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

Ø [P ® R]

|

P

Ø R

Ø P Q

Ø 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.

Øg ® Ø[p®a], Øp |- g

Øg ® Ø[p®a]

Øp

Øg

Ø[p®a] ØØg |

p Øa

5.

2. limited expressive power

can we express all valid arguments in English in Hodgise ?

today’s question

6.

Resources of Hodgise

basic tools

truth-functors ®, Ú, Ù, Ø, «

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 Ù "y [Ly ® x=y]] Ù Hx]

The Meaning of “three” ? ! ?

There are 3 muskateers

$x $y $z [[[Mx Ù My Ù M z] Ù [ Ø [x=y] ÙØ [x=z] ÙØ [y=z]] Ù

"w [Mw ®[[ w=x Ú w=y ] Ú 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.

Today’s 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 Ù Qn |- Wn

quiet ? Icabod a quiet person ???

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

Wn Ù 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

Icabod’s 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 Ù [[Wx Ù Ixn] ÙQx]] |- $x[Ex Ù [Wx Ù Ixn]]

displays validity as a matter of form

events as objects !!!

20.

Icabod ran quickly.

\ Icabod ran.

$x[Ex Ù [Qx Ù [Rx Ù Ixn]]] |- $x[Ex Ù [Rx Ù 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 Ù Mm |- Hm

Hm Ù 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 Ù Lm |- Mm

but from the premise

Mm Ù 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 Ù Lmb |- Mm

Mm Ù 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 Ù Ar] |- Ar

[Gr Ù 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 ??

couldn’t 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 Ù [Lxt Ù Rx]]

It has rained

$x [Tx Ù [Ltx Ù Rx]]

38.

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

Rt |- $x [[Tx Ù Lxt] Ù $y [Ty Ù [Lyt Ù [Lxy Ù 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 Ù [Lxt Ù [Fx Ù [!Hx]]]]

41.