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

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 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

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 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 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 ABCDMKABCDMK

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 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 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 ABCDMKABCDMK

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 donkeys Every zebra is not a donkey 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 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 ABCDMKABCDMK

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 donkeys Every zebra is not a donkey Every zebra is a zebra 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 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 ABCDMKABCDMK

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 donkies A Every zebra is not a donkey A Every zebra is a zebra B 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 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 ABCDMKABCDMK

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

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

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

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

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)

Aristotle is the father of logicFa U of A is largeLu Frank’s father is in townTf The men drinking martini in the corner is dangerousDm That is badBt 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)

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

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

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 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)

7.3E3 a. (Ia & Ba) & ∼ Ra

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

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

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

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

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)

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

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)

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)

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)

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)

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)

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

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)

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)

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)]

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)]

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)]

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

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

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

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

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

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

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

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

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

7.4E3 a. Pj ⊃ ( ∀ x)Px

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

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

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

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

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

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

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

7.4E3 a. Pj ⊃ ( ∀ x)Px b. ∼ ( ∃ x)Px ∨ Pj c. ( ∃ y)Py ⊃ (Pj & Pr) d. ∼ ( ∀ z)Pz & Pr e. ∼ Pr ⊃ ∼ ( ∃ x)Px f. ( ∃ y)Py & ∼ ( ∀ z)Pz g. (Pj ⊃ Pr) & (Pr ⊃ ( ∀ x)Px) h. ( ∼ Pj ⊃ ∼ ( ∃ x)Px) & (Pj ⊃ ( ∀ x)Px) i. ( ∀ y)Sy & ∼ ( ∀ y)Py

7.4E3 a. Pj ⊃ ( ∀ x)Px b. ∼ ( ∃ x)Px ∨ Pj c. ( ∃ y)Py ⊃ (Pj & Pr) d. ∼ ( ∀ z)Pz & Pr e. ∼ Pr ⊃ ∼ ( ∃ x)Px f. ( ∃ y)Py & ∼ ( ∀ z)Pz g. (Pj ⊃ Pr) & (Pr ⊃ ( ∀ x)Px) h. ( ∼ Pj ⊃ ∼ ( ∃ x)Px) & (Pj ⊃ ( ∀ x)Px) i. ( ∀ y)Sy & ∼ ( ∀ y)Py j. ( ∀ x)Sx ⊃ ( ∀ x)Px

7.4E3 a. Pj ⊃ ( ∀ x)Px b. ∼ ( ∃ x)Px ∨ Pj c. ( ∃ y)Py ⊃ (Pj & Pr) d. ∼ ( ∀ z)Pz & Pr e. ∼ Pr ⊃ ∼ ( ∃ x)Px f. ( ∃ y)Py & ∼ ( ∀ z)Pz g. (Pj ⊃ Pr) & (Pr ⊃ ( ∀ x)Px) h. ( ∼ Pj ⊃ ∼ ( ∃ x)Px) & (Pj ⊃ ( ∀ x)Px) i. ( ∀ y)Sy & ∼ ( ∀ y)Py j. ( ∀ x)Sx ⊃ ( ∀ x)Px k. ( ∀ x)Sx ⊃ ( ∃ y)Py

“And I haven’t seen the two Messengers either. They’ve both gone to the town. Just look along the road and tell me if you can see either of them.” “I can see nobody on the road”, said Alice. “I only wish I had such eyes”, the King remarked in a fretful tone. “To be able to see Nobody! And at that distance too!. Why it’s as much as I can do to see real people by this light!”  “Who did you pass on the road?” the King went on, holding out his hand to the Messenger for some hay. “Nobody”, said the Messenger. “Quite right”, said the King, “this young lady saw him too. So of course nobody walks slower than you.” “I do my best”, the Messenger said in a sullen tone, “I’m sure nobody walks much faster than I do!” “He can’t do that”, said the King, “or else he’d have been first.”