Updated: 29 November 2000

Logic

 

valid arguments

 

validity due to form and not content

 

created a language hodgise (first-order predicate language)

 

invented a device - tableau - for determining validity of arguments written in hodgise.

1.

 


In one sense everything is in order: For the language in question (hodgise) every semantically valid sequent is syntactically valid and vice versa.

 

Does it capture all and only those arguments expressed in a real language (English) that are valid in virtue of form?

utility depends on how good a model it provides for English

 

problems

 

1. mis-match

 

is truth-functor in hodgise but if ... then ...in English is not always a truth-functor

2.


is transitive

 

P Q, Q R |- P R

 

P Q

Q R

[P R]

|

P

R

 

 


P Q

 

 

 


Q R

3.

If then (arguably) not transitive

Given:

If A then B

If B then C

Does not follow: If A then C

Consider

If Smith dies before the election, Jones will win

If Jones wins, Smith will retire from public life after the election.

 

\If Smith dies before the election, he will retire from public after the election.

(Sainsbury p. 76)

4.

 

If God does not exist, then it is not true that if I pray, my prayers will be answered by him.

 

I do not pray.

 

Therefore, God exists.

 

g: God exists

p: I pray

a: My prayers will be answered by God.

 

g [pa], p |- g

 

g [pa]

p

g

[pa] g |

p a


5.

 

2. limited expressive power

 

can we express all valid arguments in English in Hodgise ?

 

todays question

6.

 


Resources of Hodgise

 

basic tools

 

truth-functors , , , ,

 

names n, m

predicates Fx, Gxy

quantifiers " $

identity =

 

derivative tools

 

definite descriptions

numerical quantifiers

7.

 


The Meaning of the

 

The lecturer is happy

 

$x [[[Lx "y [Ly x=y]] Hx]

 

The Meaning of three ? ! ?

 

There are 3 muskateers

 

$x $y $z [[[Mx My M z] [ [x=y] [x=z] [y=z]]

"w [Mw [[ w=x w=y ] w=z]]]

8.

 


Successes

 

objects

properties

relations

 

looking pretty good

 

all of mathematics

 

all of physics ?

9.

 

All spheres of gold are less than 10 kilometre in diameter

 

"x [Sx Lx]

 

 

All spheres of enriched uranium are less than

10 kilometres in diameter

 

"x [Ux Lx]

10.

 

 

Todays lecture is for PPE

 

limitations become evident when we come to those aspects of language that are particularly important in dealing with conscious agents

11.

 

 

Strategies in the face of limitations

 

1. be ingenious

Russell and the

 

2. extend the logic

add new symbols, new rules

 

3. move the goal posts

learn about real language by comparing it to the logical language

12.

 

 

Adverbial modification

 

Icabod went with the police quietly

 

\ Icabod went with the police

13.

 


Johnnie went with the men in white coats noisely.

 

\Johnnie went with the men in white coats.

 

Icabod went with the zemindars narcescently

 

\Icabod went with the zemindars

 

(witheringly)

14.

 


n: Icabod

Wx: x went with the police

Qx: x is quiet

 

Wn Qn |- Wn

 

quiet ? Icabod a quiet person ???

 

 

True, he went quietly but a quiet person ? never !

 

Wn Qn |- Qn

15.

 


another attempt

 

Gx : x went with the police quietly

Wx : x went with the police

 

Gn |- Wn

 

Implicit ?

"x(Gx Wx]

 

sell-out : wanted to show this via logic

16.

 

Mj |- Wj

 

"x [Mx Wx]

 

 

Ni |- Zi

 

"x [Nx Zx]

17.

 

 

What was quiet ?

 

Icabod ?

 

no - it was the manner of his going

 

Icabods going with the police

18.

 


Icabod answered the phone loudly.

 

\Icabod answered the phone.

 

Davidson

 

adverbial modification qualifies events

 

domain :

events and persons !!!

19.

 


Ex : x is an event

Qx : x is quiet

Wx : x is a going with the police

Ixy : x involves y

 

$x[Ex [[Wx Ixn] Qx]] |- $x[Ex [Wx Ixn]]

 

displays validity as a matter of form

 

events as objects !!!

20.


 

Icabod ran quickly.

\ Icabod ran.

 

$x[Ex [Qx [Rx Ixn]]] |- $x[Ex [Rx Ixn]]

 

Events as objects ! ! !

21.


 

Mickey is a happy mouse.

 

\ Mickey is a mouse.

 

Hx : x is happy.

Mx : x is a mouse.

m : Mickey

 

Hm Mm |- Hm

 

Hm Mm |- Mm

22.

 

 

 

Mickey is a large mouse.

 

\ Mickey is a mouse.

 

form not content

 

Mickey is a small moose

\ Mickey is a moose

23.

 


Lx : x is large

 

Mickey is a large mouse.

\ Mickey is a mouse.

 

Mm Lm |- Mm

 

but from the premise

 

Mm Lm |- Lm

 

we can equally infer

 

Lm

 

So Mickey is large !!!

24.


Gx : x is a large mouse

Lx : x is large

Mx : x is a mouse

 

!!! Gm |- Lm

 

!!! Gm |- Mm

 

implicit ?

 

"x [Gx Mx]

25.

 


Meet Bruce the typical mouse : b

 

Lxy : x is larger than y

 

Mm Lmb |- Mm

 

Mm Lmb |- Lmb

 

large comparative

 

no such thing as the typical mouse !

 

works with Australians

26.

 


Mickey is a large mouse.

 

Mickey is larger than most mice.

 

Most- quantifier

 

inexact

 

All mice are happy

"x [Mx Nx]

 

Most mice are Happy

Wx [Mx Nx]

27.

 

 

For most mice, Mickey is larger than them

 

Wx [Mx Lnx]

28.


Evaluative comparative

 

Ronnie is a good actor

\ Ronnie is an actor

 

Gx : x is good.

Ax : x is an actor

 

[Gr Ar] |- Ar

 

[Gr Ar] |- Gr

 

Ronnie is a good actor.

\ Ronnie is good!!!

 

no way 29.

 


good actor

 

better than the standard actor ???

 

Hugh Grant ?

Humphrey Bogart?

 

better than most actors ??

 

couldnt most of them - nearly all of them - be absolutely terrible ?

 

in which case he could be better than most but still be truly awful

30.


Propositions

 

true or false

 

Sentences

 

It is now raining

 

Truth-value changes with time

 

1. Proposition - It rains at t

t names the time now

Fixed truth-value

2. Proposition - variable truth values

We have ignored time !

31.

 

 

It is now raining.

 

It will rain.

 

It has rained.

 

It is now raining

 

\ It will be that it has rained.

32.

 

r : It is raining

 

Fr : It will be that it is raining

 

Pr : It was that it is raining

 

r |- FPr

 

Rules for the new sentence functors which are not truth functors.

33.

 

 

r T or F ???

 

1 2 3 4 5 6 7

r : F F T T F T F

 

Fr : T T T T T F F

 

Pr : F F F T T T T

34.

 

 

?? |- r FPr

 

.

 

 

The End of Time

It rains (naturally)

But no future.

So FPr is false!

35.

 


 

It is now raining.

 

Time is 12 : 32.

 

t : 12.32

 

Rx : x is rainy

 

It is now raining

 

Rt

36.


 

This is not a translation.

 

It is now raining changes truth-value through time.

 

Rt has a fixed truth value which does not change (knowledge of it may change)

37.

 

 

It is now raining : Rt

 

It will rain

$x [Tx [Lxt Rx]]

 

It has rained

$x [Tx [Ltx Rx]]

38.

 

It is now raining. \ It will be that it was raining.

 

Rt |- $x [[Tx Lxt] $y [Ty [Lyt [Lxy Rx]]]]

39.

 

 

Our final sentence functor !

 

! f : May it be the case that f

 

! Hn: May it be the case that Icabod is happy

 

! is not a truth-functor

 

! Hn does not even have a truth-value

40.

 

t: 12:50 November 29, 2000

 

Ex: x is an event

 

Lxy: x is later than y

 

Hx: x is happy

 

Fx: x is festive

 

$x [Ex [Lxt [Fx [!Hx]]]]

41.