CSC 360 Propositional Calculus

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CSC 360 Propositional Calculus

• Dr. Curry Guinn

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Todays Class

• Propositional Calculus
• Well-formed formula
• Rules of inference
• Meaning
• Truth Preservations
• First-Order Logic (First Order Predicate
  Calculus)

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Well-formed formula

• P, Q, R are atoms.
• Atoms are always well-formed
• You can add a to any atom to make a new atom
• x where x is wff
• ltx ? ygt ltx ? ygt ltx ? ygt
• Which are wff on page 182?

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Rules of Inference

• Allow us to generate new theorems from old
  theorems
• See sheet

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Interpretations

• So what do they mean?
• An interpretation in Propositional Calculus is an
  assignment of True or False to each atom.

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Truth Tables

• ltltP ? Qgt ? ltQ ? Pgtgt
• ltltP ? Qgt ? ltP ? Qgtgt

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All the given rules of inference

• are truth-preserving.
• These rules guarantee the derived statement will
  be true if the prior theorems are true
• REGARDLESS of the meaning

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Entailment and Inference

• KB ?. ? is entailed by the sentences in the
  knowledge base
• For any model where KB is true, ? must also be
  true for that model.
• KB -i ?. ? can be derived from the knowledge
  base given inference procedure i.

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Inference Procedures

• An inference procedure i is sound (or
  truth-preserving) if any sentence generated from
  a knowledge base is entailed by the knowledge
  base
• If KB -i ?, then KB ?.
• An inference procedure i is complete if, for any
  sentence that is entailed by KB, then there
  exists a proof of KB using i.
• If KB ?, then KB -i ?

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Semantics

• The meaning of a sentence is what it states about
  the world.
• An interpretation is an assignment of meaning to
  a sentence.
• Compositional semantics The meaning of a
  sentence is a function of the meaning of its
  parts.
• A sentence is true if the state of affairs it
  represents is actually the case. Depends on its
  interpretation and the state of the world.

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Inference

• A sentence is valid if it is true under all
  possible interpretations in all possible worlds
  (analytic sentences, tautologies).
• A sentence is satisfiable if there exists an
  interpretation in some world such that the
  sentence is true.
• A sentence is unsatisfiable if for every possible
  world there is no interpretation such that the
  sentence is true.

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Logics
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Propositional Logic

• Syntax
• Constants True and False. Sentences by
  themselves.
• Propositional Symbols P, Q, R, .... Sentences
  by themselves.
• Parenthesis around any sentence is a sentence.
• A sentence can formed by combining sentences
  using the logical connectives ????????????and???
  
• A literal is an atomic sentence (P, Q, R, ...) or
  its negation.

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Propositional Logic

• Semantics.
• An interpretation of a propositional symbol can
  be any arbitrary fact.
• Meaning of a complex sentence derived from its
  parts.
• Truth Table.

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Propositional Logic

• Validity and inference.
• Use a truth table to test for validity.
• ((P ? H) ? ?H) ? P
• Model Any world in which a sentence is true
  under a particular interpretation.
• A sentence ? is entailed by a knowledge base KB
  if the models of KB are all models of ?.
  (Whenever KB is true, so is a).

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Propositional Logic

• Rules of inference.
• Modus Ponens.
• And-Elimination.
• And-Introduction.
• Or-Introduction.
• Double Negation.
• Unit Resolution.
• Resolution.
• P -gt Q p V q. (Material Implication (Impl.)).
• (p AND Q) p V q. De Morgans Laws (De M.).
• (p V Q) p AND q.

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Propositional Logic

• Complexity
• Determining whether a set of sentences is
  satisfiable is NP-complete.

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Propositional Logic

• Use a truth table to determine whether the
  sentence is a tautology, self-contradictory or
  contingent
• (p -gt (p -gt q)) -gt q.
• p -gt ((p -gt q) -gt q).
• p -gt (p -gt (q AND q)).
• Prove using rules of inference
• A -gt B, (A AND B) -gt C Therefore A -gt C.

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First-Order Logic

• Also called first-order predicate calculus
• There are objects with properties and relations
  between objects.
• Syntax Constants, Predicate Symbols, Functions.
  Terms are either constants or functions.

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Sentences

• Atomic Predicate symbol followed by a list of
  terms in parenthesis.
• Complex Sentences conjoined by logical
  predicates.
• Quantifiers Allow specifying properties of
  collection of objects.
• Quantify over objects.

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Universal Qualifier (?)

• For all ...
• ??x Cat(x)???Mammal(x)???For all objects, if an
  object is a cat, then the object is a mammal.
• ??x Cat(x)???Mammal(x)??? All objects are cats
  and all objects are mammals.

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Existential Quantifier (?)

• There exists ...
• ? x Sister(x,Spot) ?? Cat(x). There exists an
  object such that the object is a sister of Spot
  and the object is a cat.

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Using First-Order Logic

• Domain a subset of the world
• The kinship domain
• ?m,c Mother(c)m ? Female(m) ? parent(m,c).
• ?p,c Parent(p,c) ? Child(c,p).
• ?g,c Grandparent(g,c) ? ?p (Parent(g,p) ?
  Parent(p,c)).
• ?x,y Sibling(x,y) ? ?(xy) ? ?p (Parent(p,x) ?
  Parent(p,y)).

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Axioms, definitions, theorems

• An axiom is a basic fact about a domain.
• We try to derive theorems from the axioms.
• An independent axiom cannot be derived from other
  axioms.
• We would like to produce a minimal set of axioms
  that are independent.
• Why might you use redundancy?
• An axiom of the form ?x,y P(x,y) ?... is often
  called a definition of P.

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For Next Class, Tuesday

• Homework due next Tuesday
• Read about the Diagonalization Argument in
  Chapter XIII of GEB
• Quiz