Updated 09 November 2000

“ It all depends on what you mean by “is” ”

President Clinton

“is” of predication

Clinton is clever

n: Clinton

Cx: x is clever

IDENTITY

Dr Jekyll is Mr Hyde.

The Prime Minister is Tony Blair.

Everest is Chomdongolinga.

n=m.

Tricky Identities

Icabod is the smartest student in the class.

1= - eⁱ^p

Sure-Fire Identities

Icabod is Icabod.

n=n

Tableau for Quantifiers.

Rule :

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

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

n: Hyde

m : Jekyll

Tx : x is tall

Tn, n=m \ Tm

Seems OK

Leibniz’ Rule

Leibniz 1646 - 1716

Indiscernibility of Identicals.

But Shakespeare had it first.

Shakespeare

1564 – 1616

Tableau

n=m

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 : don’t 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.

Hodge’s 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: don’t attempt to deal with designators which do not have a primary referent.

In particular, avoid vacuous names.

21.

Tableau for Quantifiers

"xFx

NB : “n” must be an OLD name.

“n” must already be in play in the tableau.

WEIRD

22.

$xFx

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.

Ø"xFx

$xØFx

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

24.

Ø$xFx

"xØFx

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 Ù Rx]

CES

{Sn,Rn, Ø$x[Sx Ù Rx]}

26.

Sn, Rn |- $x[Sx Ù Rx]

Ø$x[Sx Ù Rx]

"xØ[Sx Ù Rx]

Ø[Sn Ù Rn]

ØSn ØRn

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], ØRn}

28.

Cn, "x[Cx ® Rx] |- Rn

"x[Cx ® Rx]

ØRn

[Cn ® Rn]

ØCn Rn

closes, argument valid

29.

Rx : x is rich

Sx : x is sad

Domain : those in this room

$xRx, $xSx |- $x[Rx Ù Sx]

CES

{$xRx, $xSx, Ø$x[RxÙSx]}

30.

$xRx, $xSx |- $x[Rx Ù Sx]

$xRx

$xSx

Ø$x[Rx Ù Sx]

"xØ[Rx Ù Sx]

!!! Sa

Ø[Ra Ù Sa]

ØRa ØSa

!!! closes but not valid so use new names

31.

Theorem

something you can prove without using any assumptions.

P Ú Ø P

CES { Ø[P Ú ØP] }

Ø[P Ú ØP]

ØP

ØØP

32.

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

|- "x[ Fx Ú ØFx]

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 { Ø$x[x = x] }

Ø$x[x = x]

"xØ[x = x]

end of the road for Hodges

NS forges on :

Ø[b = b]

did not use an old name.

35.

Rx: x is rich

Hx : x is happy

domain: this audience

$xRx, "x[Rx®Hx] |- $xHx

CES

{$xRx, "x[Rx ® Hx], Ø$xHx}

36.

$xRx

"x[Rx ® Hx]

Ø$xHx

"xØHx

Ra ® Ha

ØRa Ha

ØHa

closes, therefore valid.

37.