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For 42 – 47: UD = everything;

For 42 – 47: UD = everything; Ax = x is an apple; Nxy = x is in y; Px = x is a pear; Rx = x is rotten; b = the basket There are apples and pears in the basket The only pear in the basket is rotten There are at least two apples in the basket There are two (and only two) apples in the basket

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For 42 – 47: UD = everything;

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  1. For 42 – 47: UD = everything; • Ax = x is an apple; Nxy = x is in y; Px = x is a pear; Rx = x is rotten; b = the basket • There are apples and pears in the basket • The only pear in the basket is rotten • There are at least two apples in the basket • There are two (and only two) apples in the basket • There are no more than two pears in the basket • 47 there are at least three apples in the basket

  2. For 42 – 47: UD = everything; • Ax = x is an apple; Nxy = x is in y; Px = x is a pear; Rx = x is rotten; b = the basket • There are apples and pears in the basket • x(Px & Nxb) & x(Ax & Nxb) • The only pear in the basket is rotten • There are at least two apples in the basket • There are two (and only two) apples in the basket • There are no more than two pears in the basket • 47 there are at least three apples in the basket

  3. For 42 – 47: UD = everything; • Ax = x is an apple; Nxy = x is in y; Px = x is a pear; Rx = x is rotten; b = the basket • There are apples and pears in the basket • x(Px & Nxb) & x(Ax & Nxb) • The only pear in the basket is rotten • x(Px & Nxb & Rx • There are at least two apples in the basket • There are two (and only two) apples in the basket • There are no more than two pears in the basket • 47 there are at least three apples in the basket

  4. For 42 – 47: UD = everything; • Ax = x is an apple; Nxy = x is in y; Px = x is a pear; Rx = x is rotten; b = the basket • There are apples and pears in the basket • x(Px & Nxb) & x(Ax & Nxb) • The only pear in the basket is rotten • x(Px & Nxb & Rx& y(Py & Nyb  y=x) ) • There are at least two apples in the basket • There are two (and only two) apples in the basket • There are no more than two pears in the basket • 47 there are at least three apples in the basket

  5. For 42 – 47: UD = everything; • Ax = x is an apple; Nxy = x is in y; Px = x is a pear; Rx = x is rotten; b = the basket • There are apples and pears in the basket • x(Px & Nxb) & x(Ax & Nxb) • The only pear in the basket is rotten • x(Px & Nxb & Rx& y(Py & Nyb  y=x) ) • There are at least two apples in the basket • xy(Ax & Nxb & Ay & Nyb • There are two (and only two) apples in the basket • There are no more than two pears in the basket • 47 there are at least three apples in the basket

  6. For 42 – 47: UD = everything; • Ax = x is an apple; Nxy = x is in y; Px = x is a pear; Rx = x is rotten; b = the basket • There are apples and pears in the basket • x(Px & Nxb) & x(Ax & Nxb) • The only pear in the basket is rotten • x(Px & Nxb & Rx& y(Py & Nyb  y=x) ) • There are at least two apples in the basket • xy(Ax & Nxb & Ay & Nyb& xy ) • There are two (and only two) apples in the basket • There are no more than two pears in the basket • 47 there are at least three apples in the basket

  7. For 42 – 47: UD = everything; • Ax = x is an apple; Nxy = x is in y; Px = x is a pear; Rx = x is rotten; b = the basket • There are apples and pears in the basket • x(Px & Nxb) & x(Ax & Nxb) • The only pear in the basket is rotten • x(Px & Nxb & Rx& y(Py & Nyb  y=x) ) • There are at least two apples in the basket • xy(Ax & Nxb & Ay & Nyb& xy ) • There are two (and only two) apples in the basket • xy(Ax & Nxb & Ay & Nyb & z(Az & Nzb  (z=y  z=x) & xy ) • There are no more than two pears in the basket • 47 there are at least three apples in the basket

  8. For 42 – 47: UD = everything; • Ax = x is an apple; Nxy = x is in y; Px = x is a pear; Rx = x is rotten; b = the basket • There are apples and pears in the basket • x(Px & Nxb) & x(Ax & Nxb) • The only pear in the basket is rotten • x(Px & Nxb & Rx& y(Py & Nyb  y=x) ) • There are at least two apples in the basket • xy(Ax & Nxb & Ay & Nyb& xy ) • There are two (and only two) apples in the basket • xy(Ax & Nxb & Ay & Nyb & z(Az & Nzb  (z=y  z=x) & xy ) • There are no more than two pears in the basket • xy(Ax & Nxb & Ay & Nyb & z(Az & Nzb  (z=y  z=x) ) • 47 there are at least three apples in the basket

  9. For 42 – 47: UD = everything; • Ax = x is an apple; Nxy = x is in y; Px = x is a pear; Rx = x is rotten; b = the basket • There are apples and pears in the basket • x(Px & Nxb) & x(Ax & Nxb) • The only pear in the basket is rotten • x(Px & Nxb & Rx& y(Py & Nyb  y=x) ) • There are at least two apples in the basket • xy(Ax & Nxb & Ay & Nyb& xy ) • There are two (and only two) apples in the basket • xy(Ax & Nxb & Ay & Nyb & z(Az & Nzb  (z=y  z=x) ) • There are no more than two pears in the basket • xy(Ax & Nxb & Ay & Nyb & z(Az & Nzb  (z=y  z=x) ) • there are at least three apples in the basket • xyz (Ax & Ay & Az & Nxb & Nyb & Nzb & xy& xz& zy )

  10. For 42 – 47: UD = everything; • Ax = x is an apple; Nxy = x is in y; Px = x is a pear; Rx = x is rotten; b = the basket • There are apples and pears in the basket • x(Px & Nxb) & x(Ax & Nxb) • The only pear in the basket is rotten • x(Px & Nxb & Rx& y(Py & Nyb  y=x) ) • There are at least two apples in the basket • xy(Ax & Nxb & Ay & Nyb& xy ) • There are two (and only two) apples in the basket • xy(Ax & Nxb & Ay & Nyb & z(Az & Nzb  (z=y  z=x) ) • There are no more than two pears in the basket • xy(Ax & Nxb & Ay & Nyb & z(Az & Nzb  (z=y  z=x) ) • there are at least three apples in the basket • xyz (Ax & Ay & Az & Nxb & Nyb & Nzb & xy& xz& zy ) • there are at most three apples in the basket • xyz (Ax & Ay & Az & Nxb & Nyb & Nzb & • & w(Aw & Nwb  w=x  w=y  w=z) )

  11. SL Truth value assignments

  12. SL Truth value assignments PL Interpretation

  13. SL Truth value assignments PL Interpretation Giving an interpretation means defining: UD

  14. SL Truth value assignments PL Interpretation Giving an interpretation means defining: UD Predicates

  15. SL Truth value assignments PL Interpretation Giving an interpretation means defining: UD Predicates Constants

  16. SL Truth value assignments PL Interpretation Giving an interpretation means defining: UD Predicates Constants Of course, we do not define variables

  17. Truth values of PL sentences are relative to an interpretation

  18. Truth values of PL sentences are • relative to an interpretation • Examples: • Fa • Fx = x is human • a = Socrates • Bab

  19. Truth values of PL sentences are • relative to an interpretation • Examples: • Fa • Fx = x is human Fx = x is handsome • a = Socrates a = Socrates • Bab

  20. Truth values of PL sentences are • relative to an interpretation • Examples: • Fa • Fx = x is human Fx = x is handsome • a = Socrates a = Socrates • Bab • Bxy = x is bigger than y • a = Himalayas • b = Alpes

  21. Truth values of PL sentences are • relative to an interpretation • Examples: • Fa • Fx = x is human Fx = x is handsome • a = Socrates a = Socrates • Bab • Bxy = x is bigger than y • a = Himalayas a = Himalayas • b = Alpes b = the moon

  22. Truth values of PL sentences are • relative to an interpretation • Examples: • Fa • Fx = x is human Fx = x is handsome • a = Socrates a = Socrates • Bab • Bxy = x is bigger than y • a = Himalayas a = Himalayas a = Himalayas • b = Alpes b = the moon b = Himalayas

  23. Truth values of PL sentences are • relative to an interpretation • Examples: • Fa • Fx = x is human Fx = x is handsome • a = Socrates a = Socrates • Bab • Bxy = x is bigger than y • a = Himalayas a = Himalayas a = Himalayas • b = Alpes b = the moon b = Himalayas • No constant can refer to more than one individual!

  24. Truth values of PL sentences are • relative to an interpretation • Examples: • Fa • Bab • ~xFx • UD = food • Fx = x is in the fridge

  25. Truth values of PL sentences are • relative to an interpretation • Examples: • Fa • Bab • ~xFx • UD = food • Fx = x is in the fridge • UD = everything • Fx = x is in the fridge

  26. Extensional definition of predicates Predicates are sets

  27. Extensional definition of predicates Predicates are sets Their members are everything they are true of

  28. Extensional definition of predicates Predicates are sets Their members are everything they are true of Predicates are defined relative to a UD

  29. Extensional definition of predicates Predicates are sets Their members are everything they are true of Predicates are defined relative to a UD Example: UD = natural numbers Ox = x is odd O = {1,3,5,7,9, ...}

  30. Extensional definition of predicates Predicates are sets Their members are everything they are true of Predicates are defined relative to a UD Example: UD = natural numbers Ox = x is odd Ox = {1,3,5,7,9, ...} Bxy = x>y Bxy = {(2,1), (3,1), (3,2), ...}

  31. Extensional definition of predicates Predicates are sets Their members are everything they are true of Predicates are defined relative to a UD Example: UD = natural numbers Ox = x is odd Bxyz = x is between y and z Ox = {1,3,5,7,9, ...} Bxyz = {(2,1,3), (3,2,4), ...} Bxy = x>y Bxy = {(2,1), (3,1), (3,2), ...}

  32. Extensional definition of predicates Predicates are sets Their members are everything they are true of Predicates are defined relative to a UD Example: UD = natural numbers Ox = x is odd Bxyz = x is between y and z Ox = {1,3,5,7,9, ...} Bxyz = {(2,1,3), (3,2,4), ...} Bxy = x>y Bxyz = y is between x and z Bxy = {(2,1), (3,1), (3,2), ...} Bxyz = {(1,2,3), (2,3,4), ...}

  33. Truth-values of compound sentences (An & Bmn) ~ Cn UD: All positive integers Ax: x is odd Bxy: x is bigger than y Cx: x is prime m: 2 n: 1

  34. Truth-values of compound sentences (An & Bmn) ~ Cn UD: All positive integers Ax: x is odd Bxy: x is bigger than y Cx: x is prime m: 2 n: 1 UD: All positive integers Ax: x is even Bxy: x is bigger than y Cx: x is prime m: 2 n: 1

  35. Truth-values of quantified sentences Birds fly UD = birds xFx

  36. Truth-values of quantified sentences Birds fly UD = birds xFx Fa Fb Fc : Ftwooty :

  37. Truth-values of quantified sentences Birds fly UD = birds UD = everything xFx x(Bx  Fx) Fa Fb Fc : Ftwooty :

  38. Truth-values of quantified sentences Birds fly UD = birds UD = everything xFx x(Bx  Fx) Fa Ba  Fa Fb Bb  Fb Fc Bc  Fc : : Ftwooty Btwootie  Ftwootie : :

  39. Truth-values of quantified sentences Birds fly Some birds don’t fly UD1 = birds UD2 = everything UD1 xFx x(Bx  Fx) x~Fx Fa Ba  Fa Fb Bb  Fb Fc Bc  Fc : : Ftwooty Btwootie  Ftwootie : :

  40. Truth-values of quantified sentences Birds fly Some birds don’t fly UD1 = birds UD2 = everything UD1 xFx x(Bx  Fx) x~Fx Fa Ba  Fa ~Ftwootie Fb Bb  Fb Fc Bc  Fc : : Ftwooty Btwootie  Ftwootie : :

  41. Truth-values of quantified sentences Birds fly Some birds don’t fly UD1 = birds UD2 = everything UD1 xFx x(Bx  Fx) x~Fx Fa Ba  Fa ~Ftwootie Fb Bb  Fb Fc Bc  Fc UD2 : : x(Bx & ~Fx) Ftwooty Btwootie  Ftwootie Bt & ~Ft : :

  42. Truth-values of quantified sentences xFx Fa & Fb & Fc & ...

  43. Truth-values of quantified sentences xFx Fa & Fb & Fc & ... xBx Fa  Fb  Fc  ...

  44. Truth-values of quantified sentences (x)(Ax  (y)Lyx)

  45. Truth-values of quantified sentences (x)(Ax  (y)Lyx) UD1: positive integers Ax: x is odd Lxy: x is less than y

  46. Truth-values of quantified sentences (x)(Ax  (y)Lyx) UD1: positive integers Ax: x is odd Lxy: x is less than y UD2: positive integers Ax: x is even Lxy: x is less than y

  47. Truth-values of quantified sentences (x)(Ax  (y)Lyx) UD1: positive integers Ax: x is odd Lxy: x is less than y UD2: positive integers Ax: x is even Lxy: x is less than y (x)(y)(Lxy & ~Ax)

  48. Va & (x) (Lxa ~ Exa) UD1: positive integers Vx: x is even Lxy: x is larger than y Exy: x is equal to y a:2 UD2: positive integers Vx: x is odd Lxy: x is less than y Exy: x is equal to y a:1 UD3: positive integers Vx: x is odd Lxy: x is larger than or equal to y Exy: x is equal to y a: 1

  49. Quantificational Truth, Falsehood, and Indeterminacy A sentence P of PL is quantificationally true if and only if P is true on every possible interpretation. A sentence P of PL is quantificationally false if and only if P is false on every possible interpretation. A sentence P of PL is quantificationally indeterminate if and only if P is neither quantificationally true nor quantificationally false.

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