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Chapter 2Logic

Chapter 2Logic

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$&

$

+$$ ,$$ '% $

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! "

# $ " %

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'& * '& * & & & &

' ' ' ' ( & ' ( & ' ( & ' ( & '

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-$ *-$ *

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!!!!

$ $ $ $

) ( ) %%% #

" *

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% & ./

All Malaysians are car-drivers (1) All people in KL are Malaysians (2)

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% & ./

% % # ($0

1 2 $(./

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7 # ' %7 # ' %

**

. ( $/0The red block is on the green block.

The green block is somewhere ABOVE theblue block.

The green block is NOT on the blue block.

The yellow block is on the green OR the blue.The blue block is on SOME other block.

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The red block is on the green block. The green block is on the yellow block.

The yellow block is on the blue block.

The blue block is on the black block.

Main conclusion

The black block is on the table.

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Rule InstancesRule Instances

#%% ('%( ' & $ '&+ % #& % $

&& !

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Sound Rules of InferenceSound Rules of Inference 3( ./

P Q P Q

--------

Q P

:' .:/ P

-------

P

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ProofProof

+ + ,

"

- +

. &

/ ++ ,

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ExampleExample

When it is raining, the ground is wet.When the ground is wet, it is slippery.

It is raining. Prove that it is slippery.

;! # *

! *

! # ,*0 ;= ! & ,*0

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Implication Introduction: s => (t => s)

Implication Distribution: (r => (s => t))

=> ((r => s) => (r => t))

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Inference RulesInference Rules (II)(II)

,$ *$

1 * $ 8 '% = $ * = % 8 !

,$ $ *

1 % * 8 = $ * =

% 8 !

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WellWell--FormedFormed Formula (wff)

TAUTOLOGY

CONTRADICTION

A complex formula that includes logical

connectives. The truth value of a wff is known

as its semantics or meaning.

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$ &

A set of well-formed formulas (wffs)constructed for any field of knowledge

is said to be the theory of that field.

0 $!$ " &

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Some other termsProposition

any statement that can be assigned a value of

true or false.Clausal form

using only connectives such as: and , or and not.

Horn clause

contains only one positive literal.

Literalis an atomic formula or the negation of an atomic

formula.

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