@ 12:15 on Wednesday, 27 October 1999
tableaus are not mechanical � need to look for implicit premises
bizarre: rules flow from truth tables (if you buy them, then t tableau rules are justified)
bracketing in arithmetic (& logic) � indicates the order in which operations are performed
e.g. I go to town
I drink beer
I drink wine
T �/span> B �/span> W
negation of a disjunction
Set up code (M: Major is sad)
Translate into our propoisitonal language (connectives)
Form CES
Construct a tableau
If the tableau closes, premises syntactically entail the conclusion
Syntactic turnstile (says that the tableau closes)
Icabod got a distinction
Icabod is sad
D but S
D and S
D �/span> S (lose some sense/connotations from the English)
There will be a picnic
It rains
There will be a picnic unless it rains
P �/span> R
If it does not rain, there will be a picnic
You work very hard
You will get a First
W � F
You will get a First if you work very hard
You will get a First only if you work very hard
If you work hard, you will get a First, and only if you work hard
[(F � W) �/span> (W � F)]
R:�� Icabod rows
F:��� Icabod gets a first
G:�� Bloggs is a good tutor
S:��� Bloggs is at st x
((R �/span> F) � G),(S � �G),
S |- (�R �/span> �F)
valid argument: if the premises are true, the conclusion must be true
use truth tables to test directly the validity according to that definition
Semantic turnstile
syntactic is a claim when you use the tableau
semantic is a claim when you use the truth table
both use the same
truth tables � more fundamental � come out of the and and or (which is where hodges gets his rules)
difficult to understand why the two are exactly the same
Sainsbury � ch 1,2 + 4
Richard Jeffrey�s� - Formal Logic: its scope & limits
hidden inconsistency � fragopan = horned pheasant from Asia � you have an open branch in the tableau � argument is valid but hidden inconsistency
test arguments for vaildity by testing their ces for cons using tableau
limited by truth-functors