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All dogs are mortalFido is a dogSo Fido is mortal
All dogs are mortalFido is a dogSo Fido is mortal
All dogs are mammalsAll mammals are mortalSo all dogs are mortal
All dogs are mortalFido is a dogSo Fido is mortal
All dogs are mammalsAll mammals are mortalSo all dogs are mortal
All As are BAll Bs are CSo all As are C
All dogs are mortalFido is a dogSo Fido is mortal
All dogs are mammalsAll mammals are mortalSo all dogs are mortal
All As are Ba is A (a has the property A)So a is B
All As are BAll Bs are CSo all As are C
All dogs are mortalFido is a dogSo Fido is mortal
All dogs are mammalsAll mammals are mortalSo all dogs are mortal
All As are Ba is A (a has the property A)So a is B
All As are BAll Bs are CSo all As are C
ABC
DMK
All dogs are mortalFido is a dogSo Fido is mortal
All dogs are mammalsAll mammals are mortalSo all dogs are mortal
No zebra is a donkey
All As are Ba is A (a has the property A)So a is B
All As are BAll Bs are CSo all As are C
All dogs are mortalFido is a dogSo Fido is mortal
All dogs are mammalsAll mammals are mortalSo all dogs are mortal
All As are Ba is A (a has the property A)So a is B
All As are BAll Bs are CSo all As are C
ABC
DMK
All dogs are mortalFido is a dogSo Fido is mortal
All dogs are mammalsAll mammals are mortalSo all dogs are mortal
No zebra is a donkeyAll zebras are not donkeyEvery zebra is not a donkey
All As are Ba is A (a has the property A)So a is B
All As are BAll Bs are CSo all As are C
All dogs are mortalFido is a dogSo Fido is mortal
All dogs are mammalsAll mammals are mortalSo all dogs are mortal
All As are Ba is A (a has the property A)So a is B
All As are BAll Bs are CSo all As are C
ABC
DMK
All dogs are mortalFido is a dogSo Fido is mortal
All dogs are mammalsAll mammals are mortalSo all dogs are mortal
No zebra is a donkeyAll zebras are not donkeyEvery zebra is not a donkeyEvery zebra is a zebra
All As are Ba is A (a has the property A)So a is B
All As are BAll Bs are CSo all As are C
All dogs are mortalFido is a dogSo Fido is mortal
All dogs are mammalsAll mammals are mortalSo all dogs are mortal
All As are Ba is A (a has the property A)So a is B
All As are BAll Bs are CSo all As are C
ABC
DMK
All dogs are mortalFido is a dogSo Fido is mortal
All dogs are mammalsAll mammals are mortalSo 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 As are Ba is A (a has the property A)So a is B
All As are BAll Bs are CSo all As are C
All dogs are mortalFido is a dogSo Fido is mortal
All dogs are mammalsAll mammals are mortalSo all dogs are mortal
All As are Ba is A (a has the property A)So a is B
All As are BAll Bs are CSo all As are C
ABC
DMK
Singular termsAristotle is the father of logicU of A is largeFrank’s father is in townThe men drinking martini in the corner is dangerousThat is badI don’t like bananas
Singular termsAristotle is the father of logicU of A is largeFrank’s father is in townThe men drinking martini in the corner is dangerousThat is badI don’t like bananas
General termsIvy universities are very largeFrank’s parents are in townPeople drinking martini in the corner are drunk
Singular termsAristotle is the father of logicU of A is largeFrank’s father is in townThe men drinking martini in the corner is dangerousThat is badI don’t like bananas
General termsIvy universities are very largeFrank’s parents are in townPeople drinking martini in the corner are drunk
Mass termsSnow is whiteMight is right
Grammatical predicatesAristotle is the father of logicU of A is largeFrank’s father is in townThe men drinking martini in the corner is dangerousThat is badI don’t like bananas
Grammatical predicatesAristotle is the father of logicU of A is largeFrank’s father is in townThe men drinking martini in the corner is dangerousThat is badI don’t like bananas
Predicates in PLAristotle 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 townThe men drinking martini in the corner is dangerousThat is badI don’t like bananas (I have the property of not liking bananas)
Aristotle is the father of logic FaU of A is large LuFrank’s father is in town TfThe men drinking martini in the corner is dangerous DmThat is bad BtI don’t like bananas ~Lv
Predicates in PLAristotle 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 townThe men drinking martini in the corner is dangerousThat is badI 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) & Idd. [(Rb Bb) Ib]∼ ∨ ∨
7.3E3
a. (Ia & Ba) & Ra∼b. (Rc & Bc) & Ic∼c. (Bd & Rd) & Idd. [(Rb Bb) Ib]∼ ∨ ∨e. Ib (Id & Ia)⊃
7.3E3
a. (Ia & Ba) & Ra∼b. (Rc & Bc) & Ic∼c. (Bd & Rd) & Idd. [(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) & Idd. [(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) & Idd. [(Rb Bb) Ib]∼ ∨ ∨e. Ib (Id & Ia)⊃f. [(Ba & Ia) & (Ib & Bb)] & (Ra Rb)∼ ∼ ∨g. Lab & Dach. (Laa & Lac) & (Dab & Dad)
7.3E3
a. (Ia & Ba) & Ra∼b. (Rc & Bc) & Ic∼c. (Bd & Rd) & Idd. [(Rb Bb) Ib]∼ ∨ ∨e. Ib (Id & Ia)⊃f. [(Ba & Ia) & (Ib & Bb)] & (Ra Rb)∼ ∼ ∨g. Lab & Dach. (Laa & Lac) & (Dab & Dad)i. (Lca Dca) & (Lcd & Dcd)∼ ∨
7.3E3
a. (Ia & Ba) & Ra∼b. (Rc & Bc) & Ic∼c. (Bd & Rd) & Idd. [(Rb Bb) Ib]∼ ∨ ∨e. Ib (Id & Ia)⊃f. [(Ba & Ia) & (Ib & Bb)] & (Ra Rb)∼ ∼ ∨g. Lab & Dach. (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) & Idd. [(Rb Bb) Ib]∼ ∨ ∨e. Ib (Id & Ia)⊃f. [(Ba & Ia) & (Ib & Bb)] & (Ra Rb)∼ ∼ ∨g. Lab & Dach. (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) & Idd. [(Rb Bb) Ib]∼ ∨ ∨e. Ib (Id & Ia)⊃f. [(Ba & Ia) & (Ib & Bb)] & (Ra Rb)∼ ∼ ∨g. Lab & Dach. (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) & Idd. [(Rb Bb) Ib]∼ ∨ ∨e. Ib (Id & Ia)⊃f. [(Ba & Ia) & (Ib & Bb)] & (Ra Rb)∼ ∼ ∨g. Lab & Dach. (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) & Idd. [(Rb Bb) Ib]∼ ∨ ∨e. Ib (Id & Ia)⊃f. [(Ba & Ia) & (Ib & Bb)] & (Ra Rb)∼ ∼ ∨g. Lab & Dach. (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) & Idd. [(Rb Bb) Ib]∼ ∨ ∨e. Ib (Id & Ia)⊃f. [(Ba & Ia) & (Ib & Bb)] & (Ra Rb)∼ ∼ ∨g. Lab & Dach. (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) & Idd. [(Rb Bb) Ib]∼ ∨ ∨e. Ib (Id & Ia)⊃f. [(Ba & Ia) & (Ib & Bb)] & (Ra Rb)∼ ∼ ∨g. Lab & Dach. (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) & Idd. [(Rb Bb) Ib]∼ ∨ ∨e. Ib (Id & Ia)⊃f. [(Ba & Ia) & (Ib & Bb)] & (Ra Rb)∼ ∼ ∨g. Lab & Dach. (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) & Idd. [(Rb Bb) Ib]∼ ∨ ∨e. Ib (Id & Ia)⊃f. [(Ba & Ia) & (Ib & Bb)] & (Ra Rb)∼ ∼ ∨g. Lab & Dach. (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.”