Hodges little trick by dfhrf555fcg

VIEWS: 11 PAGES: 33

More Info
									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
Tableau



    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.

Q: Icabod is sad.

  __ 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
 and 

 but 

It is hoped that 

 and then later 

It is not the case that 

N-S hopes that 

 because 


                            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     

 or      

 not     

 LH

 LH
              12
L : Lucan is dead.
H:Hague has a friend

  L H | LH
-------------------------
  T T         |     T
  T F         |     T
  F T         |     T
  F F         |     F


                            13
  L H | LH
------------------------
  T T         | T
  T F         | T
  F T         | T
  F F         | F

  P | P
---------------
  T |       F
  F | T
                           14
The strange story of the cruel
Dean.

P : You will pass prelims.

Q : You will be sent down.

  PQ

You pass but he sends you
down.

Unfair !

No ! The Dean is a logican


                                 15
       P Q| P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  Q)  (P  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
 | NS believes that 
---------------------------------
T |         ?
F |         ?


 | God believes that 
--------------------------------
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


  P       Q
                             31
M : Major is great.
P : There will be prosperity.

M  P, P  M

CES    {M  P, P, M}

Tableau

       MP
       P
       M

      M   P

                                32
M | M | M
--------------------------
T |         F |         T
F |         T |         F




                             33

								
To top