all dogs are mortal fido is a dog so 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 As are BAll Bs are CSo all As are C



All As are Ba is A (a has the property A)So a is B






ABC

DMK



No zebra is a donkey







ABC

DMK



No zebra is a donkeyAll zebras are not donkeyEvery zebra is not a donkey







ABC

DMK



No zebra is a donkeyAll zebras are not donkeyEvery zebra is not a donkeyEvery zebra is a zebra







ABC

DMK



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







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


General termsIvy universities are very largeFrank’s parents are in townPeople drinking martini in the corner are drunk


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


being odd

being even

being in town

not liking bananas

Two-place (binary) predicates (relations)

being grater than

being equal

Predicates in PL


being odd

being even

being in town

not liking bananas


being grater than

being equal

Three-place (ternary) predicates (relations)

being between

Predicates in PL


being odd

being even

being in town

not liking bananas


being grater than

being equal


being between

Predicates in PL

One-place (unary) predicates (properties)

being odd

being even

being in town

not liking bananas


being grater than

being equal


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


There are dogs




xDx


There are dogs




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


There are dogs




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



There are no dogs

Nothing is a dog


~xDx



There are no dogs

Nothing is a dog


~xDx

It is not the case that everything is a dog

Not everything is a dog

Not all things are dogs



There are no dogs

Nothing 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.”

all dogs are mortal fido is a dog so fido is mortal

Documents

mortalall dogs

cabcdmkall dogs

ball bs

zebra ball

dogso fido

mammalsall mammals

property aso

mortalno zebra