Updated 09 November 2000

Hodges little trick

the tableau trick

a tableau is a ‘mechanical’ device for testing sets of propostions for consistency

ergo - it tests arguments for validity

1.

OUR FIRST TABLEAU

Argument

B or O, not-O \B

Counter-example set

{B or O, not-O, not-B}

Tableau

B or O

not-0

not-B

B                                 O

2.

#### Argument

H and not-G

G or S

\ S

Counter-example set

{H and not-G, G or S, not-S}

3.

#### H and not-G

G or S

not S

|

H

not-G

G                                 S

4.

Consistent Sets

1.         Icabod is very rich.

2.         I. studied physics at Balliol.

Consistent and plausible

1.         Icabod is very rich

2.         Icabod studies PPE at Christ Church.

5.

Consistent Set

some possible circumstance in which all members of the set are true.

Inconsistent Set

no circumstances in which all the members of the set are true.

6.

Mickey Mouse Logic

very limited class of arguments

arguments whose validity depends on truth-functors

7.

SENTENCE FUNCTORS

P: It is raining.

__ and __

P and Q

sentence functor

“string of English words and variables such that if the variables are replaced by declarative sentences, the whole is a sentence.”

8.

f and y

f but y

It is hoped that f

f and then later y

It is not the case that f

N-S hopes that f

f because y

9.

TRUTH-FUNCTORS

L : Lord Lucan is dead.

H : Hague has a friend.

?          L and H

?  L or H

Some sentence functors give sentences whose truth-value is determined by the truth-values of the constituent sentences.

10.

Hodges -

truth-functor is a sentence-functor with a truth table

L                      H         |           L and H

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

T                      T          |           T

T                      F          |           F

F                      T          |           F

F                      F          |           F

11.

#### Symbol Time

and                  Ù

or                     Ú

not                   Ø

L Ù H

L Ú H

12.

H:        Hague has a friend

L                      H                     |           L Ú H

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

T                      T                      |           T

T                      F                      |           T

F                      T                      |           T

F                      F                      |           F

13.

L                      H                     |           L Ú H

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

T                      T                      |           T

T                      F                      |           T

F                      T                      |           T

F                      F                      |           F

P                      |                       ØP

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

T                      |                       F

F                      |                       T

14.

The strange story of the cruel Dean.

P:         You will pass prelims.

Q:        You will be sent down.

P Ú Q

You pass but he sends you down.

Unfair !

No ! The Dean is a logican

15.

P                      Q         |           P Ú Q

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

1                      T                      T          |           T

2                      T                      F          |           T

3                      F                      T          |           T

4                      F                      F          |           F

“Look at line 1” cries the Cruel Dean.

Inclusive versus exclusive use of “or”.

Lucky lottery.

(P Ú Q) Ù Ø(P Ù Q)

16.

### Exclusive “or”

P                      Q         |           P or Q

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

1                      T                      T          |           F

2                      T                      F          |           T

3                      F                      T          |           T

4                      F                      F          |           F

17.

G                     C         |           G because C

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

T                      T          |                       ?

T                      F          |                       F

F                      T          |                       F

F                      F          |                       F

“because” is not truth-functor

partial truth-table

18.

f         |           NS believes that f

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

T          |                       ?

F          |                       ?

f         |           God believes that f

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

T          |                       T

F          |                       F

19.

The vexed matter of conditionals.

If  .....  then ....

Sentence functor

If  P then Q

P - antecedent

Q – consequent

20.

Is “If ... then ...” a truth-functor ???

P:         Hague is has vit B deficiency in his brain.

Q:        Hague is schizophrenic.

P                      Q                     |           If p then q

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

T                      T                      |

T                      F                      |                       F

F                      T                      |

F                      F                      |

21.

R:                    The paper turns red

S:                     The solution is an acid.

R                     S          |           If R then S

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

1                      T                      T          |

2                      T                      F          |                       F

3                      F                      T          |

4                      F                      F          |

22.

P                      Q                     |           P  ® Q

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

T                      T                      |           T

T                      F                      |           F

F                      T                      |           T

F                      F                      |           T

Logicians do strange things with conditionals!!!

23.

If ice is as dense as lead, then ice floats on water.

antecedent      F

consequent     T

Conditional ???

24.

If there is economic growth then there is high inflation.

antecedent  F

consequent     F

Conditional  ???

25.

R:                    The paper turns red

S:                     The solution is an acid.

T:                    Tony Blair is in the glass

C:                    Tony Blair is in Canada

1.                     R         ®        S

2.                     R         ®        T

3.                     R         ®        C

26.

Has something gone badly wrong ?

P                      Q                     |           If P then Q

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

T                      T                      |                       ?

T                      F                      |                       F

F                      T                      |                       ?

F                      F                      |                       ?

P                      Q                     |           P ® Q

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

T                      T                      |           T

T                      F                      |           F

F                      T                      |           T

F                      F                      |           T

27.

If cobalt but no nickel is present, a brown colour appears.

Now this sentence is true precisely if either a brown colour appears, or it’s not true that ‘Cobalt but no nickel is present’.*

*In a situation where cobalt but no nickel is present, yet no brown colour appears, the sentence is false; also this is the only kind of situation which could make the sentence false. This point needs further discussion, and we shall come back to it in section 17 below.

28.

##### Section 17

Some of these translations may raise doubts. For example, surely the first sentence implies there is some kind of connection between the redness and the acidity? And surely it suggests that if the paper does not turn red, then the solution is not acid? Neither of these things is conveyed by our translation. We take the view that although somebody might well assume these things if he heard the first sentence, they are not actually stated in that sentence.

29.

“What is to be done?” (Lenin)

1.         Fancy philosophical arguments.

2.         Walk before you run.

3.         Useful nonetheless.

4.         Model.

5.         Ask your tutor. It could be interesting. It will certainly take up the hour.

30.

Back at the old tableau.

P                      Q                     |           P®Q

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

T                      T                      |           T

T                      F                      |           F

F                      T                      |           T

F                      F                      |           T

P  ® Q

ØP       Q

31.

M:       Major is great.

P:         There will be prosperity.

M ® P, ØP \ ØM

CES                 {M ® P, ØP, ØØM}

###### Tableau

M ® P

ØP

ØØM

ØM                 P

32.

M                    |           ØM                 |           ØØM

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

T                      |           F                      |           T

F                      |           T                      |           F

33.

[WN1]