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SL Truth value assignments

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

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SL Truth value assignments

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  1. SL Truth value assignments

  2. SL Truth value assignments PL Interpretation

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

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

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

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

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

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

  9. 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

  10. 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

  11. 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

  12. 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

  13. 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!

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

  15. 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

  16. Extensional definition of predicates Predicates are sets

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

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

  19. 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, ...}

  20. 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), ...}

  21. 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), ...}

  22. 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), ...}

  23. 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

  24. 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

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

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

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

  28. 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 : :

  29. 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 : :

  30. 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 : :

  31. 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 : :

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

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

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

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

  36. 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

  37. 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)

  38. 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

  39. 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.

  40. Quantificational Truth, Falsehood, and Indeterminacy A sentence P of PL is quantificationally true if and only if P is true on every possible interpretation. Explain why the following is quantificationally true. ~ (x) (Ax ≡ ~Ax)

  41. Quantificational Truth, Falsehood, and Indeterminacy A sentence P of PL is quantificationally false if and only if P is false on every possible interpretation. Explain why the following is quantificationally false: (x)Ax & (y) ~Ay

  42. Quantificational Truth, Falsehood, and Indeterminacy Show that the following is quantificationally indeterminate: (Ac & Ad) & (y) ~Ay A sentence P of PL is quantificationally indeterminate if and only if P is neither quantificationally true nor quantificationally false.

  43. Quantificational Equivalence and Consistency Sentences P and Q of PL are quantificationally equivalent if and only if there is no interpretation on which P and Q have different truth values. A set of sentences of PL is quantificationally consistent if and only if there is at least one interpretation on which all members are true. A set of sentences of PL is quantificationally inconsistent if and only if it is not quantificationally consistent, i.e. if and only if there is no interpretation on which all members have the same truth value.

  44. Quantificational Entailment and Validity A set  of sentences of PL quantificationally entails a sentence P of PL if and only if there is no interpretation on which all the members of  are true and P is false. An argument is quantificationally valid if and only if there is no interpretation on which every premise is true yet the conclusion false.

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