1 / 53

SL: Basic syntax - formulae

SL: Basic syntax - formulae. Atomic formulae – have no connectives If P is a formula, so is ~P If P and Q are formulae, so are (P&Q), (P  Q), (P  Q), (P  Q) nothing else is a formula. PL: Basic syntax – formulae of PL. Atomic formulae of PL are formulae

remedios-petty
Télécharger la présentation

SL: Basic syntax - formulae

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. SL: Basic syntax - formulae • Atomic formulae – have no connectives • If P is a formula, so is ~P • If P and Q are formulae, so are • (P&Q), (PQ), (PQ), (PQ) • nothing else is a formula

  2. PL: Basic syntax – formulae of PL • Atomic formulae of PL are formulae • If P is a formula, so is ~P • If P and Q are formulae, so are • (P&Q), (PQ), (PQ), (PQ) • If P is a formula with at least one occurrence • of x and no x-quantifier, then • xPx and xPx are formulae • nothing else is a formula

  3. PL: Basic syntax – definitions of important notions • Atomic formulae

  4. PL: Basic syntax – definitions of important notions • Atomic formulae • Formulae

  5. PL: Basic syntax – definitions of important notions • Atomic formulae • Formulae • (Main) Logical operator

  6. PL: Basic syntax – definitions of important notions • Atomic formulae • Formulae • (Main) Logical operator • Scope of a quantifier

  7. PL: Basic syntax – definitions of important notions • Atomic formulae • Formulae • (Main) Logical operator • Scope of a quantifier • Bound/Free variable

  8. PL: Basic syntax – definitions of important notions • Atomic formulae • Formulae • (Main) Logical operator • Scope of a quantifier • Bound/Free variable • Sentence

  9. PL: Basic syntax – definitions of important notions • Atomic formulae • Formulae • (Main) Logical operator • Scope of a quantifier • Bound/Free variable • Sentence • Substitution instance

  10. Aristotle

  11. Aristotle’s Syllogistic A Universal affirmative: All As are B x(Ax  Bx) E Universal negative: No As are B x(Ax  ~Bx) or ~x(Ax & Bx) I Particular affirmative: Some As are B x(Ax & Bx) O Particular negative: Some As are not B x(Ax & ~ Bx) or ~x(Ax  Bx)

  12. Logical Square

  13. Logical Square x(Sx  Px) x(Sx & ~ Px) or ~x(Sx  Px)

  14. Logical Square x(Sx  ~Px) or ~x(Sx & Bx) x(Sx  Px) x(Sx & Px) x(Sx & ~ Px) or ~x(Sx  Px)

  15. First Figure: L, M, S. 1. 1. If every M is L and every S is M, then every S is L (Barbara).2. 2. If no M is L and every S is M, then no S is L (Celarent).1. 3. If every M is L and some S is M, then some S is L (Darii).1. 4. If no M is L and some S is M, then some S is not L (Ferio). Second Figure: M, L, S. 2. 1. If no L is M and every S is 11.I, then no S is L (Cesare).2. 2. If every L is M and no S is M, then no S is L (Camestres).2. 3. If no L is M and some S is M, then some S is not L (Festino).2. 4. If every L is M and some S is not M, then some S is not L (Baroco). Third Figure: L, S, M. 3. 1. If every M is L and every M is S, then some S is L (Darapti).3. 2. If no M is L and every M is S, then some S is not L (Felapton).3. 3. If some M is L and every M is S, then some S is L (Disamis).3. 4. If every M is L and some M is S, then some S is L (Datisi).3. 5. If some M is not L and every M is S, then some S is not L (Bocardo).3. 6. If no M is L and some M is S, then some S is not L (Ferison). All Syllogisms

  16. Barbara • 1. If every M is L and every S is M, then every S is L (Barbara). • All Ms are L • All Ss are M • So all Ss are L

  17. Barbara • 1. If every M is L and every S is M, then every S is L (Barbara). • All Ms are L A • All Ss are M A bArbArA • So all Ss are L A

  18. Chrysippus

  19. Greece

  20. 7.6E2 a. (∀y)(By ⊃ Ly)

  21. 7.6E2 a. (∀y)(By ⊃ Ly) b. (∀x)(Rx ⊃ Sx)

  22. 7.6E2 a. (∀y)(By ⊃ Ly) b. (∀x)(Rx ⊃ Sx) c. (∀z)(Rz ⊃ ∼ Lz)

  23. 7.6E2 a. (∀y)(By ⊃ Ly) b. (∀x)(Rx ⊃ Sx) c. (∀z)(Rz ⊃ ∼ Lz) d. (∃w)(Rw & Cw)

  24. 7.6E2 a. (∀y)(By ⊃ Ly) b. (∀x)(Rx ⊃ Sx) c. (∀z)(Rz ⊃ ∼ Lz) d. (∃w)(Rw & Cw) e. (∃x)Bx & (∃x)Rx

  25. 7.6E2 a. (∀y)(By ⊃ Ly) b. (∀x)(Rx ⊃ Sx) c. (∀z)(Rz ⊃ ∼ Lz) d. (∃w)(Rw & Cw) e. (∃x)Bx & (∃x)Rx f. (∃y)Oy & (∃y) ∼ Oy

  26. 7.6E2 a. (∀y)(By ⊃ Ly) b. (∀x)(Rx ⊃ Sx) c. (∀z)(Rz ⊃ ∼ Lz) d. (∃w)(Rw & Cw) e. (∃x)Bx & (∃x)Rx f. (∃y)Oy & (∃y) ∼ Oy g. [(∃z)Bz & (∃z)Rz] & ∼ (∃z)(Bz & Rz)

  27. 7.6E2 a. (∀y)(By ⊃ Ly) b. (∀x)(Rx ⊃ Sx) c. (∀z)(Rz ⊃ ∼ Lz) d. (∃w)(Rw & Cw) e. (∃x)Bx & (∃x)Rx f. (∃y)Oy & (∃y) ∼ Oy g. [(∃z)Bz & (∃z)Rz] & ∼ (∃z)(Bz & Rz) h. (∀w)(Rw ⊃ Sw) & (∀x)(Gx ⊃ Ox)

  28. 7.6E2 a. (∀y)(By ⊃ Ly) b. (∀x)(Rx ⊃ Sx) c. (∀z)(Rz ⊃ ∼ Lz) d. (∃w)(Rw & Cw) e. (∃x)Bx & (∃x)Rx f. (∃y)Oy & (∃y) ∼ Oy g. [(∃z)Bz & (∃z)Rz] & ∼ (∃z)(Bz & Rz) h. (∀w)(Rw ⊃ Sw) & (∀x)(Gx ⊃ Ox) i. (∃y)By & [(∃y)Sy & (∃y)Ly]

  29. 7.6E2 a. (∀y)(By ⊃ Ly) b. (∀x)(Rx ⊃ Sx) c. (∀z)(Rz ⊃ ∼ Lz) d. (∃w)(Rw & Cw) e. (∃x)Bx & (∃x)Rx f. (∃y)Oy & (∃y) ∼ Oy g. [(∃z)Bz & (∃z)Rz] & ∼ (∃z)(Bz & Rz) h. (∀w)(Rw ⊃ Sw) & (∀x)(Gx ⊃ Ox) i. (∃y)By & [(∃y)Sy & (∃y)Ly] j. ∼ (∃z)(Rz & Lz)

  30. 7.6E2 a. (∀y)(By ⊃ Ly) b. (∀x)(Rx ⊃ Sx) c. (∀z)(Rz ⊃ ∼ Lz) d. (∃w)(Rw & Cw) e. (∃x)Bx & (∃x)Rx f. (∃y)Oy & (∃y) ∼ Oy g. [(∃z)Bz & (∃z)Rz] & ∼ (∃z)(Bz & Rz) h. (∀w)(Rw ⊃ Sw) & (∀x)(Gx ⊃ Ox) i. (∃y)By & [(∃y)Sy & (∃y)Ly] j. ∼ (∃z)(Rz & Lz) k. (∀w)(Cw ⊃ Rw) & ∼ (∀w)(Rw ⊃ Cw)

  31. 7.6E2 a. (∀y)(By ⊃ Ly) b. (∀x)(Rx ⊃ Sx) c. (∀z)(Rz ⊃ ∼ Lz) d. (∃w)(Rw & Cw) e. (∃x)Bx & (∃x)Rx f. (∃y)Oy & (∃y) ∼ Oy g. [(∃z)Bz & (∃z)Rz] & ∼ (∃z)(Bz & Rz) h. (∀w)(Rw ⊃ Sw) & (∀x)(Gx ⊃ Ox) i. (∃y)By & [(∃y)Sy & (∃y)Ly] j. ∼ (∃z)(Rz & Lz) k. (∀w)(Cw ⊃ Rw) & ∼ (∀w)(Rw ⊃ Cw) l. (∀z)Rx & [(∃y)Cy & (∃y) ∼ Cy]

  32. 7.6E2 a. (∀y)(By ⊃ Ly) b. (∀x)(Rx ⊃ Sx) c. (∀z)(Rz ⊃ ∼ Lz) d. (∃w)(Rw & Cw) e. (∃x)Bx & (∃x)Rx f. (∃y)Oy & (∃y) ∼ Oy g. [(∃z)Bz & (∃z)Rz] & ∼ (∃z)(Bz & Rz) h. (∀w)(Rw ⊃ Sw) & (∀x)(Gx ⊃ Ox) i. (∃y)By & [(∃y)Sy & (∃y)Ly] j. ∼ (∃z)(Rz & Lz) k. (∀w)(Cw ⊃ Rw) & ∼ (∀w)(Rw ⊃ Cw) l. (∀z)Rx & [(∃y)Cy & (∃y) ∼ Cy] m. (∀y)Ry ∨ [(∀y)By ∨ (∀y)Gy]

  33. 7.6E2 a. (∀y)(By ⊃ Ly) b. (∀x)(Rx ⊃ Sx) c. (∀z)(Rz ⊃ ∼ Lz) d. (∃w)(Rw & Cw) e. (∃x)Bx & (∃x)Rx f. (∃y)Oy & (∃y) ∼ Oy g. [(∃z)Bz & (∃z)Rz] & ∼ (∃z)(Bz & Rz) h. (∀w)(Rw ⊃ Sw) & (∀x)(Gx ⊃ Ox) i. (∃y)By & [(∃y)Sy & (∃y)Ly] j. ∼ (∃z)(Rz & Lz) k. (∀w)(Cw ⊃ Rw) & ∼ (∀w)(Rw ⊃ Cw) l. (∀z)Rx & [(∃y)Cy & (∃y) ∼ Cy] m. (∀y)Ry ∨ [(∀y)By ∨ (∀y)Gy] n. ∼ (∀x)Lx & (∀x)(Lx ⊃ Bx)

  34. 7.6E2 a. (∀y)(By ⊃ Ly) b. (∀x)(Rx ⊃ Sx) c. (∀z)(Rz ⊃ ∼ Lz) d. (∃w)(Rw & Cw) e. (∃x)Bx & (∃x)Rx f. (∃y)Oy & (∃y) ∼ Oy g. [(∃z)Bz & (∃z)Rz] & ∼ (∃z)(Bz & Rz) h. (∀w)(Rw ⊃ Sw) & (∀x)(Gx ⊃ Ox) i. (∃y)By & [(∃y)Sy & (∃y)Ly] j. ∼ (∃z)(Rz & Lz) k. (∀w)(Cw ⊃ Rw) & ∼ (∀w)(Rw ⊃ Cw) l. (∀z)Rx & [(∃y)Cy & (∃y) ∼ Cy] m. (∀y)Ry ∨ [(∀y)By ∨ (∀y)Gy] n. ∼ (∀x)Lx & (∀x)(Lx ⊃ Bx) o. (∃w)(Rw & Sw) & (∃w)(Rw & ∼ Sw)

  35. 7.6E2 a. (∀y)(By ⊃ Ly) b. (∀x)(Rx ⊃ Sx) c. (∀z)(Rz ⊃ ∼ Lz) d. (∃w)(Rw & Cw) e. (∃x)Bx & (∃x)Rx f. (∃y)Oy & (∃y) ∼ Oy g. [(∃z)Bz & (∃z)Rz] & ∼ (∃z)(Bz & Rz) h. (∀w)(Rw ⊃ Sw) & (∀x)(Gx ⊃ Ox) i. (∃y)By & [(∃y)Sy & (∃y)Ly] j. ∼ (∃z)(Rz & Lz) k. (∀w)(Cw ⊃ Rw) & ∼ (∀w)(Rw ⊃ Cw) l. (∀z)Rx & [(∃y)Cy & (∃y) ∼ Cy] m. (∀y)Ry ∨ [(∀y)By ∨ (∀y)Gy] n. ∼ (∀x)Lx & (∀x)(Lx ⊃ Bx) o. (∃w)(Rw & Sw) & (∃w)(Rw & ∼ Sw) p. [(∃w)Sw & (∃w)Ow] & ∼ (∃w)(Sw & Ow)

  36. 7.6E2 a. (∀y)(By ⊃ Ly) b. (∀x)(Rx ⊃ Sx) c. (∀z)(Rz ⊃ ∼ Lz) d. (∃w)(Rw & Cw) e. (∃x)Bx & (∃x)Rx f. (∃y)Oy & (∃y) ∼ Oy g. [(∃z)Bz & (∃z)Rz] & ∼ (∃z)(Bz & Rz) h. (∀w)(Rw ⊃ Sw) & (∀x)(Gx ⊃ Ox) i. (∃y)By & [(∃y)Sy & (∃y)Ly] j. ∼ (∃z)(Rz & Lz) k. (∀w)(Cw ⊃ Rw) & ∼ (∀w)(Rw ⊃ Cw) l. (∀z)Rx & [(∃y)Cy & (∃y) ∼ Cy] m. (∀y)Ry ∨ [(∀y)By ∨ (∀y)Gy] n. ∼ (∀x)Lx & (∀x)(Lx ⊃ Bx) o. (∃w)(Rw & Sw) & (∃w)(Rw & ∼ Sw) p. [(∃w)Sw & (∃w)Ow] & ∼ (∃w)(Sw & Ow) q. (∃x)Ox & (∀y)(Ly ⊃ ∼ Oy)

  37. 1 Everyone that Michael likes likes either Henry or Sue 2 Michael likes everyone that both Sue and Rita like 3 Michael likes everyone that either Sue or Rita like 4 Rita doesn’t like Michael but she likes everyone that Michael likes 5 Grizzly bears are dangerous but black bears are not 6 Grizzly bears and polar bears are dangerous but black bears are not 7 Every self-respecting polar bear is a good swimmer

  38. 1 Everyone that Michael likes likes either Henry or Sue • x(Lmx  LxsLxh) • Michael likes everyone that both Sue and Rita like • 3 Michael likes everyone that either Sue or Rita like • 4 Rita doesn’t like Michael but she likes everyone that Michael likes • 5 Grizzly bears are dangerous but black bears are not • 6 Grizzly bears and polar bears are dangerous but black bears are not • 7 Every self-respecting polar bear is a good swimmer

  39. 1 Everyone that Michael likes likes either Henry or Sue • x(Lmx  LxsLxh) • Michael likes everyone that both Sue and Rita like • x(Lsx&Lrx  Lmx) • 3 Michael likes everyone that either Sue or Rita like • 4 Rita doesn’t like Michael but she likes everyone that Michael likes • 5 Grizzly bears are dangerous but black bears are not • 6 Grizzly bears and polar bears are dangerous but black bears are not • 7 Every self-respecting polar bear is a good swimmer

  40. 1 Everyone that Michael likes likes either Henry or Sue • x(Lmx  LxsLxh) • Michael likes everyone that both Sue and Rita like • x(Lsx&Lrx  Lmx) • Michael likes everyone that either Sue or Rita like • x(LsxLrx  Lmx) • 4 Rita doesn’t like Michael but she likes everyone that Michael likes • 5 Grizzly bears are dangerous but black bears are not • 6 Grizzly bears and polar bears are dangerous but black bears are not • 7 Every self-respecting polar bear is a good swimmer

  41. 1 Everyone that Michael likes likes either Henry or Sue • x(Lmx  LxsLxh) • Michael likes everyone that both Sue and Rita like • x(Lsx&Lrx  Lmx) • Michael likes everyone that either Sue or Rita like • x(LsxLrx  Lmx) • Rita doesn’t like Michael but she likes everyone that Michael likes • ~Lrm & x(Lmx  Lrx) • 5 Grizzly bears are dangerous but black bears are not • 6 Grizzly bears and polar bears are dangerous but black bears are not • 7 Every self-respecting polar bear is a good swimmer

  42. 1 Everyone that Michael likes likes either Henry or Sue • x(Lmx  LxsLxh) • Michael likes everyone that both Sue and Rita like • x(Lsx&Lrx  Lmx) • Michael likes everyone that either Sue or Rita like • x(LsxLrx  Lmx) • Rita doesn’t like Michael but she likes everyone that Michael likes • ~Lrm & x(Lmx  Lrx) • Grizzly bears are dangerous but black bears are not • x(Gx  Dx) & x(Bx  ~Dx) • 6 Grizzly bears and polar bears are dangerous but black bears are not • 7 Every self-respecting polar bear is a good swimmer

  43. 1 Everyone that Michael likes likes either Henry or Sue • x(Lmx  LxsLxh) • Michael likes everyone that both Sue and Rita like • x(Lsx&Lrx  Lmx) • Michael likes everyone that either Sue or Rita like • x(LsxLrx  Lmx) • Rita doesn’t like Michael but she likes everyone that Michael likes • ~Lrm & x(Lmx  Lrx) • Grizzly bears are dangerous but black bears are not • x(Gx  Dx) & x(Bx  ~Dx) • Grizzly bears and polar bears are dangerous but black bears are not • x(Gx  Px  Dx) & x(Bx  ~Dx) • 7 Every self-respecting polar bear is a good swimmer

  44. 1 Everyone that Michael likes likes either Henry or Sue • x(Lmx  LxsLxh) • Michael likes everyone that both Sue and Rita like • x(Lsx&Lrx  Lmx) • Michael likes everyone that either Sue or Rita like • x(LsxLrx  Lmx) • Rita doesn’t like Michael but she likes everyone that Michael likes • ~Lrm & x(Lmx  Lrx) • Grizzly bears are dangerous but black bears are not • x(Gx  Dx) & x(Bx  ~Dx) • Grizzly bears and polar bears are dangerous but black bears are not • x(Gx  Px  Dx) & x(Bx  ~Dx) • Every self-respecting polar bear is a good swimmer • x(Px & Rxx  Sx)

  45. UD = people 9 Anyone who likes Sue likes Rita x(Lxs  Lxr) 10 Everyone who likes Sue likes Rita 11 If anyone likes Sue, Michael does 12 If everyone likes Sue, Michael does 13 If anyone likes Sue, he or she likes Rita 14 Michael doesn’t like everyone 15 Michael doesn’t like anyone 16 If someone likes Sue, then s/he likes Rita 17 If someone likes Sue, then someone likes Rita

  46. UD = people • 9 Anyone who likes Sue likes Rita • x(Lxs  Lxr) • Everyone who likes Sue likes Rita • the same as 9 • 11 If anyone likes Sue, Michael does • 12 If everyone likes Sue, Michael does • 13 If anyone likes Sue, he or she likes Rita • 14 Michael doesn’t like everyone • 15 Michael doesn’t like anyone • 16 If someone likes Sue, then s/he likes Rita • 17 If someone likes Sue, then someone likes Rita

  47. UD = people • 9 Anyone who likes Sue likes Rita • x(Lxs  Lxr) • Everyone who likes Sue likes Rita • the same as 9 • If anyone likes Sue, Michael does • xLxs  Lms • 12 If everyone likes Sue, Michael does • 13 If anyone likes Sue, he or she likes Rita • 14 Michael doesn’t like everyone • 15 Michael doesn’t like anyone • 16 If someone likes Sue, then s/he likes Rita • 17 If someone likes Sue, then someone likes Rita

  48. UD = people • 9 Anyone who likes Sue likes Rita • x(Lxs  Lxr) • Everyone who likes Sue likes Rita • the same as 9 • If anyone likes Sue, Michael does • xLxs  Lms • If everyone likes Sue, Michael does • xLxs  Lms • 13 If anyone likes Sue, he or she likes Rita • 14 Michael doesn’t like everyone • 15 Michael doesn’t like anyone • 16 If someone likes Sue, then s/he likes Rita • 17 If someone likes Sue, then someone likes Rita

  49. UD = people • 9 Anyone who likes Sue likes Rita • x(Lxs  Lxr) • Everyone who likes Sue likes Rita • the same as 9 • If anyone likes Sue, Michael does • xLxs  Lms • If everyone likes Sue, Michael does • xLxs  Lms • If anyone likes Sue, he or she likes Rita • x(Lxs  Lxr) • 14 Michael doesn’t like everyone • 15 Michael doesn’t like anyone • 16 If someone likes Sue, then s/he likes Rita • 17 If someone likes Sue, then someone likes Rita

  50. UD = people • 9 Anyone who likes Sue likes Rita • x(Lxs  Lxr) • Everyone who likes Sue likes Rita • the same as 9 • If anyone likes Sue, Michael does • xLxs  Lms • If everyone likes Sue, Michael does • xLxs  Lms • If anyone likes Sue, he or she likes Rita • x(Lxs  Lxr) • 14 Michael doesn’t like everyone ~xLmx • 15 Michael doesn’t like anyone • 16 If someone likes Sue, then s/he likes Rita • 17 If someone likes Sue, then someone likes Rita

More Related