Propositional Logic

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Propositional Logic

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

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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

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Alphabet

• Non-Logical Symbols
• (meaning given by interpretation)
• Propositions
• P, Q, R,
• atomic statements (facts) about the world
• R its-raining-now
• neednt be a single letter
• Logical Symbols
• (fixed meaning)

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Alphabet

• Logical Symbols
• Connectives
• not (?)
• and (?)
• or (?)
• implies (?)
• equivalent (?)
• Punctuation Symbols ( , )
• Truth symbols
• TRUE, FALSE

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Well-formed formulae (wffs)

• Sentences
• just like in a programming language, there are
  rules (syntax) for legally creating compound
  statements
• remember were 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)

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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

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Are these WFFs?

• P Q R
• (P ? Q) ? (R ? S)
• P ? ? (Q ? R)

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Semantics

• KB Q
• KB - Set of wffs
• Q- a wff
• Entailment
• Compositional
• Two-Valued

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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

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Truth tables

• Connectives Semantics

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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

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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

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• 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

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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
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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) isnt valid
• when P is true Q is true, the formula isnt
  true
• in order to not be valid, there only need exist
  one counter-example
• also called a tautology

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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