Updated 09 November 2000

$xFx

 

Is that true ?

 

To answer -

 

interpretation

 

domain

 

meaning of predicates

1.

 


$x[Sx Rx]

 

Interpretation

 

Domain : residents Oxford

Sx : x is a student

Rx : x is rich

 

 

Interpretation

 

Domain: dons Oxford

 

Sx : x is smart

 

Rx : x is rich

2.

 


Predicates

 

intension (meaning)

 

extension

 

Determining whether something is true or false is a matter of going through the intension to the extension

 

$xFx

 

Fx : x is funny

can you find ?

what is the extension ?

3.

 


$x[Fx Gx]

 

Interpretation

 

Domain {a, b, c}

 

Fx : {a, b}

 

Gx : {a}

 

so the wff is true

4.

 


$x[Fx Gx]

 

Domain {a, b, c, d}

 

Fx: {a, b}

 

Gx: {c, d}

 

The wff is false

 

$ " F x P

5.

 


Propositional Logic

 

consistency

 

P1, P2, P3 |- C

(via tableau)

 

then

 

P1, P2, P3 |= C

(via truth tables)

6.

 


Completeness

 

 

P1, P2, P3 |= C

(via truth tables)

 

then

 

P1, P2, P3 |- C

(via tableau)

7.


Consistency

 

 

If P1, P2, P3 |- C

 

then P1, P2, P3 |= C

8.

 

 

Completeness

 

 

If P1, P2, P3 |= C

 

then P1, P2, P3 |- C

9.

 

 

P1, P2, P3 |- C

 

just in case

 

P1, P2, P3 |= C

10.

 


Theorems

 

|- P P

 

Can be established by tableau without premises

11.

 

 

Tautologies |= P P

 

P | P P

----------------

T | T F

F | F T

|- f just in case |= f

12.


 

Predicate logic

 

 

no truth tables

 

true in an interpretation

 

 

|= f just in case f is true in every interpretation

 

 

every interpretation !?!?

13.

 

 

Domains:

 

students,

 

counting numbers 1,2,3,,

 

clever and honest politicians,

 

ballroom dancers,

 

fundamental particles,

 

copies of yesterdays Independent,

14.

 

Predicates:

 

x is a student;

x is a Zemindar,

x is twice as large as y and z together,

x is made of silver,

15.

 


P1, P2, P3 |= C

 

Every interpretation which makes P1, P2 and P3 true also makes C true.

 

P1, P2, P3 |- C

 

Tableau closes

 

Completeness

16.


 

P1, P2, P3 |- C

 

then

 

P1, P2, P3 |= C

 

Consistency 17.

 


Completeness

 

 

Proved by Godel.

 

 

Warming up exercise for Gdels Theorem

 

See Gdel, Escher, Bach.

18.

 


n : Dobaci

 

Lxy : x loves y

 

$x"yLxy |- Lnn

 

CES

 

{ $x"yLxy, Lnn }

19.

 


$x"yLxy |- Lnn

 

 

$x"yLxy

Lnn

"yLay

Laa

Lan

"yLby

Lba

Lbn

Lbb

"yLcy

Lcb

Lcn

Lca

Lcc

20.


$x"yLxy |- Lnn

 

Interpretation

Domain : this audience

+ God + Dobaci

Lxy : x loves y

n : Dobaci

 

Dobaci hates himself, God loves everyone one. So premise true, conclusion false.

\ not valid

21.

 

 

$x"yLxy |- Lnn

 

 

Interpretation

 

Domain : { a, b }

n b

 

Lxy {<a, a> , <a, b>}

22.


Referring

 

Names : n, m, ...

(non-vacuous)

 

Pronouns

He is a spy

 

Let n be the name of the person picked out by use of He.

 

Sn

 

Not equivalent in meaning but good enough.

23.

 


Definite Descriptions

 

The lecturer is happy.

 

Hx : x is happy.

n : The lecturer

 

Hn

 

The lecturer is happy. \ There is a lecturer.

 

Hn |- $xLx

 

tableau wont close !

24.

 


Hn : The lecturer is happy.

 

First Problem: we lost information.

 

Second Problem: designators must be non-vacuous.

25.

 

 

 

Something exists

 

$x [x=x]

 

The King does not exist.

 

n : The King

 

$x[x=n]

26.

 

 

 

But $x[x=n] is a theorem.

 

CES {$x[x = n] }

 

tableau

$x[x = n]

"x[x=n]

[n=n]

 

So it is true come what may that $x[x=n]. But the king does not exist !!!

 

Dont treat definite descriptions as names.

27.

 

 

Russells Theory of Definite Descriptions.

 

The lecturer is bored.

 

1. There is a lecturer.

$xLx

 

2. There is at most one lecturer

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

 

3. He is bored. "x[Lx Bx]

28.

 


Better

 

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

 

 

Hodges (bless him]

 

$x"y[ x=y Lx] "x[Lx Bx]

 

 

Domains need particular care.

29.

 


The train is late.

 

Tx : x is a train

 

Lx : x is late

 

$x[[Tx "y[Ty x=y]] Lx]

 

One and only one train !!!

 

Soon but not yet.

30.

 


1. Make the domain small

 

Domain : trains expected at Oxford from London in the next hour.

 

2. Take a larger domain but use an extra predicate.

 

Domain : trains in the UK

 

Nx : train with engine no. 7374

 

$x[[[Tx Nx] "y[[Ty Ny] x=y]] Lx]

31.


 

Do not apply Russells theory mechanically.

 

 

The whale is a mammal.

 

$x[[Wx "y[Wy x=y]] Mx]

 

 

"x[ Wx Mx]

32.

 


The lecturer is mad. \ There is a lecturer.

 

 

Mn |- $xLx

 

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

 

|- $xLx

 

 

CES

 

{$x[[Lx "y[Ly x=y]] Mx], $xLx }

33.

 


$x[[Lx"y[Lyx=y]]Mx]

 

$xLx

 

"xLx

 

[[La"y[Lya=y]]Ma]

 

[La "y[Ly a=y]

 

Ma

 

La

"y[Ly a=y]

La

34.

 


Our second problem

The King does not exist.

 

domain : residents of UK

n : the King

 

$x[x=n]

 

Wont do because $x[x=n] is a theorem.

 

The King exists

$x[Kx "y[Ky x=y]]

 

The King does not exist

$x[Kx "y[Kyx=y]

35.

 


Icabod does not sing.

Domain : all humans

Sx : x sings

n : Icabod

 

Sn

 

The Truth Revealed !

 

Icabod does not exist.

 

Ex : x exists

En

36.

 


En

 

Absurd. There is Icabod lacking the property of existence !

 

Icabod lacks singing abilility and also existence.

 

Existence is not a predicate

 

Existence is expressed by the existential quantifier.

37.

 


So we cannot write : En.

 

But still this Icabod does not exist.

 

Who is Icabod ?

 

Icabod is the Balliol student who is clever and who is over 10 feet tall.

 

Drop names in favour of definite descriptions.

38

 


Cerberus exists.

 

Who is Cerberus ?

 

Gx : x is a dog who guards the gates of Hell.

 

$x[[Gx "y[Gyx=y]]

 

Cerberus does not exist.

 

$x[[Gx"y[Gyx=y]]

 

Translation ?

Device ?

39.

 

 

How many heads has Cerberus ?

 

Domain : animals

Tx : x has three heads.

 

$x[[Gx "y[Gyx=y]] Tx].

 

Domain : fictional animals

 

True in the domain of fictional animals. False in the domain of real animals.

40.

 


Cerberus has three heads but Cerberus does not exist !

 

1. Cerberus does not exist but people imagine that he does exist and that he has three heads.

 

2. Domain of real and fictional animals.

Fx : x is fictional.

 

$x[[Gx"y[Gyx=y]] Fx]

Fido : the dog of Billy Hague

 

Fido is not fictional.

 

$x[[Mx "y[My x=y]] Fx]

41.

 


There is at least one student.

 

$xSx

 

There are at least two students.

 

$x$y[[Sx Sy] [x=y]]

 

There are at least three students.

 

$x$y$z[[[SxSy]Sz]] [[[[x=y] [x=z]] [y=z]]].

42.


 

There is at most one student.

 

"x"y[[Sx Sy] [x=y]]

 

 

There are at most two students.

 

"x"y"z[[[Sx Sy] Sz] [[x=y] [x=z]]

43.

 

 

There is exactly one student.

 

$x[Sx "y[Sy x=y]]

 

There are exactly two students.

 

$x$y[[Sx Sy] "z[Sz [z=x] [z=y]]].

 

And so on and so forth without limit.

 

Can we get rid of the numerals in this way?

44.