SL: Basic syntax - formulae

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