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 donkeysEvery 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 donkeysEvery 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 donkies 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∀ ⊃ ∃