Reasoning in Propositional Logic

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Reasoning in Propositional Logic

Transcript of Reasoning in Propositional Logic

Page 1: Reasoning in Propositional Logic

Reasoning inPropositional Logic

Page 2: Reasoning in Propositional Logic

Logic reasoning (inference)

Logical entailment (implication)

Logical deduction (inference algorithms)

BC a

BC a

Sentences Sentences

Facts Facts

Representation

Modeled world

Se

man

tics

Se

man

tics

logicallyentails

emerge

INFERENCE

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Logical implication

One thing follows another

KB |= a

A KB logically entails the sentence a if and only if a is true in the world where the KB is true.

Models: Formally structured worlds with respect to which

truth can be evaluated.

m is a model of a sentence a if a is true in m.

M(a) is the set of all the models of a .

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Example of logical entailment

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Possible models

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Models for the KB

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Logically entailed sentence

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Sentence no logically entailed

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Inferential processes

Characteristics

• Sound or truth preserving: an inference algorithm that derives only entailed sentences.

• Complete: an inference algorithm that derive any sentence that is entailed.

Example: Model Checking

1. List the models (combinations according to the propositional symbols).

2.Mark those in which KB is true.

3.Verify if in these models the sentence to be inferred is also true.

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Logical concepts

Logical equivalence

Two sentences α and β are logically

equivalent if they are true in the same set of

models.

Validity

A sentence is valid if it is true

in all models.

Satisfiability

A sentence is satisfiable if it is

true in, or satisfied by, some model.

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Standard logical equivalences

Associativity Double-negation elimination

ContrapositionCommutativity

PQQP

PQQP

PP )( PQQP

)(())((

))(())((

RQPRQP

RQPRQP

Distributivity De MorganImplication elimination

Biconditional elimination

)()()(

)()()(

RPQPRQP

RPQPRQP

QPQP

QPQP

)(

)(QPQP )()( PQQPQP

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Validity

A sentence is valid (tautology) if it is always true.

qp qqp )(p q

T

F

T

F

T

F

T

T

T

T

F

F

F

F

F

T

T

T

T

T

pqqp ))((

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