All dogs are mortal Fido is a dog So Fido is mortal

1. All dogs are mortal Fido is a dog So Fido is mortal

2. All dogs are mortal Fido is a dog So Fido is mortal All dogs are mammals All mammals are mortal So all dogs are mortal

3. All dogs are mortal Fido is a dog So Fido is mortal All dogs are mammals All mammals are mortal So all dogs are mortal All As are B All Bs are C So all As are C

4. All As are B a is A (a has the property A) So a is B All As are B All Bs are C So all As are C All dogs are mortal Fido is a dog So Fido is mortal All dogs are mammals All mammals are mortal So all dogs are mortal

5. All As are B a is A (a has the property A) So a is B All As are B All Bs are C So all As are C A B C D M K All dogs are mortal Fido is a dog So Fido is mortal All dogs are mammals All mammals are mortal So all dogs are mortal

6. All As are B a is A (a has the property A) So a is B All As are B All Bs are C So all As are C All As are B a is A (a has the property A) So a is B All As are B All Bs are C So all As are C A B C D M K All dogs are mortal Fido is a dog So Fido is mortal All dogs are mammals All mammals are mortal So all dogs are mortal No zebra is a donkey All dogs are mortal Fido is a dog So Fido is mortal All dogs are mammals All mammals are mortal So all dogs are mortal

7. All As are B a is A (a has the property A) So a is B All As are B All Bs are C So all As are C All As are B a is A (a has the property A) So a is B All As are B All Bs are C So all As are C A B C D M K All dogs are mortal Fido is a dog So Fido is mortal All dogs are mammals All mammals are mortal So all dogs are mortal No zebra is a donkey All zebras are not donkey Every zebra is not a donkey All dogs are mortal Fido is a dog So Fido is mortal All dogs are mammals All mammals are mortal So all dogs are mortal

8. All As are B a is A (a has the property A) So a is B All As are B All Bs are C So all As are C All As are B a is A (a has the property A) So a is B All As are B All Bs are C So all As are C A B C D M K All dogs are mortal Fido is a dog So Fido is mortal All dogs are mammals All mammals are mortal So all dogs are mortal No zebra is a donkey All zebras are not donkey Every zebra is not a donkey Every zebra is a zebra All dogs are mortal Fido is a dog So Fido is mortal All dogs are mammals All mammals are mortal So all dogs are mortal

9. All As are B a is A (a has the property A) So a is B All As are B All Bs are C So all As are C All As are B a is A (a has the property A) So a is B All As are B All Bs are C So all As are C A B C D M K All dogs are mortal Fido is a dog So Fido is mortal All dogs are mammals All mammals are mortal So all dogs are mortal No zebra is a donkey A All zebras are not donkey A Every zebra is not a donkey A Every zebra is a zebra B All dogs are mortal Fido is a dog So Fido is mortal All dogs are mammals All mammals are mortal So all dogs are mortal

10. Singular terms Aristotle is the father of logic U of A is large Frank’s father is in town The men drinking martini in the corner is dangerous That is bad I don’t like bananas

11. Singular terms Aristotle is the father of logic U of A is large Frank’s father is in town The men drinking martini in the corner is dangerous That is bad I don’t like bananas General terms Ivy universities are very large Frank’s parents are in town People drinking martini in the corner are drunk

12. Singular terms Aristotle is the father of logic U of A is large Frank’s father is in town The men drinking martini in the corner is dangerous That is bad I don’t like bananas General terms Ivy universities are very large Frank’s parents are in town People drinking martini in the corner are drunk Mass terms Snow is white Might is right

13. Grammatical predicates Aristotle is the father of logic U of A is large Frank’s father is in town The men drinking martini in the corner is dangerous That is bad I don’t like bananas

14. Grammatical predicates Aristotle is the father of logic U of A is large Frank’s father is in town The men drinking martini in the corner is dangerous That is bad I don’t like bananas Predicates in PL Aristotle is the father of logic (Aristotle has the property of being the father of logic) U of A is large (U of A has the property of being large) Frank’s father is in town The men drinking martini in the corner is dangerous That is bad I don’t like bananas (I have the property of not liking bananas)

15. Aristotle is the father of logic Fa U of A is large Lu Frank’s father is in town Tf The men drinking martini in the corner is dangerous Dm That is bad Bt I don’t like bananas ~Lv Predicates in PL Aristotle is the father of logic (Aristotle has the property of being the father of logic) U of A is large (U of A has the property of being large) Frank’s father is in town The men drinking martini in the corner is dangerous That is bad I don’t like bananas (I have the property of not liking bananas)

16. Predicates in PL One-place (unary) predicates being odd being even being in town not liking bananas

17. Predicates in PL One-place (unary) predicates being odd being even being in town not liking bananas Two-place (binary) predicates (relations) being grater than being equal

18. Predicates in PL One-place (unary) predicates being odd being even being in town not liking bananas Two-place (binary) predicates (relations) being grater than being equal Three-place (ternary) predicates (relations) being between

19. Predicates in PL One-place (unary) predicates being odd being even being in town not liking bananas Two-place (binary) predicates (relations) being grater than being equal Three-place (ternary) predicates (relations) being between Predicates in PL One-place (unary) predicates (properties) being odd being even being in town not liking bananas Two-place (binary) predicates (relations) being grater than being equal Three-place (ternary) predicates (relations) being between n –place (n-ary) predicates (relations)

20. 7.3E3 a. (Ia & Ba) & ∼ Ra

21. 7.3E3 a. (Ia & Ba) & ∼ Ra b. (Rc & Bc) & ∼ Ic

22. 7.3E3 a. (Ia & Ba) & ∼ Ra b. (Rc & Bc) & ∼ Ic c. (Bd & Rd) & Id

23. 7.3E3 a. (Ia & Ba) & ∼ Ra b. (Rc & Bc) & ∼ Ic c. (Bd & Rd) & Id d. ∼ [(Rb ∨ Bb) ∨ Ib]

24. 7.3E3 a. (Ia & Ba) & ∼ Ra b. (Rc & Bc) & ∼ Ic c. (Bd & Rd) & Id d. ∼ [(Rb ∨ Bb) ∨ Ib] e. Ib ⊃ (Id & Ia)

25. 7.3E3 a. (Ia & Ba) & ∼ Ra b. (Rc & Bc) & ∼ Ic c. (Bd & Rd) & Id d. ∼ [(Rb ∨ Bb) ∨ Ib] e. Ib ⊃ (Id & Ia) f. [(Ba & Ia) & (Ib & ∼ Bb)] & ∼ (Ra ∨ Rb)

26. 7.3E3 a. (Ia & Ba) & ∼ Ra b. (Rc & Bc) & ∼ Ic c. (Bd & Rd) & Id d. ∼ [(Rb ∨ Bb) ∨ Ib] e. Ib ⊃ (Id & Ia) f. [(Ba & Ia) & (Ib & ∼ Bb)] & ∼ (Ra ∨ Rb) g. Lab & Dac

27. 7.3E3 a. (Ia & Ba) & ∼ Ra b. (Rc & Bc) & ∼ Ic c. (Bd & Rd) & Id d. ∼ [(Rb ∨ Bb) ∨ Ib] e. Ib ⊃ (Id & Ia) f. [(Ba & Ia) & (Ib & ∼ Bb)] & ∼ (Ra ∨ Rb) g. Lab & Dac h. (Laa & Lac) & (Dab & Dad)

28. 7.3E3 a. (Ia & Ba) & ∼ Ra b. (Rc & Bc) & ∼ Ic c. (Bd & Rd) & Id d. ∼ [(Rb ∨ Bb) ∨ Ib] e. Ib ⊃ (Id & Ia) f. [(Ba & Ia) & (Ib & ∼ Bb)] & ∼ (Ra ∨ Rb) g. Lab & Dac h. (Laa & Lac) & (Dab & Dad) i. ∼ (Lca ∨ Dca) & (Lcd & Dcd)

29. 7.3E3 a. (Ia & Ba) & ∼ Ra b. (Rc & Bc) & ∼ Ic c. (Bd & Rd) & Id d. ∼ [(Rb ∨ Bb) ∨ Ib] e. Ib ⊃ (Id & Ia) f. [(Ba & Ia) & (Ib & ∼ Bb)] & ∼ (Ra ∨ Rb) g. Lab & Dac h. (Laa & Lac) & (Dab & Dad) i. ∼ (Lca ∨ Dca) & (Lcd & Dcd) j. ∼ (Adc ∨ Abc) & (Acd & Acb)

30. 7.3E3 a. (Ia & Ba) & ∼ Ra b. (Rc & Bc) & ∼ Ic c. (Bd & Rd) & Id d. ∼ [(Rb ∨ Bb) ∨ Ib] e. Ib ⊃ (Id & Ia) f. [(Ba & Ia) & (Ib & ∼ Bb)] & ∼ (Ra ∨ Rb) g. Lab & Dac h. (Laa & Lac) & (Dab & Dad) i. ∼ (Lca ∨ Dca) & (Lcd & Dcd) j. ∼ (Adc ∨ Abc) & (Acd & Acb) k. Acb  (Sbc & Rb)

31. 7.3E3 a. (Ia & Ba) & ∼ Ra b. (Rc & Bc) & ∼ Ic c. (Bd & Rd) & Id d. ∼ [(Rb ∨ Bb) ∨ Ib] e. Ib ⊃ (Id & Ia) f. [(Ba & Ia) & (Ib & ∼ Bb)] & ∼ (Ra ∨ Rb) g. Lab & Dac h. (Laa & Lac) & (Dab & Dad) i. ∼ (Lca ∨ Dca) & (Lcd & Dcd) j. ∼ (Adc ∨ Abc) & (Acd & Acb) k. Acb  (Sbc & Rb) l. (Aab & Aad) & ∼ (Lab ∨ Lad)

32. 7.3E3 a. (Ia & Ba) & ∼ Ra b. (Rc & Bc) & ∼ Ic c. (Bd & Rd) & Id d. ∼ [(Rb ∨ Bb) ∨ Ib] e. Ib ⊃ (Id & Ia) f. [(Ba & Ia) & (Ib & ∼ Bb)] & ∼ (Ra ∨ Rb) g. Lab & Dac h. (Laa & Lac) & (Dab & Dad) i. ∼ (Lca ∨ Dca) & (Lcd & Dcd) j. ∼ (Adc ∨ Abc) & (Acd & Acb) k. Acb  (Sbc & Rb) l. (Aab & Aad) & ∼ (Lab ∨ Lad) m. (Sdc & Sca) ⊃ Sda

33. 7.3E3 a. (Ia & Ba) & ∼ Ra b. (Rc & Bc) & ∼ Ic c. (Bd & Rd) & Id d. ∼ [(Rb ∨ Bb) ∨ Ib] e. Ib ⊃ (Id & Ia) f. [(Ba & Ia) & (Ib & ∼ Bb)] & ∼ (Ra ∨ Rb) g. Lab & Dac h. (Laa & Lac) & (Dab & Dad) i. ∼ (Lca ∨ Dca) & (Lcd & Dcd) j. ∼ (Adc ∨ Abc) & (Acd & Acb) k. Acb  (Sbc & Rb) l. (Aab & Aad) & ∼ (Lab ∨ Lad) m. (Sdc & Sca) ⊃ Sda n. (Abd & Adb) ⊃ (Lbd & Ldb)

34. 7.3E3 a. (Ia & Ba) & ∼ Ra b. (Rc & Bc) & ∼ Ic c. (Bd & Rd) & Id d. ∼ [(Rb ∨ Bb) ∨ Ib] e. Ib ⊃ (Id & Ia) f. [(Ba & Ia) & (Ib & ∼ Bb)] & ∼ (Ra ∨ Rb) g. Lab & Dac h. (Laa & Lac) & (Dab & Dad) i. ∼ (Lca ∨ Dca) & (Lcd & Dcd) j. ∼ (Adc ∨ Abc) & (Acd & Acb) k. Acb  (Sbc & Rb) l. (Aab & Aad) & ∼ (Lab ∨ Lad) m. (Sdc & Sca) ⊃ Sda n. (Abd & Adb) ⊃ (Lbd & Ldb) o. (Lcb & Lba) ⊃ (Dca & Sca)

35. 7.3E3 a. (Ia & Ba) & ∼ Ra b. (Rc & Bc) & ∼ Ic c. (Bd & Rd) & Id d. ∼ [(Rb ∨ Bb) ∨ Ib] e. Ib ⊃ (Id & Ia) f. [(Ba & Ia) & (Ib & ∼ Bb)] & ∼ (Ra ∨ Rb) g. Lab & Dac h. (Laa & Lac) & (Dab & Dad) i. ∼ (Lca ∨ Dca) & (Lcd & Dcd) j. ∼ (Adc ∨ Abc) & (Acd & Acb) k. Acb  (Sbc & Rb) l. (Aab & Aad) & ∼ (Lab ∨ Lad) m. (Sdc & Sca) ⊃ Sda n. (Abd & Adb) ⊃ (Lbd & Ldb) o. (Lcb & Lba) ⊃ (Dca & Sca) p. ∼ [(Rc ∨ Bc) ∨ Ic] ⊃ ∼ [(Lac ∨ Lbc) ∨ (Lcc ∨ Ldc)]

36. 7.3E3 a. (Ia & Ba) & ∼ Ra b. (Rc & Bc) & ∼ Ic c. (Bd & Rd) & Id d. ∼ [(Rb ∨ Bb) ∨ Ib] e. Ib ⊃ (Id & Ia) f. [(Ba & Ia) & (Ib & ∼ Bb)] & ∼ (Ra ∨ Rb) g. Lab & Dac h. (Laa & Lac) & (Dab & Dad) i. ∼ (Lca ∨ Dca) & (Lcd & Dcd) j. ∼ (Adc ∨ Abc) & (Acd & Acb) k. Acb  (Sbc & Rb) l. (Aab & Aad) & ∼ (Lab ∨ Lad) m. (Sdc & Sca) ⊃ Sda n. (Abd & Adb) ⊃ (Lbd & Ldb) o. (Lcb & Lba) ⊃ (Dca & Sca) p. ∼ [(Rc ∨ Bc) ∨ Ic] ⊃ ∼ [(Lac ∨ Lbc) ∨ (Lcc ∨ Ldc)] q. Rd & ∼ [Ra ∨ (Rb ∨ Rc)]

37. 7.3E3 a. (Ia & Ba) & ∼ Ra b. (Rc & Bc) & ∼ Ic c. (Bd & Rd) & Id d. ∼ [(Rb ∨ Bb) ∨ Ib] e. Ib ⊃ (Id & Ia) f. [(Ba & Ia) & (Ib & ∼ Bb)] & ∼ (Ra ∨ Rb) g. Lab & Dac h. (Laa & Lac) & (Dab & Dad) i. ∼ (Lca ∨ Dca) & (Lcd & Dcd) j. ∼ (Adc ∨ Abc) & (Acd & Acb) k. Acb  (Sbc & Rb) l. (Aab & Aad) & ∼ (Lab ∨ Lad) m. (Sdc & Sca) ⊃ Sda n. (Abd & Adb) ⊃ (Lbd & Ldb) o. (Lcb & Lba) ⊃ (Dca & Sca) p. ∼ [(Rc ∨ Bc) ∨ Ic] ⊃ ∼ [(Lac ∨ Lbc) ∨ (Lcc ∨ Ldc)] q. Rd & ∼ [Ra ∨ (Rb ∨ Rc)] r. (Rd & Id) & ∼ [((Ra & Ia) ∨ (Rb & Ib)) ∨ (Rc & Ic)]

38. Quantifiers many a few few most some at least one little much at most three all every any each none a(n) not one whatever whichever

39. Existential quantifier There are dogs At least one thing is a dog There is at least one thing which is a dog There is at least one thing which has the property of being a dog

40. Existential quantifier There are dogs At least one thing is a dog There is at least one thing which is a dog There is at least one thing which has the property of being a dog xDx

41. Existential quantifier There are dogs At least one thing is a dog There is at least one thing which is a dog There is at least one thing which has the property of being a dog xDx Universal quantifier Everything is a dog Each thing is a dog Each and every thing is a dog All things are dogs Anything is a dog

42. Existential quantifier There are dogs At least one thing is a dog There is at least one thing which is a dog There is at least one thing which has the property of being a dog xDx Universal quantifier Everything is a dog Each thing is a dog Each and every thing is a dog All things are dogs Anything is a dog xDx

43. Negations of simple quantified sentences It is not the case that there are dogs There are no dogs Nothing is a dog There is no thing that is a dog

44. Negations of simple quantified sentences It is not the case that there are dogs There are no dogs Nothing is a dog There is no thing that is a dog ~xDx

45. Negations of simple quantified sentences It is not the case that there are dogs There are no dogs Nothing is a dog There is no thing that is a dog ~xDx It is not the case that everything is a dog Not everything is a dog Not all things are dogs

46. Negations of simple quantified sentences It is not the case that there are dogs There are no dogs Nothing is a dog There is no thing that is a dog ~xDx It is not the case that everything is a dog Not everything is a dog Not all things are dogs ~xDx

47. 7.4E3 a. Pj ⊃ (∀x)Px

48. 7.4E3 a. Pj ⊃ (∀x)Px b. ∼ (∃x)Px ∨ Pj

49. 7.4E3 a. Pj ⊃ (∀x)Px b. ∼ (∃x)Px ∨ Pj c. (∃y)Py ⊃ (Pj & Pr)

50. 7.4E3 a. Pj ⊃ (∀x)Px b. ∼ (∃x)Px ∨ Pj c. (∃y)Py ⊃ (Pj & Pr) d. ∼ (∀z)Pz & Pr