Updated 09 November 2000

 

It all depends on what you mean by is

President Clinton

1.


is of predication

 

Clinton is clever

 

n: Clinton

 

Cx: x is clever

 

Cn

2.

 

 

IDENTITY

 

Dr Jekyll is Mr Hyde.

 

The Prime Minister is Tony Blair.

 

Everest is Chomdongolinga.

 

n=m.

3.


 

Tricky Identities

 

Icabod is the smartest student in the class.

 

1= - eip

 

Sure-Fire Identities

 

Icabod is Icabod.

 

n=n

4.

 

 

Tableau for Quantifiers.

 

Rule :

 

Whenever (d=d) occurs on a branch, close that branch.

5.


 

Dobaci is even smarter than the tallest person in the room.

 

Dobaci is Icabod

 

\ Icabod is even smarter than the tallest person in the room

 

i: Icabod

d: Dobaci

Sx: x is even smarter than the tallest person in the room

 

Sd, d=i |- Si

6.

 

n: Hyde

m : Jekyll

 

Tx : x is tall

 

Tn, n=m \ Tm

 

Seems OK

7.

 


Leibniz Rule

 

Leibniz 1646 - 1716

 

Indiscernibility of Identicals.

 

But Shakespeare had it first.

 

Shakespeare

1564 1616

8.


 

Tableau

.

.

.

Tn

n=m

|

Tm

9.

 

 

Shakespeare in Trouble !

 

Tn, n=m |- Tm

 

Does not always work !

10.

 

 

Lxy : x was in love with y

n : The Master of Balliol

m : Andrew Graham

f : Florence Nightingale

 

Lnf, n=m |- Lmf

 

As so often in affairs of the heart, something has gone wrong.

11.

 

1)                                          Lnf

2)                                          n=m

n designates different things in 1) and 2)

1)      Jowett 2) Graham

12.


 

Build Leibniz/ Shakespeare into our logic.

 

Exclude any cases in which it fails.

 

My policy : dont go where Leibniz fears to follow.

13.


 

NS believes that Marseille is the capital of France.

 

Paris is the capital of France.

 

\ NS believes that Marseille is Paris

 

Propositional attitude (intentional contexts)

14.

 

 

It could have been the Margaret Thatcher is the Prime Minister.

 

The Prime Minister is Tony Blair

 

\It could have been that Margaret Thatcher is Tony Blair.

 

invalid.

 

Modal contexts.

15.


 

Problems for Leibniz

 

1. Temporal shifts

 

2. Propositional attitudes

 

3. Modalities

 

We simply set these asides for a later occasion.

16.


 

Hodges Policy : second assumption

 

we assume every designator is purely referential

 

Test : re-write with the designator at the front

 

The Master of Balliol is such that he was in love with Florence.

17.


 

If it expresses the same as the original, the designator is purely referential.

 

In the front, it picks out the primary referent (the shopping list referent).

 

So this case is not purely referential.

18.

 


The Prime Minister is such that he/she could have been Margaret Thatcher.

 

Difference in what it expresses. So not purely referential.

 

obsurities in this procedure (expresses the same thing?)

19.

 

 

The Honest Approach

 

avoid arguments in which Leibniz law fails.

20.

 

 

Hodges First Assumption

 

When dealing with sentences in which designator is purely referential, we assume that it has a primary referent.

 

NS: dont attempt to deal with designators which do not have a primary referent.

 

In particular, avoid vacuous names.

21.

 

 

Tableau for Quantifiers

 

"xFx

|

Fn

 

NB : n must be an OLD name.

 

n must already be in play in the tableau.

 

WEIRD

22.


$xFx

|

Fn

 

NB : n must be a new name.

 

n must not have already occured in the tableau.

 

This is fair enough as we will see.

23.

 

"xFx

|

$xFx

 

If it is false that everyone is funny, then you can find someone who is not funny.

24.

 

$xFx

|

"xFx

 

If it is false that you can find someone who is funny, then take anyone you like he or she will not be funny.

25.


 

Sx : x is a student

Rx : x is rich

n : Icabod

 

Domain : residents of Oxford

 

Sn, Rn |- $x[Sx Rx]

 

CES

{Sn,Rn, $x[Sx Rx]}

26.

 


 

Sn, Rn |- $x[Sx Rx]

 

Sn

Rn

$x[Sx Rx]

|

"x[Sx Rx]

|

[Sn Rn]

 

Sn Rn

tableau closes, argument valid

27.


 

Cx : x is at Christ Church

Rx : x is rich

n : Icabod

Domain : students in Oxford

 

Cn, "x[Cx Rx] |- Rn

 

CES

{Cn, "x[Cx Rx], Rn}

28.


 

Cn, "x[Cx Rx] |- Rn

 


Cn

 

"x[Cx Rx]

 

Rn

|

[Cn Rn]

Cn Rn

 

closes, argument valid

29.


 

Rx : x is rich

Sx : x is sad

Domain : those in this room

 

$xRx, $xSx |- $x[Rx Sx]

 

CES

 

{$xRx, $xSx, $x[RxSx]}

30.


$xRx, $xSx |- $x[Rx Sx]

 

$xRx

$xSx

$x[Rx Sx]

|

"x[Rx Sx]

Ra

!!! Sa

|

[Ra Sa]

 

Ra Sa

!!! closes but not valid so use new names

31.

 


Theorem

 

something you can prove without using any assumptions.

 

P P

 

CES { [P P] }

 

[P P]

P

P

P

32.

 

 

Theorems represent truths of logic. They are true come what may.

 

|- "x[ Fx Fx]

33.


 

Something green exists.

$xGx

Something exists.

$x[x = x]

Cannot just write : $x

not even a sentence

34.

 


Consider whether it should be a theorem of logic that $x[x = x]

 

CES { $x[x = x] }

 

$x[x = x]

|

"x[x = x]

 

end of the road for Hodges

 

NS forges on :

[b = b]

did not use an old name.

35.

 

Rx: x is rich

Hx : x is happy

domain: this audience

 

$xRx, "x[RxHx] |- $xHx

 

CES

 

{$xRx, "x[Rx Hx], $xHx}

36.

 


$xRx

"x[Rx Hx]

$xHx

|

"xHx

|

Ra

|

Ra Ha

 

Ra Ha

|

Ha

 

closes, therefore valid.

37.