** **

**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**

**Ø**** [P ****®**** R]**

**|**

**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.**

** **

**Ø****g ****®**** ****Ø****[p****®****a], ****Ø****p ****|****-**** g**

** **

**Ø****g ****®**** ****Ø****[p****®****a]**

**Ø****p**

**Ø****g**

** **

** ****Ø****[p****®****a]
**__ØØ____g__** ****|**** **

**p
**__Ø____a__

**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] ****Ù****Ø**** [x=z] ****Ù****Ø**** [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**

** **

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

** **

** **

**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**

*W***x [Mx ****®**** Nx]**

**27.**

** **

** **

**For most mice,
Mickey is larger than them**

** **

** Wx
[Mx **

**28.**

**Evaluative
comparative**

** **

**Ronnie is a
good actor**

**\**** Ronnie is an
actor**

** **

**Gx : x is
good.**

** **

** ****[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**

** **

** **

**It is now
raining**

** **

**Truth-value
changes with time**

** **

**1. Proposition
- It rains at t**

**t – names the
time now**

**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**

** **

*F***r : It will be that it is
raining**

** **

*P***r : 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**

** **

*F***r : T T T T T F F**

** **

*P***r : F F F T T T T**

**34.**

** **

** **

?? |- r ® *FP*r

…….

The End of Time

It rains
(naturally)

But no future.

So *FP*r 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.**

** **

** **