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 Ù P] ® ØE], E |= [ØG Ú ØP]

G P E | [[GÙP]®ØE] | E || [ØGÚØP]

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

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 Ù P] ®ØE], E, Ø [ØG Ú ØP] }

[[G Ù P] ® ØE]

E

Ø[ ØG Ú ØP]

|

ØØG

ØØP

Ø[G Ù P] ØE

ØG ØP

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, ØR}

tableau

P

Q

ØR

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

NS’s 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 doesn’t 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.

"xØHx

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 can’t 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 Ù Rx] WRONG

No students are rich

"x[Sx ® ØRx]

40.

Some students are rich.

$x[Sx Ù Rx]

Some students are not rich

$x[Sx Ù ØRx]

41.

“All” goes with ®

“Some” goes with Ù

!!! $x[Sx ® Rx]

P ® Q

ØP Ú Q

$x[ ØSx Ú 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 ® ØRx]

Some students are rich

$x[Sx Ù Rx]

Some students are not rich

$x[Sx Ù ØRx]

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 Ú ØR

CES: { Ø[R Ú ØR] }

Tableau

Ø[R Ú ØR ]

ØR

ØØR

Theorems : proved without assumptions.

45.

Tautologies

R | [R Ú ØR]

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

T | T F

F | F T

Tautologies - true come what may.

You don’t need a weather person to ...

|= R Ú Ø 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.