Contradiction Elimination

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?

Hi roba,
You actually answered the question yourself in the last sentence , when you stated "the last 2 lines are nonsensical".
Pse note that in your truth table "contr" is just any label (like "A", and it does not stand at all for a contradiction!
Your truth table must be built starting with A and PHY (4 lines), then (A AND NOT A) [which is a contradiction!] , and finally (A and NOT A) -> PHY
Also, in the previous years you learned that the material implication is always true when the premise is false (= contradiction).

My point was not about the theory of the rules. Any parrot can recite those. My problem was the causal relationship; that there doesn't seem to be one. It all seems like a "free lunch" and that makes me uncomfortable.

robanaurochs Wrote:
-------------------------------------------------------
> My point was not about the theory of the rules.
> Any parrot can recite those. My problem was the
> causal relationship; that there doesn't seem to be
> one. It all seems like a "free lunch" and that
> makes me uncomfortable.

Remember though that propositional logic has absolutely nothing to do with causal relationships. It is all about the implication of propositional atoms in conjunction with the truth functional connectives. Now if you look at it that way, what does a contradiction imply in a particular domain of discourse?

The following shows why an arbitrary conclusion drawn from a contradition is valid by using a simple example.

Quote

1. ~ A · A (premise)
2. A (from 1)
3. ~A (from 1)
4. A | B (from 2)
5. (A | B ) · ~A (from 3 and 4)
6. B (from 5).

Is the adduced inference completely valid? The logician, who, for
some reasons would like to answer this question negatively should point
out those steps of the inference which were made incorrectly.

The above example is taken from www.filozof.uni.lodz.pl/bulletin/pdf/17_34_8.pdf

Also remember that all contradictions are ultimately of the form P ^ ~P so the above example demonstrates why you can introduce an arbitrary conclusion from all contraditions.

I found pages 7 and 8 of the study guide cleared up Rob's question for me.