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.
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.
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.
L: Lucan is dead.
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.