INTRO LOGIC DAY 17 Translations in PL 3

INTRO LOGIC DAY 17 Translations in PL 3 1

REVIEW of DAY 1 and DAY 2 2

Existential Quantifier someone is happy there is someone who is happy there is some x : x is happy x Hx 3

Existential-Negative Quantifier someone is unhappy there is someone who is not happy there is some x : x is not happy x Hx 4

Universal Quantifier everyone is happy no matter who you are happy no matter who x is happy x Hx 5

Universal-Negative Quantifier everyone is unhappy no matter who you are not happy no matter who x is not happy x Hx 6

Negative-Existential Quantifier no one is happy there is no one who is happy there is no x : x is happy x Hx 7

Negative-Universal Quantifier not everyone is happy not: no matter who you are happy not: no matter who x is happy x Hx 8

Quantifier-Specification – ‘some’ some Freshman is Happy there is someone who is F and who is H there is some x x is F x ( Fx and & x is H Hx ) 9

Quantifier-Specification – ‘no’ no Freshman is Happy there is no one who is F and who is H there is no x x is F x ( Fx and & x is H Hx ) 10

Quantifier-Specification – ‘every’ every Freshman is Happy no matter who you are IF you are F THEN you are H no matter who x is IF x is F x ( Fx THEN x is H Hx ) 11

new material for day 3 12

Multiple Quantification sentences with more than one quantifier GENERAL STRATEGY (1) Count the number of quantifiers in original sentence. (2) Determine the overall structure of the sentence. (3) Work on constituents separately. (4) Substitute constituents back into overall formula. (5) Count the number of quantifiers in final formula. (6) Compare (5) with (1). 13

Example 1 everyone is FRIENDLY, but not everyone is HAPPY everyone is F but not everyone is H x Fx & x Hx x Fx x Hx 14

Example 2 every CAT is a PET, but not every PET is a CAT every C is P but not every P is C x ( Cx Px ) & x ( Px Cx ) x ( Cx Px ) x ( Px Cx ) 15

Example 3 if everyone is FRIENDLY, then everyone is HAPPY if everyone is F then everyone is H x. Fx x. Hx 16

Example 4 if every STUDENT is FRIENDLY, then every STUDENT is HAPPY if every S is F then every S is H x(Sx Fx) x(Sx Hx) x(Sx Fx) x(Sx Hx) 17

‘Any’ versus ‘Every’ Basic Principle both ‘any’ and ‘every’ are universal quantifiers, BUT they are usually not inter-changeable. 18

Some times they are interchangeable any one can Dance every one can Dance if I can Dance, then any one can if I can Dance, then every one can x. Dx Di x. Dx 19

Usually, they are not interchangeable is every one here? Jay doesn’t respect every one if every one can fix your car, then I can no one respects every one is any one here? Jay doesn’t respect any one if any one can fix your car, then I can no one respects any one 20

Difference between ‘every’ and ‘any’ the scope of ‘every’ is narrow the scope of ‘any’ is wide 21

Example — Not-Every Jay doesn’t respect everyone not! Jay respects everyone x. Rjx ‘not’ ( ) has wide scope ‘every’ ( ) has narrow scope 22

Example — Not-Any Jay doesn’t respect anyone does Jay respect a? does Jay respect b? does Jay respect c? ? Rja ? Rjb ? Rjc no! no! Rja Rjb Rjc etc. no matter who you are Jay does not respect you no matter who x is Jay does not respect x x Rjx ‘any’ ( ) has wide scope ‘not’ ( ) has narrow scope 23

Not-Any = None = Not-Some Jay respects no one there is no one whom Jay respects there is no x : Jay respects x x Rjx Recall = x Rjx = x Rjx Jay respects Jay doesn’t = no one respect anyone 24

Example — IF-EVERY if everyone fails, then satan wins if everyone fails then x. Fx satan wins Ws Ws ‘every’ has narrow scope ‘if…then’ has wide scope 25

How do we SHOW such a formula? (1) : x. Fx Ws CD (2) x. Fx (3) : Ws As 26

Example — IF-ANY if anyone fails, then satan wins if a fails then satan wins if b fails then satan wins if c fails then satan wins etc. if anyone fails then satan wins 27

In Other Words no matter who you are if no matter who x is x if ( you fail then satan wins x fails then satan wins Fx Ws ) ‘any’ has wide scope ‘if…then’ has narrow scope 28

How do we SHOW such a formula? (1) : x ( Fx Ws ) (2) : Fa Ws (3) Fa (4) : Ws UD CD As UD = Universal Derivation (later!) 29

Special Note Sometimes (but not always) ‘if-any’ = ‘if-some’ x ( Fx ) = x. Fx provided has no free occurrence of ‘x’ 30

THE END 31