Hi all
I've picked up COS361 this year after doing COS261 back in 2006. Excuse me for being a bit rusty but I just don't seem to get the justification for introducing anything if you've proven a contradiction.
I get and understand the causal relationship for not introduction (p22), which we learnt in the first two years as
reductio ad absurdum but there doesn't seem to be a causal relationship between the premise and the conclusion in the middle of p21. As a consequence, I can't see how the middle assumption block is closed on the left-hand argument of the example proof at the bottom of p21. I don't understand how you can magically conjure up q as a consequence of the contradiction in line 4.
The rule states that
contr |-
phi
From prior in the chapter, it says that any sequent can be converted into a theorem by conjoining all the premises and then implying the conclusion.
ie. p1, p2, p3 |- c
can be rewritten:
|- (p1 ^ p2 ^ p3) --> c
Thus the rule can also be rewritten in the same way as:
|- (
contr) --> (
phi)
Thus it should follow the same truth table as any other implication
Language: PHP
A | B || A -> B
----------------------
F | F || T
F | T || T
T | F || F
T | T || T
If you do the same thing for the the contradiction rule, you get:
Language: PHP
contr | phi || contr -> phi
------------------------------------
F | F || T
F | T || T
T | F || F
T | T || T
The last two lines are nonsensical in terms of a contradiction since contradictions are always false. In this case, the rule cannot complete the truth table for the implication thus it doesn't make sense that the implication is a valid operator. Thus, it doesn't follow that an arbitrary variable can be conjured up if a contradiction occurs.
What does anybody else think?