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## INTRO LOGIC

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**INTRO LOGIC**Translations in PL5 DAY 19**Review – 1 Quantifier, 1 Predicate**Jay respects someone x Rjx Jay respects everyone x Rjx Jay respects no one x Rjx some one respects Kay x Rxk everyone respects Kay x Rxk no-one respects Kay x Rxk**Review – 2 Quantifiers, 1 Predicate**everyone respects everyone x y Rxy someone respects someone x y Rxy no-one respects everyone x y Rxy no-one is respected by everyone x y Ryx everyone respects someone (or other) x y Rxy there is someone whom everyone R’s x y Ryx there is someone who respects no-one x y Rxy there is someone whom no-one respects x y Ryx**New Material**1 quantifier, 2 predicates2 quantifiers, 2 predicates2 quantifiers, 3 predicates**Example 1**every student respects Kay no matter who you are IF you are a student, THEN you respect Kay no matter who x is IF x is a student, THEN x respects Kay x ( Sx Rxk )**Example 2**no student respects Jay there is no-one who is a S and who R's j there is no x : x is a S and x R's j x ( Sx & Rxj)**Example 3**Jay respects some (at least one) student there is someone who is a S (whom) j respects who is a S and whom j R's there is someone x x is a S and j R's x x (Sx & Rjx)**Example 4**no-one respects every politician there is no one who respects every politician there is no x : x respects every politician x x respects every politician**Example 4b**x x respects every politician no matter who you are IF you are P, THEN x R’s you no matter who y is IF y is P, THEN x R’s y y Py Rxy y ( Py Rxy ) x y ( Py Rxy )**Example 5**every Student respects someone or other no matter who you are IF you are S, THEN you R someone no matter who x is IF x is S, THEN x R's someone x Sx x R's someone x ( Sx x R's someone )**Example 5b**x ( Sx x R's someone ) there is someone who is R'ed by x whom x R's there is some y x R's y y Rxy x ( Sx y Rxy )**Example 6**there is a politician whom no-one respects there is someone who is a P whom no-one R‘s who is P AND whom no-one R's there is some x : x is P AND no-one R's x x Px & no-one R's x x ( Px & no-one R's x )**Example 6b**x ( Px & no one R's x ) there is no one who R’s x there is no y y R's x y Ryx x ( Px & y Ryx )**Example 7**there is a Politicianwhom every Citizen Respects there is someone who is a P whom every C R's who is P AND whom every C R's there is some x x is P AND every C R's x x Px & every C R's x x ( Px & every C R's x )**Example 7b**x ( Px & every C R's x ) no matter who you are if you are C then you R x no matter who y is if y is C then y R’s x y Cy Ryx y ( Cy Ryx ) x ( Px & y ( Cy Ryx ) )**Example 8**every Citizen Respects some Politician (or other) no matter who you are if you are C, then you R some P no matter who x is if x is C, then x R's some P x Cx x R's some P x ( Cx x R's some P )**Example 8b**x ( Cx x R's some P ) there is someone who is a P R'ed by x who is a P AND who is R'ed by x who is P AND whom x R's there is some y y is P AND x R's y y Py & Rxy y ( Py & Rxy ) x ( Cx y ( Py & Rxy ) )**Example 9**there is a Citizen who Respects no Politician there is someone who is a C who R's no P who is a C AND who R's no P there is some x : x is a C AND x R's no P x Cx & x R's no P x ( Cx & x R's no P )**Example 9b**x ( Cx & x R's no P ) there is no one who is a P AND who is R'ed by x there is no y : y is P AND y is R’ed by x y is P AND x R's y y Py & Rxy y ( Py & Rxy ) x ( Cx & y ( Py & Rxy ) )