" / / ' ' fal sum " toll om " " absurdum " - PowerPoint PPT Presentation

Lecture # 2 Propositional Logic

proposition " a declarative sentence that is either " : True ( T / 1) or False ( F / / ) Ip ' ' " / / ' ' fal sum " toll om " " absurdum

" ) stand in for propositions propositional variables ( " " " p e.g. , g , us focus on the logic ( rather than the propositions , themselves ) and help use variables to refer to atomic propositions , we often prefer to which cannot be expressed in terms of simpler propositions .

we can quality or combine propositions of logical operators : - negation " not p " T ft :c : : p - - conjunction and of p n q " " p : - : : : : cinnamon : : : : : : TP n of in order of decreasing precedence

disjunction is the typical programming language " " or def foo ( x , y ) e. g : . if x so ok y s o : raise Exception L " no negative uipnts " ) up the English exclusive or is often implied " or " ice cream for dessert . " you can have cake of " e - g .

we use truth tables to show the value of a proposition for all combinations of values taken by its variables . v q n p p ⑤ of e. g p . ¥¥T÷¥F¥±¥¥÷÷¥'

rows in a truth table for a proposition of a¥¥fil :L : :/ :L :/ :3 variables ? how N many n - 4 zoo N =3 N e.g N - - e.g - Z e.g . Pqr_ pqrs_ Pq l l T T l l . l l l O O l o l O o l O o l l O O O O O l O O O O ÷

Applying logic to English propositions : " " I love cats p = " " science is of science = computer a " 4 s too " r = qt ⇒ p n of read : = Flt ¥f¥¥E ⑧ v ng n p n ( q v n r ) p - pm p = T gun of

A tautology is a proposition that is always true . T e. g . v - p p n q ) v - p v e of ( p A contradiction is a proposition that is always false . F / t e. g . p r e p t ( Cpr g) v - q ) u r p

a tautology ? n q ) v Tp req How to phone ( p is ÷fi÷÷¥÷ T

a conditional statement a proportion known is p → as of . implication ) ( aka " if p , then q read " 1 a hypothesis / ↳ ndusion / consequent antecedent

the logic conditional statement is NOI caution : " " if equivalent to the statement wi unpinatim programming ! card { if e. g not a proposition ! . } #

" if the Ac is on , then l 'll be cold e. g . d = the Ac is " ifkmdmmgenogh € on p be cold l 'll q - . - Pgpfq → ueip can left this then I → go truth table for p ¥1 ¥ IF we oozed ?gw÷q ← T F F

can lift this wight I strong enough If km then l l l ? can if this weight only if km strong enough I .

conditional in English other ways to many express . some tricky ones for p → q : p is sufficient for q " " " I a I is necessary for p " " q " q unless 7 p "

we equus using just ' ' ? , V can , n n → ⇒ ¥¥¥T÷÷ ' ←→ alert " Ymir if Tp then p T ② ← is equivalent otherwise p → g so p → qI#