Logic and Knowledge Representation P r o p o s i t i o n - PowerPoint PPT Presentation

Logic and Knowledge Representation P r o p o s i t i o n a l L o g i c 4 M a y 2 0 1 8 G i o v a n n i S i l e n o g s i l e n o @e n s t . f r T é l é c o m P a r i s T e c h , P a r i s - D a u p h i n e U n i v e r s i t y

T h i s i s a n i c e v a r i a t i o n o f E p i m e n i d e s p a r a d o x , t h e s t o r y o f a C r e t a n s a y i n g “ A l l C r e t a n s a r e l i a r s ” .

Logic: a long history

Overview on (Western) logic ● G r e e k L o g i c – S t o i c s – A r i s t o t l e – l o g i c i n a r g u m e n t a t i o n – s y l l o g i s m

Overview on (Western) logic ● G r e e k L o g i c – S t o i c s – A r i s t o t l e – l o g i c i n a r g u m e n t a t i o n – s y l l o g i s m ● Me d i e v a l a n d t r a d i t i o n a l l o g i c – T h o m a s A q u i n a s ( 1 2 2 5 - 1 2 7 4 ) – m o d a l l o g i c – Wi l l i a m o f O c k h a m ( 1 2 8 8 - 1 3 4 8 ) – l a w s o f d e M o r g a n – t e r n a r y l o g i c – L o g i c o f P o r t R o y a l ● A n t o i n e A r n a u l d & P i e r r e N i c o l e ( 1 6 6 2 )

All bankers are athletes. No consultant is a banker. Therefore….

All bankers are athletes. No consultant is a banker. Therefore…. s o m e a t h l e t e i s n o t a c o n s u l t a n t ( ? ? ? )

All bankers are athletes. No consultant is a banker. Therefore…. s o m e a t h l e t e i s n o t a c o n s u l t a n t ! ● V a l i d , a n d n o t t r i v i a l .

All bankers are athletes. No consultant is a banker. Therefore…. A B s o m e a t h l e t e i s n o t a c o n s u l t a n t . C ● V a l i d , a n d n o t t r i v i a l .

All bankers are athletes. No consultant is a banker. Therefore…. A B s o m e a t h l e t e i s n o t a c o n s u l t a n t . C ● V a l i d , a n d n o t t r i v i a l .

All bankers are athletes. No consultant is a banker. Therefore…. A B s o m e a t h l e t e i s n o t a c o n s u l t a n t . C ● V a l i d , a n d n o t t r i v i a l .

All bankers are athletes. No consultant is a banker. Therefore…. A B s o m e a t h l e t e i s n o t a c o n s u l t a n t . C ● V a l i d , a n d n o t t r i v i a l .

Overview on (Western) logic ● Mo d e r n l o g i c – D e s c a r t e s , L e i b n i z – G e o r g e B o o l e ( 1 8 4 8 ) – G o t t l o b F r e g e , B r e g r i fg s c h r i f t ( 1 8 7 9 ) ● Q u a n t i fj c a t i o n – C h a r l e s P e i r c e ● R e a s o n i n g a n d l o g i c – G u i s e p p e P e a n o ● L o g i c a l A x i o m a t i z a t i o n o f A r t i h m e t i c s – B e r t a n d R u s s e l l & A l f r e d N . Wh i t e h e a d , P r i n c i p i a M a t h e m a t i c a ( 1 9 2 5 ) ● L o g i c a l A x i o m i z a t i o n o f M a t h e m a t i c s

Propositional logic

A language consists of symbols , ... ● a l p h a b e t – p r o p o s i t i o n a l s y m b o l s p 1 , p 2 , . . .

A language consists of symbols , ... ● a l p h a b e t – p r o p o s i t i o n a l s y m b o l s p 1 , p 2 , . . . – c o n n e c t i v e s ● n ⊥ u l l a r y : T , ( t o p , b o t t o m ) ● u n a r y : ¬ ( n e g a t i o n ) ● b ∧ ∨ ⊃ ⊂ ↑ ↓ ⊅ ⊄ ≡ ≠ i n a r y : , , , , , , , , , ( a n d , o r , i m p l i e s , o n l y - i f , n a n d ( i n c o m p a t i b l e ) , n o r , n o t - i m p l i e s , n o t - o n l y - i f , e q u i v a l e n t , n o t - e q u i v a l e n t )

A language consists of a syntax (rules to aggregate symbols), ... ● s e t A o f a t o m i c f o r m u l a s – A c o n t a i n s a l l p r o p o s i t i o n a l s y m b o l s – A ⊥ c o n t a i n s t h e n u l l a r y c o n n e c t i v e s T ,

A language consists of a syntax (rules to aggregate symbols), ... ● s e t A o f a t o m i c f o r m u l a s – A c o n t a i n s a l l p r o p o s i t i o n a l s y m b o l s – A ⊥ c o n t a i n s t h e n u l l a r y c o n n e c t i v e s T , ● s e t P o f ( w e l l - f o r m e d ) p r o p o s i t i o n a l f o r m u l a s – P c o n t a i n s a t o m i c f o r m u l a s – i f F i s i n P , t h e n ¬ F i s i n P – i f F a n d G a r e i n P , t h e n ( F o G ) i s i n P , w h e r e o i s a ∧ ∨ ⊃ ⊂ ↑ ↓ ⊅ , ⊄ ≡ ≠ b i n a r y c o n n e c t i v e (, , , , , , , , ) . – P i s t h e s m a l l e s t s e t t h a t h a s t h e s e p r o p e r t i e s ( e q u i v a l e n t l y , t h e r e i s n o t h i n g i n P t h a t d o e s n o t s a t i s f y t h e s e p r o p e r t i e s )

A language consists of a semantic (rules to interpret its expressions) ● S e m a n t i c s s h o u l d t e l l u s h o w t h e m e a n i n g o f t h e c o n s t i t u e n t p a r t s o f a d i s c o u r s e , a n d t h e i r m o d e o f c o m b i n a t i o n , d e t e r m i n e t h e o v e r a l l m e a n i n g .

A language consists of a semantic (rules to interpret its expressions) ● S e m a n t i c s s h o u l d t e l l u s h o w t h e m e a n i n g o f t h e c o n s t i t u e n t p a r t s o f a d i s c o u r s e , a n d t h e i r m o d e o f c o m b i n a t i o n , d e t e r m i n e t h e o v e r a l l m e a n i n g . ● B m e a n i n g u t w h a t d o w e m e a n b y ?

A language consists of a semantic (rules to interpret its expressions) ● S e m a n t i c s s h o u l d t e l l u s h o w t h e m e a n i n g o f t h e c o n s t i t u e n t p a r t s o f a d i s c o u r s e , a n d t h e i r m o d e o f c o m b i n a t i o n , d e t e r m i n e t h e o v e r a l l m e a n i n g . correspondence semantics ...that a dog is out there ● B m e a n i n g u t w h a t d o w e m e a n b y ? e x . “ t h e r e i s a d o g ”

A language consists of a semantic (rules to interpret its expressions) ● S e m a n t i c s s h o u l d t e l l u s h o w t h e m e a n i n g o f t h e c o n s t i t u e n t p a r t s o f a d i s c o u r s e , a n d t h e i r m o d e o f c o m b i n a t i o n , d e t e r m i n e t h e o v e r a l l m e a n i n g . correspondence semantics ...that a dog is out there ● B m e a n i n g u t w h a t d o w e m e a n b y ? e x . “ t h e r e i s a d o g ” truth-conditional semantics ..that that proposition is true “ t h e r e i s a d o g ” i s t r u e

A language consists of a semantic (rules to interpret its expressions) ● S e m a n t i c s s h o u l d t e l l u s h o w t h e m e a n i n g o f t h e c o n s t i t u e n t p a r t s o f a d i s c o u r s e , a n d t h e i r m o d e o f c o m b i n a t i o n , d e t e r m i n e t h e o v e r a l l m e a n i n g . correspondence semantics ...that a dog is out there ● B m e a n i n g u t w h a t d o w e m e a n b y ? e x . “ t h e r e i s a d o g ” cognitive semantics ..that the locutor believes that.. truth-conditional semantics ..that that proposition is true “ t h e r e i s a d o g ” i s t r u e

Strange effects... ● “ a d o g i s a d o g ” ● “ a d o g i s a m a m m a l ”

Strange effects... Each sentence is assigned to a truth value ● “ a d o g i s a d o g ” TRUE ● “ a d o g i s a m a m m a l ” TRUE