Updated 09 November 2000

 

The two tests for validity

 

entailment - what the premises do to the conclusion in a valid argument

1.

 

 

semantic entailment

 

semantic turnstile

 

test via truth-tables

 

P1, P2, P3 |= C

2.


 

syntactic entailment

 

syntactic turnstile

 

test via tableau

 

P1, P2, P3 |- C

3.

 

 

The Big Surprise

 

P1, P2, P3 |= C

 

just in case

 

P1, P2, P3 |- C

4.

 


Semantics = meaning

Syntax

 

P Q | P and Q

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

T T | T

T F | F

F T | F

F F | F

 

 

P Q | P or Q

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

T T | T

T F | T

F T | T

F F | F

5.

 


G: God is all good.

P : God is all powerful.

E : There is evil.

 

[[G P] E], E |= [G P]

 

G P E | [[GP]E] | E || [GP]

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

T T T | T F F | T || F F F

T T F | T T T | F || F F F

T F T | F T F | T || F T T

T F F | F T T | F || F T T

F T T | F T F | T || T T F

F T F | F T T | F || T T F

F F T | F T F | T || T T T

F F F | F T T | F || T T T

6.


CES

 

{ [[G P] E], E, [G P] }

 

[[G P] E]

E

[ G P]

|

G

 

P

 

[G P] E

 

G P

 

tableau closed

ces inconsistent

rgument valid

7.


 

Why bother to do it twice?

 

truth-tables more basic

 

why not stick to them ?

 

not applicable to more sophisticated logical language

 

need a proof technique

 

alternatives -

natural deduction

axiomatic method

8.

 

 

P : All dons are boring.

Q : Icabod is a don.

R : Icabod is boring.

 

P, Q |- R

 

CES { P, Q, R}

 

tableau

P

Q

R

9.

 


Inside the Proposition

 

Icabod is happy.

 

Clinton is funny.

 

William is a loser.

 

10.

 

 

1. Referring

picking out something

Icabod, for instance

 

2. Predicating

using a predicate to

ascribe a property

is happy

11.

 


Referring Devices

 

1. Proper Names

William Hague

 

2. Pronouns

She

 

3. Demonstratives

This, That

 

4. Definite Descriptions

The rich student

The best logician

 

Hodges : designators

12.


 

Designators

 

Hodges includes non-count nouns

 

butter

beer

bacon

 

Many are words for stuff

13.


 

Primary Reference

 

When a designator can be used on its own in a situation so as to refer to something, we call that thing the primary reference of the designator

 

Hodges p. 156

14.


 

Primary Reference

 

shopping list approach

 

primary reference is what you would fetch if I gave you a bare list of designators

15.

 

 

Margaret Thatcher

 

The President of OUSU

 

The fastest car in Oxford

 

Superman

 

NSs list of designators

 

The Master of Balliol

 

10391

16.

 


The Master of Balliol

 

fetch me that new Labour economist who presides over the only college more ugly than Keble

 

also known as Andrew Graham

 

The Master of Balliol was in love with Florence Nightingale

17.


 

Andrew Graham was never in love with Florence Nightingale.

Benjamin Jowett was however.

 

boring correspondence

 

The Master of Balliol does not always designate its primary reference.

18.

 

 

Purely Referential occurrence of a Designator

 

Designator refers to its primary referent

19.


 

Purely Referential

 

In fact there are many roles a designator may play in a sentence, besides referring to its primary reference. Sometimes it refers to something else; sometimes it doesnt refer to anything at all.

 

Hodges p.156

20.

 


Policy 1 on Reference

 

Restrict attention to designators that have a primary reference.

 

No vacuous names.

 

Ignore names that do not stand for anything

 

Gandalf

Cerberus

Superman

21.

 

 

Policy 2 on reference

 

Designators designate their primary references.

 

If they do not, we ignore them.

 

Hodges puts this more obscurely.

22.


 

Another restriction

 

we will focus on things and ignore stuff

 

Things can be counted -

students, trees, Porsches

 

You cannot count stuff

coffee, snow, butter

 

Stuff is picked out by non-count nouns.

23.


 

Proper Names

 

m, n, o, ...

 

n : Icabod

 

n is happy.

24.

 


Predicates

 

Icabod is happy.

 

----- is happy

 

Isabel is smarter than

Icabod.

 

---is smarter than ~~~

 

use variables to show the blanks

 

x is happy

x is smarter than y

25.

Hx : x is happy

Hn : Icabod is happy.

Sxy : x is smarter than y

 

m: Isabel

 

Smn

 

Snm

 

Predicates that need two or more names are called relational predicates.

26.

 

 

Hodges

 

a string of words and variables such that if you replace the variables with names the whole is a declarative sentence

27.

 

 

More complex predicates

 

Reading is between Oxford and London.

 

n: Reading

o : Oxford

m : London

Bxyz : x is between y and z

Bnom

28.

 


Quantifiers

 

another way of forming an indicative sentence from a predicate

 

Everyone

 

Everyone is happy

29.

 

 

Quantifiers have a domain which must be specified.

 

Quantifiers indicate something about number of things in the domain

30.

 

 

All

Some (someone)

No one

At least one

Exactly two

At most three

 

~~~~~~~~~~~~~~~~

Most

Few

Many

31.

 

 

Universal Quantifier

 

All

 

"x___

 

"x + Hx

 

"xHx

 

For all x, Hx

 

Take anything you like (in the domain) it is such that it is H.

32.

 


Nobody is happy.

 

n: nobody

Hx : x is happy

 

Hn

 

We are all sad but at least Nobody is happy.

 

"xHx

 

Take anyone you like in the domain, she or he is not happy.

33.

 

 

Existential Quantifier

 

$x

 

Hx

 

$xHx

 

You can find something (someone) such that it is happy.

34.

 


Domain : persons in this room

 

Lxy : x loves y

n : Icabod

m: Isabel

 

Lnn

 

Lmn

 

Lnm

 

Lmm

35.

 

 

Lxy : x loves y

n : Icabod

 

$xLnx

 

$xLxn

 

$x$yLxy

 

"xLnx

 

"xLxn

 

"x"yLxy

 

"y"xLxy

36.

 

 

Someone loves everyone.

 

$x"yLxy

 

Everyone is loved by someone.

 

"y$xLxy

These are very different !

37.

 

 

Someone loves everyone.

 

$x"yLxy

b

a

c

Everyone is loved by someone.

 

"y$xLxy

 

b

 


a c

 

These are very different !

38.

 

 

Domain : 1, 2, 3, ...

 

Bxy : x is bigger than y

 

"x$yByx

 

true - the numbers go on forever

 

$y"xByx

 

false - you cant find a number which is bigger than any number!

39.

 


Domain : people in Oxford

 

Sx : x is a student

 

Rx : x is rich

 

All students are rich.

 

"x[Sx Rx]

 

?? "x[Sx Rx] WRONG

 

No students are rich

 

"x[Sx Rx]

40.

 

 

Some students are rich.

 

$x[Sx Rx]

 

 

Some students are not rich

 

$x[Sx Rx]

41.

 

 

All goes with

Some goes with

 

 

!!! $x[Sx Rx]

 

P Q

 

P Q

 

$x[ Sx Rx]

 

one rich don is enough even though all students are poor!!

42.

 

 

All students are rich

 

"x[Sx Rx]

 

No students are rich

 

"x[Sx Rx]

 

Some students are rich

 

$x[Sx Rx]

 

Some students are not rich

 

$x[Sx Rx]

43.

 

 

"x[Sx Rx]

 

All students are rich.

 

Any student is rich.

 

Every student is rich.

 

Students are rich.

 

Each student is rich.

44.

 


Theorems

 

|- R R

 

CES: { [R R] }

 

Tableau

 

[R R ]

 

R

R

Theorems : proved without assumptions.

45.

 

 

Tautologies

 

R | [R R]

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

T | T F

F | F T

 

Tautologies - true come what may.

 

You dont need a weather person to ...

 

|= R R

46.

 

 

Something is a tautology just in case it is a theorem

 

|= A if and only if |- A

47.

 

 

Keynes

 


 

Domain

 

residents of Oxford

49.