus

Alphabet Non-Logical Symbols (meaning given by interpretation) Propositions P, Q, R,… atomic statements (facts) about the world R : it’s-raining-now needn’t be a single letter Logical Symbols (fixed meaning)

Alphabet Logical Symbols • Connectives: not () and () or () implies () equivalent () • Punctuation Symbols: ( , ) • Truth symbols: TRUE, FALSE

Well-formed formulae (wffs) • Sentences • just like in a programming language, there are rules (syntax) for legally creating compound statements • remember: we’re always stating a truth about the world, hence every wff is something that has a Boolean value (it is either a true or a false statement about the world)

Syntax rules • Propositions (P, Q, R, …) are wffs • Truth symbols (TRUE, FALSE) are wffs • If A is a wff, so is A • If A and B are wffs, so are • A  B • A  B • A B • A  B There are no other wffs. • Language: set of all wffs

Are these WFFs? • P Q R • (P  Q)  (R  S) • P   (Q  R)

Semantics KB |= Q KB - Set of wffs Q- a wff |= Entailment Compositional Two-Valued

What is an interpretation? • An interpretation gives meaning to the non-logical symbols of the language. • An assignment of facts to atomic wffs • a fact is taken to be either true or false about the world • thus, by providing an interpretation, we also provide the truth value of each of the atoms example • P : it-is-raining-here-now • since this is either a true or false statement about the world, the value of P is either true or false a function that maps atomic formulas to truth values

Truth tables Connectives Semantics

How to evaluate a wff • ((P  U)  R)  (S  V) • First, we need an interpretation • P : T; U : F; R : T; S : F; V : T • Then using this interpretation, evaluate formula according to the fixed meanings of the connectives • P  U : T • (P  U)  R : T • S  V : F • whole formula : F

Satisfiability and Models • An interpretation I satisfies a wff iff I assigns the wff the value T • An interpretation I satisfies a set of S of wffs iff I satisfies every wff in S. • An interpretation that satisfies a (set of) wff is said to be a model of it. • A (set of) wff is satisfiable iff there exists some interpretation that satisfies it

Examples: • P is satisfiable • simply let P be true • P P is unsatisfiable • if P is false, the formula is false • if P is true, P is false, the formula is false • P  Q is satisfiable • three ways: P is true, Q is true; etc. • A wff that is unsatisfiable is called a contradiction • for example, a model for {A  B, B  C} is • A : true, B : true, C : true • note: there may be more than one model for a (set of) wff

Entailment (Logical Consequence) • KB |= Q iff for every interpretation I, • If I satisfies KB then I satisfies Q. • That is, if every model of KB is also a model of Q. • For example: • A  B, A |= B

Validity • A formula G is valid if it is true for every interpretation • P  P is valid • if P is true, then the formula is true • if P is false, then ~P is true and the formula is true • (P Q)  (P  Q) isn’t valid • when P is true & Q is true, the formula isn’t true • in order to not be valid, there only need exist one counter-example • also called a tautology

Some important Theorems a) KB |= Q iff KB U { Q} is unsatisfiable b) KB , A |= B iff KB |= (A  B) c) Monotonicity: if KB  KB’ then {Q | KB |= Q}  {Q | KB’ |= Q}

Propositional Logic