Updated 09 November 2000
Sentence functors

 

 

F and y

 

F because y

 

Icabod hopes that F

 

It is false that y

1.

 

 

Truth functors

 

 

F and y

 

F or y

 

?? If F then y.

2.

 


P: There is economic progress.

 

E: The electorate will seek a change.

 

B: Blair will lose the election.

 

[P E], P |- B

 

CES

{[P E], P, B}

 

TABLEAU

[P E]

P

B

P E

3.


Implicit Premise

[E B]

 

[P E], P, [E B] |- B

 

{[P E], P, [E B], B }

 

 

[P E]

P

[E B]

B

P E

E B

4.

 

TRUTH TABLE

 

F y | F y

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

T T | T

T F | T

F T | T

F F | F

 

 

TABLEAU RULE

 

f y

 

f y

5.


 

BRACKETING

 

 

3 * 4 + 2

 

3 * (4 + 2)

 

(3 * 4) + 2

 

brackets give the order in which operations are preformed.

6.

 


T: I go to town.

B: I drink beer.

W: I drink wine.

 

T B W

 

1 [T [B W]]

2 [[T B] W]

 

brackets tell us the order of the operations -

 

brackets are said to indicate the scope of the truth-functor

7.


 

For each basic truth functor we have two tableau rules -

 

 

f y

 

 


f y

8.

 

 

Truth-table for [f y]

 

f y | [ f y ]

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

T T | T

T F | T

F T | T

F F | F

 

 

TABLEAU RULE [f y]

|

f

y

9.

 

 

f y

|

f

y

 

 

? [ f y]

10.

 


Truth table

 

f y | [ f y]

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

T T | T

T F | F

F T | F

F F | F

 

 

TABLEAU RULE

 

[f y]

 


f y

11.

 

 

f y | [f y]

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

T T | T

T F | F

F T | T

F F | T

 

 

RULE

 

[f y]

 

f y

12.

 

 

[f y]

 

f y | [f y]

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

T T | T

T F | F

F T | T

F F | T

 
TABLEAU RULE
[f y]

|

f

y

13.

 

 

T: The Tories win

H: Hague is happy

B: Blair is sad

 

[T H], [H B] |- [T B]

 

 

COUNTER EXAMPLE SET

 

! { [T H], [H B],

[T B] }

14.


 

Sequent

 

[T H] , [H B] |- [T B]

 

CES

 

{[T M] , [M B] , [T B]}

15.

 


TABLEAU

 

[T H]

 

[H B]

 

[T B]

|

T

 

B

 

T H

H B

16.


 

TESTING FOR VALIDITY

 

Set up code ( M : Major is sad)

 

Translate into our propositional language (, , , ...)

 

Form counter example set

 

Construct a tableau

 

If tableau closes, the premises syntactically entail the conclusion.

 

[L M], L |- M

 

syntactic sequent

 

syntactic turnstile |-

17.


 

Translation

 

D : Icabod got a distinction.

S : Icabod is sad.

 

D but S

 

D and S

 

[D S]

 

This difference makes no difference to validity.

18.

 


 

There will be a picnic unless it rains.

 

P : There will be a picnic.

R : It rains.

 

? Hodges : [P R]

 

If it does not rain, there will be a picnic.

 

[ R P]

19.


 

P R | [R P]

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

T T | F TT

T F | T TT

F T | F TF

F F | T FF

 

 

 

P R | [P R]

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

T T | T

T F | T

F T | T

F F | F

20.

F W | [FW] [WF]

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

T T | T T

T F | F T

F T | T F

F F | T T

21.

 

 

P R | [P R]

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

T T | T

T F | T

F T | T

F F | F

22.

W : You work very hard.

F : You will get a First.

 

[W F]

 

Maybe you are smart and lucky.

 

Tutor : You will get a First only if you work hard.

 

[F W]

23.

 

 

If you work hard, you will get a first and only if you work hard.

 

[F W] [W F]

 

 

F W | [ F W]

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

T T | T

T F | F

F T | F

F F | T

24.

 

 

f y | [f y]

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

T T | T

T F | F

F T | F

F F | T

 

 

Tableau Rule

 

[f y]

 


f f

y y

 

25.

 

f y | [f y]

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

T T | F T

T F | T F

F T | T F

F F | F T

 

 

RULE

[ f y]

 


f f

y y

26.


 

R : Icabod rows

F: Icabod gets a First.

G: Bloggs is a good tutor.

S: Bloggs is at St.Xs.

 

 

[[R F] G], [S G], S |- [R F]

 

 

Counter example set

 

{ [[R F] G], [S G], S, [R F] }

27.

 

 

[[R F ] G]

[S G]

S

[R F]

|

R

 

F

[R F] G

 

R F S G

28.


A second test for validity!

 

Valid argument

 

If the premises are true, the conclusion must be true.

 

Test via truth-tables

 

Truth-tables allow us to survey the conditions under which sentences are true.

29.


 

B: NS went to Budapest

O: NS stayed in Oxford

 

[B O], O \ B

 

 

B O | [B O] | O || B

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

1 T T | T | F || T

2 T F | T | T || T

3 F T | T | F || F

4 F F | F | T || F

 

Both premises true only in line 2.

Conclusion true in line 2.

Therefore valid argument

30.


I: Icabod is a Balliol student

M: Icabod is modest

 

[B M], M \ B

 

B M | [B M] | M || B

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

1 T T | T | T || F

2 T F | T | F || F

3 F T | T | T || T

4 F F | F | F || T

 

Lines 1 and 3 make both premises true.

 

But conclusion is false in line 1.

So argument is invalid !

31.

 

 

Exclusive or ?

 

 

B M | [[BM][BM]] | M || B

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

T T | T F | T || F

T F | T T | F || F

F T | T T | T || T

F F | F T | T || T

 

 

Both premises are true only in the third line.

 

Conclusion is true for that line.

 

\ argument is valid.

32.

 

 

Semantic entailment

 

 

P1, P2, P3 |= C

 

Just in case truth-tables show that whenever the premises are true the conclusion is true.

Semantic turnstile.

33.

 


Two distinct but related ideas.

 

P1, P2 |- C

 

P1, P2 |= C

 

The Big Surprise

 

P1, P2 |- C

 

exactly when

 

P1, P2 |= C

 

truth-tables vs tableau

34.

 

 

more formal

 

 

Richard Jeffreys Formal Logic its Scope and Limits

 

 

more philosophical

 

Mark Sainsburys Logical Forms

35.

 


 

P : Icabod is a pigeon

Q : Icabod is a fragopan

S: Icabod has no horns

 

[P Q], S |- P

 

CES {[P Q], S, P}

 

[P Q]

 

S

P

 

P Q

 

non-pigeon, tragopan, no horns

36.

 

 

what is a tragopan ?

 

a horned pheasant from Asia

37.

 

 

Test arguments for validity by testing their counter-example sets for consistency using tableau.

 

Limited to arguments whose validity depends on truth-functors.

38.

 

 

tableau for proposition logic (truth functional logic)

 

purely mechanical device

 

mechanical check of validity

 

mechanical generation of proof of validity

 

computer program

39.